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Vertical Graph Analytics for Complex Data Modeling

This article explores vertical graph analytics, which aims to improve the speed and accuracy of analyzing complex data models. Topics covered include vertical structuring of graph data, connectivity component partitioning, community mining, and graph and hypergraph clustering.

williamgeorge
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Vertical Graph Analytics for Complex Data Modeling

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  1. Vertical Graph Analytics (summarizes SatNotes 06_15-pres)Most complex data is modelled as a graph or hypergraph (a table is a graph without edges, so ALL data is modelled as a graph!). We strive for max speed and accuracy in our graph analytics by using vertical structure. We consider the following topics: • Vertical structuring of graph data (Edge pTree (E), PathPtree(PP), ShortestPathTrees…). • Connectivity Component Partitioning. • Community Mining (k-plexes, which include cliques as 0-plexes; k-cores, Density-communities, Degree-communities, • Community existence theorems (determine if a given Induced SubGraph is a community) and community mining algorithms (find all communities) include: Vertex Count based Existence Thms. Inheritance (downward or upward closure based existence thms). Density Difference. Degree Difference. • Graph and HyperGraph Clustering (Community based, Vertex betweenness, Edge betweenness Clustering). • MultiPART graphs, HyperGraphs, MultiPART Hypergraphs nnd the Clique Tree construct (cTree) for MultiPART graphs and hypergraphs. IntDeg(C) kCint = vC kvint PP(G), the Path Ptree of graph, G, is a vertical representation of all paths in G and is used to find diameter, shortest paths, communities, motifs... By modifying data structures (from horizontal to vertical) the analytics fit hardware strengths and allow do NP-hard/complete problems. k-plex existence: C = k-plex iff vC|Cv|  |VC|2–k2 k-plex inheritance: An induced subgraph of a k-plex is a k-plex. A Path is a sequence of edges connecting a sequence of vertices, distinct except for end-vertices. A Simple Path (assumed) excludes loops, (v,v). We’ll always program using the pop-count (produces 1-counts during ANDs/ORs for free, timewise). C is a clique iff all C level 1counts are |VC|-1. k-core inheritance:If  cover by induced k-cores, G is k-core. k-core existence:C = k-core iff vC,|VC|  k. ExtDeg(C), kCext =vC kvext vC, kvint =#edges v to C; kvext=#edges v to C’. COMMUNITIES (=~ a subgraph with more edges than expected): A k-plex is a [max] subgraph in which each vertex is adjacent to all subgraph vertices except at most k of them. A 0-plex is called a clique. A k-core is a [max] subgraph in which each vertex is adjacent to at least k subgraph vertices. An n-clique is a [max] subgraph s.t. the geodesic distance between any vertex pair is n. An n-clan is a [max] n-clique with diameter n. An n-club is [max] subgraph of diam=n. External Density of C δext(C)= edges(C,C’)|/(nc(n-nc)). InternalDensity of C δint(C)=|edges(C,C)|/(nc(nc−1)/2) ExtDenC*n(n-nC)/2=ExtDegC IntDenC<<IntDegC. Clique Existence: When is an induced SG a clique? Edge Count existence thm (EC): |EC||PUC|=COMB(|VC|,2)|VC|!/((|VC|-2)!2!) SubGraph existence theorem: (VC,EC) is a k-clique iff every induced k-1 subgraph, (VD,ED) is a (k-1)-clique. A Clique Mining alg: Finds all cliques in a graph- uses an ARM-Apriori-like downward closure property: CLQkkCliqueSet, CCLQk+1Candk+1CliqueSet By SG, CCLQk+1= all s of CLQk-pairs having k-1 common vertices. Let CCCLQk+1 be a union of two k-cliques with k-1 common vertices. Let v,w be the kth vertices of the k-cliques, then CCLQk+1 iff (PE)(v,w)=1. (Just need to check a single bit in PE.) A good tradeoff between large δint(C) and small δext(C) is goal of density community mining algs. A simple approach is to maximize differences. Density Difference alg for Communities: δint(C)−δext(C) >Thresh? Degree Difference kCint – kCext> Thresh? Easy to compute even for Big Graphs. Giant Yahoo Data Dump Aims to Help Computers Know What You Want: (see “Here’s What Developers Are Doing With Google’s AI Brain”).

  2. A graph is a set of vertices, V, and a set of edges, E, each connecting a pair of those vertices. An edge from vertex h to vertex k is realized as the unordered set, {h,k} (or just hk), and can be viewed as an undirected line from h to k. We can either list the edges in a two column table (the Edge Table) or we can use a 3 column table in which the first 2 columns list all possible vertex pairs (in raster order) and the third column is a bit map indicating with a 1-bit the pairs that are edges and with a 0-bit the pairs that are not edges. This second option is called the edge map or edge mask and is shown below for a small graph, G1. The edge map obviously has |V|2 rows. If the raster ordering is always assumed, the edge map is just a single column of bits. The edge map can be compressed into a pTree (predicate Tree) by dividing the bits up into “strides” of |V| bits each (4 for G1). This forms the lowest level of the pTree (level_0) and an upper level (level_1), indicates the truth of the predicate, “Not Purely Zeros” for the respective level_0 pTrees, and can be used to avoid retrieving level_0 pTrees that are purely zeros. We use the notation Ek for the kth level_0 pTree (which bitmaps the endpoints of the edges adjacent to vertex k.) and call it the Edge pTree of k A Pathis a sequence of edges connecting a sequence of vertices, distinct, except for endpts. A Simple Path (assumed throughout) disallows simple loops, (v,v) Next we build a ShortestPathtree, SPG1 for G1 2 2-Level Stride=4, Edge pTree 1 Level1 1 1 1 1 Edge Map Vertex Masks Edges V1 V2 3 4 E 0 0 1 1_ 0 0 0 1_ 1 0 0 1_ 1 1 1 0 M1 1 0 0 0 M2 0 1 0 0 M4 0 0 0 1 M3 0 0 1 0 1,1 1,2 1,3 1,4_ 2,1 2,2 2,3 2,4_ 3,1 3,2 3,3 3,4_ 4,1 4,2 4,3 4,4 E2 0 0 0 1 E3 1 0 0 1 E4 1 1 1 0 E1 0 0 1 1 It starts with Level_0 of the EdgeTree.  vertex, k, this gives us a mask, Sk, of the end pts of edges adjacent to vertex k (shortest path of Length 1 starting at k). The complement of Ek (with k turned off) gives us the endpoints that never need to be considered again (since all shortest paths from k to these vertices hve been found). We call these pTrees the “Not Reached Yet masks” or “N masks”. We can avoid these calculations by noting Ct(N14 )=0. We use the notation, Shk for the map of the endpts of Shortest Paths thru h then k (obviously of length=2) and NLv for the map of vertices not reached by lengthL shortest paths from vertex v N11 0 1 0 0 1 N12 1 0 1 0 2 N13 0 1 0 0 1 N14 0 0 0 0 0 S1 0 0 1 1 2 S2 0 0 0 1 1 S3 1 0 0 1 2 S4 1 1 1 0 3 S13 =N11&E3 0 0 0 0 0 S14 =N11&E4 0 1 0 0 1 S24 =N12&E4 1 0 1 0 2 S31 =N13&E1 0 0 0 0 0 S34 =N13&E4 0 1 0 0 1 S41 =N14&E1 0 0 0 0 0 S42 =N14&E2 0 0 0 0 0 S43 =N14&E3 0 0 0 0 0 N21=N11& (S13|S14)’ 0 0 0 0 0 N23=N13& (S31|S34)’ 0 0 0 0 0 N24=N14& (S41|S42|S43)’ 0 0 0 0 0 N22=N12& (S24)’ 0 0 0 0 0 This entire level is unnecessary to construct since |N2k|=0 k. The SPTreeis shown by the green links. S142 =N21&E2 0 0 0 0 0 S243 =N22&E3 0 0 0 0 0 S241 =N22&E1 0 0 0 0 0 S312 =N23&E1 0 0 0 0 0 S342 =N23&E2 0 0 0 0 0 The connectivity components can be deduced from the zero set of the final NLks. Girvan and Newman started a flurry of research by suggesting the graph could be edge labelled by an edge_between-ness measurement (which counts the shortest path participations of the edge) and that a graph could be usefully partitioned (into strongly connected components) by the divisive hierarchical clustering of removing edges in desc order of between-ness. Btwn14=1 Btwn24=2 Btwn34=1

  3. N1 Counting SP participations: If a12= the count of 12 participations and a21=the count of 21 participations, then the full participation is BtwnGN=a12.. + a21.. + a12..a21.. + 1 since  21 SP, 1—2 will participate in the middle of another participation extending a 12 participation. So the problem is computing all ahk.. Correctly (We don’t need the +1) Btwn1.5 = a12k + a21h Btwn2.5 = a12k + a21h + a12h * a21h hListEkShk=Ek&N1h 1 0 0 0 0 1 0 1 2 2 0 0 0 0 1 1 1 3 3 0 0 0 0 1 1 1 3 4 0 0 0 0 1 1 1 3 5 1 1 1 1 0 0 1 5 6 0 1 1 1 0 0 0 3 7 1 1 1 1 1 0 0 5 i N2k= N1k& (ORhListEkSkh)’ So N21= S12|S13|S14|S16)’ a e 6 G2 h c 9 4 1 2 g b 3 8 We can now deduce the graph is connected, |N21|=0  CC1=all. d 5 f 7 S S 5 6 1 0 1 1 1 0 0 0 3 7 6 1 0 1 1 1 0 0 0 3 5 6 7 0 0 0 0 0 0 0 0 7 6 5 0 0 0 0 0 0 0 0 S 1 0 1 1 1 0 1 0 4 2 1 0 1 1 0 0 0 3 3 1 1 0 1 0 0 0 3 4 1 1 1 0 0 0 0 3 5 0 0 0 0 0 1 0 1 6 1 0 0 0 1 0 1 3 7 0 0 0 0 0 1 0 1 S S S S S S S S S 6 5 0 0 0 0 0 0 0 0 6 1 0 1 1 1 0 0 0 3 6 7 1 0 0 0 0 0 0 0 N2 1 0 0 0 0 0 0 0 0 N2 2 0 0 0 0 1 0 1 2 N2 3 0 0 0 0 1 0 1 2 N2 4 0 0 0 0 1 0 1 2 3 1 0 0 0 0 0 1 0 1 1 2 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 1 0 1 7 6 1 0 0 0 1 0 0 2 2 1 6 0 0 0 0 1 0 1 2 4 1 0 0 0 0 0 1 0 1 3 1 6 0 0 0 0 1 0 1 2 5 6 1 0 0 0 0 0 1 2 4 1 6 0 0 0 0 1 0 1 2 4 2 0 0 0 0 0 0 0 0 2 3 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 3 2 0 0 0 0 0 0 0 0 1 4 0 0 0 0 0 0 0 0 4 3 0 0 0 0 0 0 0 0 3 4 0 0 0 0 0 0 0 0 2 4 0 0 0 0 0 0 0 0 1 6 0 0 0 0 1 0 1 2 N2 5 0 1 1 1 0 0 0 3 N2 6 0 0 0 0 0 0 0 0 SP N2 7 0 1 1 1 0 0 0 3 2 4 8 So Btwn1—2 = 8 + 8 + 64 + 1 81 Btwn1—b = Btwn1—c = Btwn2—3 = Btwn2—4 = 3 + 14 + 42 + 1 = 60 Btwnb—d = Btwnc—e = Btwn3—5 = Btwn4—6 = 2 + 14 + 28 + 1 = 45 N3 2 0 0 0 0 0 0 0 0 N3 3 0 0 0 0 0 0 0 0 N3 4 0 0 0 0 0 0 0 0 N3 5 0 0 0 0 0 0 0 0 N3 6 0 0 0 0 0 0 0 0 Btwnd—f=Btwnd—g=Btwne—h=Btwne—i=Btwnr—7=Btwn5—8=Btwn6—9=Btwn6—a=0+16+0+1= 17 Radius(x)MAXyExLength(SP(x,y)) is an interesting vertex label. MIN(ahk , akh ) is an interesting edge label. The min # of SP hops from an edge in either direction is an edge radius (like vertex radius?). 6 6 5 1 0 1 1 1 0 1 0 4 2 1 0 1 1 0 0 0 3 3 1 1 0 1 0 0 0 3 4 1 1 1 0 0 0 0 3 5 0 0 0 0 0 1 0 1 6 1 0 0 0 1 0 1 3 7 0 0 0 0 0 1 0 1 5 7 7 Alternatively, DATON (deleted all ties or none (none if a vertex is isolated))? BtwnGN|1.5|2.5_DNIv_DATON Both have the same ordering as BtwnGN Btwn1.5 2 3 4 6 7 1 2 2 2 6 2 1 1 3 1 5 3 6 3 Btwn2.5 2 3 4 6 7 1 2 2 2 c 2 1 1 3 1 5 3 6 3 BtwnGN 2 3 4 6 7 1 4 4 4 b 2 1 1 3 1 5 6 6 6 6 5 7 2 2 2 1 1 1 1 6 0 0 0 0 1 0 1 2 2 1 0 0 0 0 0 1 0 1 3 1 0 0 0 0 0 1 0 1 4 1 0 0 0 0 0 1 0 1 5 6 1 0 0 0 0 0 1 2 6 1 0 1 1 1 0 0 0 3 7 6 1 0 0 0 1 0 0 2 4 3 BtwnGN_DNIv Btwn1_DNIv_DATON Btwn1_DNIv  V 6 6 5 5 7 7 2 2 1 1 4 3 4 3 4 3 4 3 2 1 6 0 0 0 0 1 0 1 2 3 1 6 0 0 0 0 1 0 1 2 4 1 6 0 0 0 0 1 0 1 2 5 6 1 0 1 1 1 0 0 0 3 7 6 1 0 1 1 1 0 0 0 3 1 0 0 0 1 0 0 0 1 2 0 0 0 1 0 0 0 1 3 0 0 0 1 0 0 0 1 4 1 1 1 0 0 0 0 3 5 0 0 0 0 0 1 0 1 6 0 0 0 0 1 0 1 2 7 0 0 0 0 0 1 0 1 Btwn1 = |Ph-1|+|Pk-1|+1 2 3 4 6 7 1 6 6 6 6 2 5 5 3 5 5 3 6 3 One way to get good dendogram partitions is to stop at large BtwnGN gaps: e.g., gaps>2*avg BtwnGNAvg=10/9=1.1 so ThreshGN=2.2 Btwn1.5Avg=10/5=2 so Thresh1.5=4 (there are no large Btwn1.5 gaps) Btwn2.5Avg=10/11=0.9 so Thresh2.5=1.8

  4. hListEkShk = Ek&N1h NkNk&(ORhListEkSkh)’ Counting SP participations in G5 N 1 0 0 1 1 0 1 0 1 4 N 2 0 0 1 0 1 1 1 1 5 N 3 1 1 0 1 1 0 1 0 5 N 4 1 0 1 0 1 1 1 1 6 N 5 0 1 1 1 0 1 0 1 5 N 6 1 1 0 1 1 0 1 0 5 N 7 0 1 1 1 0 1 0 1 5 N 8 1 1 0 1 1 0 1 0 5 S 1 0 1 0 0 1 0 1 0 3 S 1 0 1 0 0 1 0 1 0 3 S 2 1 0 0 1 0 0 0 0 2 S 2 1 0 0 1 0 0 0 0 2 S 3 0 0 0 0 0 1 0 1 2 S 3 0 0 0 0 0 1 0 1 2 S 4 0 1 0 0 0 0 0 0 1 S 4 0 1 0 0 0 0 0 0 1 S 5 1 0 0 0 0 0 1 0 2 S 5 1 0 0 0 0 0 1 0 2 S 6 0 0 1 0 0 0 0 1 2 S 6 0 0 1 0 0 0 0 1 2 S 7 1 0 0 0 1 0 0 0 2 S 7 1 0 0 0 1 0 0 0 2 S 8 0 0 1 0 0 1 0 0 2 S 8 0 0 1 0 0 1 0 0 2 BTWNGN_DNIv BTWNGN 2 3 4 5 6 7 8 1 6 3 3 2 4 3 1 1 4 5 1 6 1 7 8 1 2 8 3 7 4 S 1 2 0 0 0 1 0 0 0 0 1 S 1 5 0 0 0 0 0 0 0 0 0 S 1 7 0 0 0 0 0 0 0 0 0 S 2 1 0 0 0 0 1 0 1 0 2 S 2 4 0 0 0 0 0 0 0 0 0 S 3 6 0 0 0 0 0 0 0 0 0 S 3 8 0 0 0 0 0 0 0 0 0 S 4 2 1 0 0 0 0 0 0 0 1 5 6 S 6 3 0 0 0 0 0 0 0 0 0 S 6 8 0 0 0 0 0 0 0 0 0 S 7 1 0 1 0 0 0 0 0 0 1 S 7 5 0 0 0 0 0 0 0 0 0 S 5 1 0 1 0 0 0 0 0 0 1 S 5 7 0 0 0 0 0 0 0 0 0 S 8 3 0 0 0 0 0 0 0 0 0 S 8 6 0 0 0 0 0 0 0 0 0 N 1 0 0 1 0 0 1 0 1 3 N 2 0 0 1 0 0 1 0 1 3 N 3 1 1 0 1 1 0 1 0 5 N 4 0 0 1 0 1 1 1 1 5 N 5 0 0 1 1 0 1 0 1 4 N 6 1 1 0 1 1 0 1 0 5 N 7 0 0 1 1 0 1 0 1 4 N 8 1 1 0 1 1 0 1 0 5 S 1 2 0 0 0 1 0 0 0 0 1 S 2 1 0 0 0 0 1 0 1 0 2 S 4 2 1 0 0 0 0 0 0 0 1 S 7 1 0 1 0 0 0 0 0 0 1 S 5 1 0 1 0 0 0 0 0 0 1 BTWN1.5_DNIv BTWN1.5 2 3 4 5 6 7 8 1 6 2 2 2 2 3 1 1 4 5 1 6 1 7 8 1 2 8 3 7 4 G5 5 6 S 4 2 1 0 0 0 0 1 0 1 0 2 S 5 1 2 0 0 0 1 0 0 0 0 1 S 7 1 2 0 0 0 1 0 0 0 0 1 S 2 1 5 0 0 0 0 0 0 0 0 0 S 2 1 7 0 0 0 0 0 0 0 0 0 S 1 2 4 0 0 0 0 0 0 0 0 0 S 4 2 1 0 0 0 0 1 0 1 0 2 S 5 1 2 0 0 0 1 0 0 0 0 1 S 7 1 2 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 3 0 0 0 0 0 0 0 1 1 6 0 0 0 0 0 0 0 1 1 2 0 0 0 1 0 0 0 0 1 5 0 0 0 0 0 0 1 0 1 7 1 0 0 0 1 0 0 0 2 4 0 1 0 0 0 0 0 0 1 8 0 0 1 0 0 1 0 0 2 BTWNGN_DNIv_DATON N 1 0 0 1 0 0 1 0 1 3 N 2 0 0 1 0 0 1 0 1 3 N 3 1 1 0 1 1 0 1 0 5 N 4 0 0 1 0 1 0 1 0 3 N 5 0 0 1 0 0 1 0 1 3 N 6 1 1 0 1 1 0 1 0 5 N 7 0 0 1 0 0 1 0 1 3 N 8 1 1 0 1 1 0 1 0 5 BTWN1.5_DNIv_DATON 1 2 1 1 2 2 3 8 8 3 3 8 4 7 7 4 7 4 5 6 5 6 5 6 CC1= N’1= N’2= N’4= N’5= N’7= 1 1 0 1 1 0 1 0 5 S 4 2 1 5 0 0 0 0 0 0 0 0 0 S 4 2 1 7 0 0 0 0 0 0 0 0 0 S 5 1 2 4 0 0 0 0 0 0 0 0 0 S 7 1 2 4 0 0 0 0 0 0 0 0 0 CC3= N’3= N’6= N’8= 0 0 1 0 0 1 0 1 3 N 1 0 0 1 0 0 1 0 1 3 N 2 0 0 1 0 0 1 0 1 3 N 3 1 1 0 1 1 0 1 0 5 N 4 0 0 1 0 1 0 1 0 3 N 5 0 0 1 0 0 1 0 1 3 N 6 1 1 0 1 1 0 1 0 5 N 7 0 0 1 0 0 1 0 1 3 N 8 1 1 0 1 1 0 1 0 5

  5. SP partic for G6 1 5 E 1 0 1 1 1 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 0 0 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 4 1 1 0 0 0 0 1 0 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 6 0 0 0 0 1 0 1 1 0 0 0 0 7 0 0 0 1 1 1 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 9 0 0 0 0 0 0 0 1 0 1 1 1 c 0 0 1 0 0 0 0 0 1 1 1 0 8 0 0 0 0 0 1 0 0 1 1 0 0 a 0 0 0 0 0 0 0 1 1 0 1 1 S 1 0 1 1 1 0 0 0 0 0 0 0 0 3 S 2 1 0 1 1 0 0 0 0 0 0 0 0 3 S 3 1 1 0 0 0 0 0 0 0 0 0 1 3 S 4 1 1 0 0 0 0 1 0 0 0 0 0 3 S 5 0 0 0 0 0 1 1 0 0 0 0 0 2 S 6 0 0 0 0 1 0 1 1 0 0 0 0 3 S 7 0 0 0 1 1 1 0 0 0 0 0 0 3 N 1 0 0 0 0 1 1 1 1 1 1 1 1 8 N 1 0 0 0 0 0 0 0 0 0 0 0 0 0 N 1 0 0 0 0 1 1 0 1 1 1 1 0 6 N 1 0 0 0 0 0 0 0 1 0 0 0 0 1 N 2 0 0 0 0 1 1 1 1 1 1 1 1 8 N 2 0 0 0 0 0 0 0 0 0 0 0 0 0 N 2 0 0 0 0 1 1 0 1 1 1 1 0 6 N 2 0 0 0 0 0 0 0 1 0 0 0 0 1 N 3 0 0 0 0 1 1 1 1 0 0 0 0 4 N 3 0 0 0 0 0 0 0 0 0 0 0 0 0 N 3 0 0 0 0 1 1 0 0 0 0 0 0 2 N 3 0 0 0 1 1 1 1 1 1 1 1 0 8 N 4 0 0 0 0 0 0 0 0 0 0 0 0 0 N 4 0 0 0 0 0 0 0 1 1 1 1 1 5 N 4 0 0 0 0 0 0 0 0 1 1 1 0 3 N 4 0 0 1 0 1 1 0 1 1 1 1 1 8 N 5 0 0 0 0 0 0 0 0 0 0 0 0 0 N 5 1 1 1 1 0 0 0 1 1 1 1 1 9 N 5 1 1 1 0 0 0 0 0 1 1 1 1 7 N 5 0 0 1 0 0 0 0 0 0 0 1 1 3 N 6 1 1 1 1 0 0 0 0 1 1 1 1 8 N 6 0 0 1 0 0 0 0 0 0 0 0 0 1 N 6 1 1 1 0 0 0 0 0 0 0 1 1 5 N 6 0 0 0 0 0 0 0 0 0 0 0 0 0 N 7 0 0 0 0 0 0 0 0 0 0 1 1 2 N 7 1 1 1 0 0 0 0 1 1 1 1 1 8 N 7 0 0 0 0 0 0 0 0 0 0 0 0 0 N 7 0 0 1 0 0 0 0 0 1 1 1 1 5 N b 0 0 0 1 1 0 1 0 0 0 0 0 3 N b 1 1 0 1 1 1 1 0 0 0 0 0 8 N b 0 0 0 0 0 0 0 0 0 0 0 0 0 N b 1 1 1 1 1 1 1 1 0 0 0 0 8 G6 S 9 0 0 0 0 0 0 0 1 0 1 1 1 4 S c 0 0 1 0 0 0 0 0 1 1 1 0 4 N 9 0 0 0 0 0 0 0 0 0 0 0 0 0 N 9 1 1 0 1 1 0 1 0 0 0 0 0 5 N 9 0 0 0 1 0 0 0 0 0 0 0 0 1 N 9 1 1 1 1 1 1 1 0 0 0 0 0 7 N c 0 0 0 0 1 0 1 0 0 0 0 0 2 N c 1 1 0 1 1 1 1 1 0 0 0 0 7 N c 0 0 0 0 0 0 0 0 0 0 0 0 0 N c 0 0 0 1 1 1 1 0 0 0 0 0 4 S 8 0 0 0 0 0 1 0 0 1 1 0 0 3 S a 0 0 0 0 0 0 0 1 1 0 1 1 4 N 8 1 1 1 1 0 0 0 0 0 0 0 0 4 N 8 1 1 0 0 0 0 0 0 0 0 0 0 2 N 8 0 0 0 0 0 0 0 0 0 0 0 0 0 N 8 1 1 1 1 1 0 1 0 0 0 1 1 8 N a 1 1 0 1 1 0 1 0 0 0 0 0 5 N a 0 0 0 0 0 0 0 0 0 0 0 0 0 N a 0 0 0 1 0 0 0 0 0 0 0 0 1 N a 1 1 1 1 1 1 1 0 0 0 0 0 7 S b 0 0 0 0 0 0 0 0 1 1 0 1 3 4 2 6 7 3 c 9 b 8 a S 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 S 2 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 2 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S 3 1 0 0 0 1 0 0 0 0 0 0 0 0 1 S 3 2 0 0 0 1 0 0 0 0 0 0 0 0 1 S 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 S 1 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 1 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S 3 c 0 0 0 0 0 0 0 0 1 1 1 0 3 S 4 1 0 1 1 0 0 0 0 0 0 0 0 0 2 S 4 2 0 0 1 0 0 0 0 0 0 0 0 0 1 S 4 7 0 0 0 0 1 1 0 0 0 0 0 0 2 S 5 6 0 0 0 0 0 0 0 1 0 0 0 0 1 S 5 7 0 0 0 1 0 0 0 0 0 0 0 0 1 S 6 8 0 0 0 0 0 0 0 0 1 1 0 0 2 S 6 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 6 7 0 0 0 1 0 0 0 0 0 0 0 0 1 S 7 4 1 1 0 0 0 0 0 0 0 0 0 0 2 S 7 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 7 6 0 0 0 0 0 0 0 1 0 0 0 0 1 S 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S 9 b 0 0 0 0 0 0 0 0 0 0 0 0 0 S a b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 8 9 0 0 0 0 0 0 0 0 0 0 1 1 2 S 9 c 0 0 1 0 0 0 0 0 0 0 0 0 1 S a 9 0 0 0 0 0 0 0 0 0 0 0 0 0 S a c 0 0 1 0 0 0 0 0 0 0 0 0 1 S 8 a 0 0 0 0 0 0 0 0 0 0 1 1 2 S 9 8 0 0 0 0 0 1 0 0 0 0 0 0 3 S 9 a 0 0 0 0 0 0 0 0 0 0 0 0 0 S a 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S b 9 0 0 0 0 0 0 0 1 0 0 0 0 1 S b c 0 0 1 0 0 0 0 0 0 0 0 0 1 S b a 0 0 0 0 0 0 0 1 0 0 0 0 1 S c 3 1 1 0 0 0 0 0 0 0 0 0 0 2 S c 9 0 0 0 0 0 0 0 1 0 0 0 0 1 S c a 0 0 0 0 0 0 0 1 0 0 0 0 1 S c b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 1 3 c 0 0 0 0 0 0 0 0 1 1 1 0 3 S 2 3 c 0 0 0 0 0 0 0 0 1 1 1 0 3 S 2 4 7 0 0 0 0 1 1 0 0 0 0 0 0 2 S 3 1 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S 1 4 7 0 0 0 0 1 1 0 0 0 0 0 0 2 S 8 9 c 0 0 1 0 0 0 0 0 0 0 0 0 1 S 8 a c 0 0 1 0 0 0 0 0 0 0 0 0 1 S a 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S 5 6 8 0 0 0 0 0 0 0 0 1 1 0 0 2 S 7 6 8 0 0 0 0 0 0 0 0 1 1 0 0 2 S b 9 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S 3 2 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S 4 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 S 4 1 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 4 2 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 4 7 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 4 7 6 0 0 0 0 0 0 0 1 0 0 0 0 1 S 5 7 4 1 1 0 0 0 0 0 0 0 0 0 0 2 S 6 7 4 1 1 0 0 0 0 0 0 0 0 0 0 2 S 8 6 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 8 6 7 0 0 0 1 0 0 0 0 0 0 0 0 1 S 9 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S 9 c 3 1 1 0 0 0 0 0 0 0 0 0 0 2 S a c 3 1 1 0 0 0 0 0 0 0 0 0 0 2 S 8 9 b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 8 a b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 3 c 9 0 0 0 0 0 0 0 1 0 0 0 0 1 S c a 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S 3 c a 0 0 0 0 0 0 0 1 0 0 0 0 1 S 6 8 9 0 0 0 0 0 0 0 0 0 0 1 1 2 S b a 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S c 9 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S 6 8 a 0 0 0 0 0 0 0 0 0 0 1 1 2 S 3 c b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 7 4 1 0 1 1 0 0 0 0 0 0 0 0 0 1 S 7 4 2 1 0 1 0 0 0 0 0 0 0 0 0 1 S b c 3 1 1 0 0 0 0 0 0 0 0 0 0 2 S c 3 1 0 0 0 1 0 0 0 0 0 0 0 0 1 S c 3 2 0 0 0 1 0 0 0 0 0 0 0 0 1 S 1 3 c 9 0 0 0 0 0 0 0 1 0 0 0 0 1 S 1 4 7 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 1 4 7 6 0 0 0 0 0 0 0 1 0 0 0 0 1 S 2 3 c 9 0 0 0 0 0 0 0 1 0 0 0 0 1 S 3 2 4 7 0 0 0 0 1 0 0 0 0 0 0 0 1 S 3 c a 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S 4 2 3 c 0 0 0 0 0 0 0 0 1 1 1 0 3 S 5 6 8 9 0 0 0 0 0 0 0 0 0 0 1 1 2 S 6 7 4 1 0 0 1 0 0 0 0 0 0 0 0 0 1 S 6 7 4 2 0 0 1 0 0 0 0 0 0 0 0 0 1 S 6 8 9 c 0 0 1 0 0 0 0 0 0 0 0 0 1 S 7 4 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 S 7 4 1 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 7 4 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 S 7 4 2 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 7 6 8 9 0 0 0 0 0 0 0 0 0 0 1 1 2 S 8 9 c 3 1 1 0 0 0 0 0 0 0 0 0 0 2 S b 9 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S b a 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S c 3 2 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S c 9 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S 3 c 9 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S 4 7 6 8 0 0 0 0 0 0 0 0 1 1 0 0 2 S 1 3 c a 0 0 0 0 0 0 0 1 0 0 0 0 1 S 2 3 c a 0 0 0 0 0 0 0 1 0 0 0 0 1 S 5 6 8 a 0 0 0 0 0 0 0 0 0 0 1 1 2 S 7 6 8 a 0 0 0 0 0 0 0 0 0 0 1 1 2 S 2 4 7 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 2 4 7 6 0 0 0 0 0 0 0 1 0 0 0 0 1 S 3 1 4 7 0 0 0 0 1 0 0 0 0 0 0 0 1 S 4 1 3 c 0 0 0 0 0 0 0 0 1 1 1 0 3 S 5 7 4 1 0 0 1 0 0 0 0 0 0 0 0 0 1 S 5 7 4 2 0 0 1 0 0 0 0 0 0 0 0 0 1 S 8 6 7 4 1 1 0 0 0 0 0 0 0 0 0 0 2 S 8 a c 3 0 0 0 0 0 0 0 0 0 0 0 0 0 S 9 8 6 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 9 8 6 7 0 0 0 1 0 0 0 0 0 0 0 0 1 S 9 c 3 1 0 0 0 1 0 0 0 0 0 0 0 0 1 S 9 c 3 2 0 0 0 1 0 0 0 0 0 0 0 0 1 S a 8 6 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S a 8 6 7 0 0 0 1 0 0 0 0 0 0 0 0 1 S a c 3 1 0 0 0 1 0 0 0 0 0 0 0 0 1 S a c 3 2 0 0 0 1 0 0 0 0 0 0 0 0 1 S b c 3 1 0 0 0 1 0 0 0 0 0 0 0 0 1 S b c 3 2 0 0 0 1 0 0 0 0 0 0 0 0 1 S c 3 1 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S c 9 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S 1 3 c b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 2 3 c b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 6 8 9 b 0 0 0 0 0 0 0 0 0 0 0 0 0 Diameter(G6)=4, G6 is one connected component. The radius of each point is 4.

  6. BTWNGN with DNIv BTWN1 with DNIv V BTWN1 with DNIv ^ BTWN1 with DNIv and DATON S 1 0 1 1 1 0 0 0 0 0 0 0 0 3 S 2 1 0 1 1 0 0 0 0 0 0 0 0 3 S 3 1 1 0 0 0 0 0 0 0 0 0 1 3 S 4 1 1 0 0 0 0 1 0 0 0 0 0 3 S 5 0 0 0 0 0 1 1 0 0 0 0 0 2 S 6 0 0 0 0 1 0 1 1 0 0 0 0 3 S 7 0 0 0 1 1 1 0 0 0 0 0 0 3 S 9 0 0 0 0 0 0 0 1 0 1 1 1 4 S c 0 0 1 0 0 0 0 0 1 1 1 0 4 S 8 0 0 0 0 0 1 0 0 1 1 0 0 3 S a 0 0 0 0 0 0 0 1 1 0 1 1 4 S b 0 0 0 0 0 0 0 0 1 1 0 1 3 1 1 1 1 5 5 5 5 G6 G6 G6 G6 4 4 2 2 4 2 4 6 6 2 7 7 6 7 6 7 3 3 3 3 c c c c 9 9 b b 9 b 9 b 8 8 8 8 a a a a A way to get good dendogram partitions is to stop at large BtwnGN gaps: e.g., gaps>2*avg Avg=47/19=2.5 At this partition gap=42-35=7>5. BTWNGN(hk)=|hk..|+|kh..|+|hk..||kh..|+1 2 3 4 5 6 7 8 9 a b c 1 1 21 35 2 21 30 3 42 4 42 5 8 6 6 48 48 7 8 42 20 9 1 1 31 a 1 18 b 6 S 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 S 2 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 2 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S 3 1 0 0 0 1 0 0 0 0 0 0 0 0 1 S 3 2 0 0 0 1 0 0 0 0 0 0 0 0 1 S 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 S 1 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 1 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S 3 c 0 0 0 0 0 0 0 0 1 1 1 0 3 S 4 1 0 1 1 0 0 0 0 0 0 0 0 0 2 S 4 2 0 0 1 0 0 0 0 0 0 0 0 0 1 S 4 7 0 0 0 0 1 1 0 0 0 0 0 0 2 S 5 6 0 0 0 0 0 0 0 1 0 0 0 0 1 S 5 7 0 0 0 1 0 0 0 0 0 0 0 0 1 S 6 8 0 0 0 0 0 0 0 0 1 1 0 0 2 S 6 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 6 7 0 0 0 1 0 0 0 0 0 0 0 0 1 S 7 4 1 1 0 0 0 0 0 0 0 0 0 0 2 S 7 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 7 6 0 0 0 0 0 0 0 1 0 0 0 0 1 S 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S 9 b 0 0 0 0 0 0 0 0 0 0 0 0 0 S a b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 8 9 0 0 0 0 0 0 0 0 0 0 1 1 2 S 9 c 0 0 1 0 0 0 0 0 0 0 0 0 1 S a 9 0 0 0 0 0 0 0 0 0 0 0 0 0 S a c 0 0 1 0 0 0 0 0 0 0 0 0 1 S 8 a 0 0 0 0 0 0 0 0 0 0 1 1 2 S 9 8 0 0 0 0 0 1 0 0 0 0 0 0 3 S 9 a 0 0 0 0 0 0 0 0 0 0 0 0 0 S a 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S b 9 0 0 0 0 0 0 0 1 0 0 0 0 1 S b c 0 0 1 0 0 0 0 0 0 0 0 0 1 S b a 0 0 0 0 0 0 0 1 0 0 0 0 1 S c 3 1 1 0 0 0 0 0 0 0 0 0 0 2 S c 9 0 0 0 0 0 0 0 1 0 0 0 0 1 S c a 0 0 0 0 0 0 0 1 0 0 0 0 1 S c b 0 0 0 0 0 0 0 0 0 0 0 0 0 Here, 30-21=9. Here, 18-8=10. 2 3 4 5 6 7 8 9 a b c BTWN1 1 5 5 5 2 5 5 3 6 4 5 5 4 4 6 5 5 7 8 6 6 9 7 6 7 a 6 7 b 6 All Btwn1 and Btwn1.5 gaps=1. Lexical orderings for breaking ties aren’t semantic; they depend on the number order of vertices, which is artificial. So DATON is better! S 1 3 c 0 0 0 0 0 0 0 0 1 1 1 0 3 S 2 3 c 0 0 0 0 0 0 0 0 1 1 1 0 3 S 2 4 7 0 0 0 0 1 1 0 0 0 0 0 0 2 S 3 1 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S 1 4 7 0 0 0 0 1 1 0 0 0 0 0 0 2 S 8 9 c 0 0 1 0 0 0 0 0 0 0 0 0 1 S 8 a c 0 0 1 0 0 0 0 0 0 0 0 0 1 S a 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S 5 6 8 0 0 0 0 0 0 0 0 1 1 0 0 2 S 7 6 8 0 0 0 0 0 0 0 0 1 1 0 0 2 S b 9 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S 3 2 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S 4 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 S 4 1 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 4 2 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 4 7 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 4 7 6 0 0 0 0 0 0 0 1 0 0 0 0 1 S 5 7 4 1 1 0 0 0 0 0 0 0 0 0 0 2 S 6 7 4 1 1 0 0 0 0 0 0 0 0 0 0 2 S 8 6 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 8 6 7 0 0 0 1 0 0 0 0 0 0 0 0 1 S 9 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S 9 c 3 1 1 0 0 0 0 0 0 0 0 0 0 2 S a c 3 1 1 0 0 0 0 0 0 0 0 0 0 2 S 8 9 b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 8 a b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 3 c 9 0 0 0 0 0 0 0 1 0 0 0 0 1 S c a 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S 3 c a 0 0 0 0 0 0 0 1 0 0 0 0 1 S 6 8 9 0 0 0 0 0 0 0 0 0 0 1 1 2 S b a 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S c 9 8 0 0 0 0 0 1 0 0 0 0 0 0 1 S 6 8 a 0 0 0 0 0 0 0 0 0 0 1 1 2 S 3 c b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 7 4 1 0 1 1 0 0 0 0 0 0 0 0 0 1 S 7 4 2 1 0 1 0 0 0 0 0 0 0 0 0 1 S b c 3 1 1 0 0 0 0 0 0 0 0 0 0 2 S c 3 1 0 0 0 1 0 0 0 0 0 0 0 0 1 S c 3 2 0 0 0 1 0 0 0 0 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c BTWN1.5 1 1 3 4 2 3 3 3 6 4 5 5 2 2 6 5 5 7 8 6 4 9 1 2 3 a 2 3 b 2 gaps>2*avg Avg=6-1/19=.26 All gaps are 1 so they all qualify. Stop after 6’s, no partition!. S 1 3 c 9 0 0 0 0 0 0 0 1 0 0 0 0 1 S 1 4 7 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 1 4 7 6 0 0 0 0 0 0 0 1 0 0 0 0 1 S 2 3 c 9 0 0 0 0 0 0 0 1 0 0 0 0 1 S 3 2 4 7 0 0 0 0 1 0 0 0 0 0 0 0 0 S 3 c a 8 0 0 0 0 0 1 0 0 0 0 0 0 0 S 4 2 3 c 0 0 0 0 0 0 0 0 1 1 1 0 3 S 5 6 8 9 0 0 0 0 0 0 0 0 0 0 1 1 2 S 6 7 4 1 0 0 1 0 0 0 0 0 0 0 0 0 1 S 6 7 4 2 0 0 1 0 0 0 0 0 0 0 0 0 1 S 6 8 9 c 0 0 1 0 0 0 0 0 0 0 0 0 1 S 7 4 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 S 7 4 1 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 7 4 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 S 7 4 2 3 0 0 0 0 0 0 0 0 0 0 0 1 1 S 7 6 8 9 0 0 0 0 0 0 0 0 0 0 1 1 2 S 8 9 c 3 1 1 0 0 0 0 0 0 0 0 0 0 2 S b 9 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S b a 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S c 3 2 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S c 9 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S 3 c 9 8 0 0 0 0 0 1 0 0 0 0 0 0 0 S 4 7 6 8 0 0 0 0 0 0 0 0 1 1 0 0 2 S 1 3 c a 0 0 0 0 0 0 0 1 0 0 0 0 1 S 2 3 c a 0 0 0 0 0 0 0 1 0 0 0 0 1 S 5 6 8 a 0 0 0 0 0 0 0 0 0 0 1 1 2 S 7 6 8 a 0 0 0 0 0 0 0 0 0 0 1 1 2 S 2 4 7 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 2 4 7 6 0 0 0 0 0 0 0 1 0 0 0 0 1 S 3 1 4 7 0 0 0 0 1 0 0 0 0 0 0 0 0 S 4 1 3 c 0 0 0 0 0 0 0 0 1 1 1 0 3 S 5 7 4 1 0 0 1 0 0 0 0 0 0 0 0 0 1 S 5 7 4 2 0 0 1 0 0 0 0 0 0 0 0 0 1 S 8 6 7 4 1 1 0 0 0 0 0 0 0 0 0 0 2 S 8 a c 3 0 0 0 0 0 0 0 0 0 0 0 0 0 S 9 8 6 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S 9 8 6 7 0 0 0 1 0 0 0 0 0 0 0 0 1 S 9 c 3 1 0 0 0 1 0 0 0 0 0 0 0 0 1 S 9 c 3 2 0 0 0 1 0 0 0 0 0 0 0 0 1 S a 8 6 5 0 0 0 0 0 0 0 0 0 0 0 0 0 S a 8 6 7 0 0 0 1 0 0 0 0 0 0 0 0 1 S a c 3 1 0 0 0 1 0 0 0 0 0 0 0 0 1 S a c 3 2 0 0 0 1 0 0 0 0 0 0 0 0 1 S b c 3 1 0 0 0 1 0 0 0 0 0 0 0 0 1 S b c 3 2 0 0 0 1 0 0 0 0 0 0 0 0 1 S c 3 1 4 0 0 0 0 0 0 1 0 0 0 0 0 1 S c 9 8 6 0 0 0 0 1 0 1 0 0 0 0 0 2 S 1 3 c b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 2 3 c b 0 0 0 0 0 0 0 0 0 0 0 0 0 S 6 8 9 b 0 0 0 0 0 0 0 0 0 0 0 0 0 Stop after 5’s. BTWN1.5 with DNIv_DATON Stop after 1’s (e.g., at the end). 1 5 G6 4 2 6 7 3 c 9 b 8 a BTWN1 with DNIv V 1 5 G6 4 2 6 7 3 c 9 b 8 a

  7. 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 1 9 a s s s t t 3 1 2 4 8 9 a e s t 6 7 w y y o p y w y 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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w 1 1 2 4 1 2 4 1 2 1 1 1 2 4 1 2 2 1 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 6 7 5 3 3 2 a e 1 4 4 1 1 1 1 3 2 1 1 8 8 2 1 8 2 1 8 2 8 8 8 2 1 8 2 2 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 p p p p p p p p p p w w p p p q s s s w w w w 1 1 p q s w o 3 o y 1 t x y 6 7 3 1 3 4 1 6 1 c e 0 8 c 1 1 r r r r r r r r r r r y y y y r r r r r r r r r r r u y y y y y y 9 9 w w r r u u y y y y y y y y y x 9 9 s t v w 1 1 1 1 r u y o x 9 a e k o s t v w 3 1 3 3 3 2 1 6 7 6 7 2 2 e 1 1 2 1 4 2 1 2 1 1 3 1 9 1 1 1 3 9 1 1 1 1 q q q q q q q q q q q w w q q q o o o o p w w w w 1 1 q o p w s u x y s 1 t x y 6 7 3 4 1 4 2 3 9 e 2 f 2 9 e 1 1 s s s s s 3 3 s s s s 3 3 3 1 1 s 3 o p y 1 2 4 6 7 4 9 3 2 e 8 2 1 1 1 t t t t t t t t 3 3 w w t t t 3 3 3 w 1 1 1 1 t 3 w y 1 2 4 1 6 7 6 7 3 9 4 e 8 2 1 8 8 8 3 3 u u u u u u u u u u u u u u u u u u u u u u u u u u x x x x x x y y y y y y y y u u u u u u u u u u x x x x x y y y y y y y 3 3 9 9 w w 9 9 e e k k w w u u u u o o x x x y y y y y 3 3 3 9 w 9 e e e k k w 1 1 1 1 1 1 1 1 1 1 1 1 1 1 u o r x y q s 3 9 w 9 e k s w 1 2 4 1 1 1 1 2 4 1 2 1 6 7 6 7 6 7 6 7 6 7 6 7 6 7 4 3 1 9 d 1 1 4 1 2 1 3 2 1 2 8 2 1 8 8 8 8 2 1 8 2 8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 v v v v v v v v v v v v 2 2 9 9 v v v v 2 2 9 x x y y y 1 1 1 1 v 2 9 x y 1 4 1 o w o s w 6 7 6 7 4 8 2 9 e 6 1 6 1 2 1 1 2 1 1 1 1 x x x x x x x x x x x x x x x x x x x x x x 3 3 9 9 w w x x x x x x x x x x x 3 3 3 9 9 f g j l n o u v v w w 1 1 1 1 1 1 x 3 9 f g j l n o u v w 1 2 4 1 y y y y y y y y 2 y 1 y 6 7 6 7 6 7 b 8 2 1 1 1 1 1 3 2 2 5 9 3 1 9 1 1 1 1 1 1 1 1 3 1 9 1 1 1 1 1 1 1 w w w w w w w w 1 1 w 1 p q t x y 6 7 6 f 1 1 1 a e 1 1 y y y y y y y y y y y y y y y 9 9 k k w w y y y y y y y y y y y y y y y 9 9 a k k s t v w 1 1 1 1 1 1 y 9 a e f g j k l n o s t u v w 1 3 3 1 2 3 3 2 1 6 7 6 7 6 7 g 3 1 4 1 1 1 2 1 1 2 2 1 1 2 4 9 1 1 9 3 1 1 3 9 1 1 1 1 1 1

  12. n Btwn1.5 Btwn2.5 y e 19 e y 79 y w 18 w y 74 9 y 17 x w 65 9 1 17 1 3 59 k y 17 9 1 59 y a 16 y 9 59 y n 16 y k 47 s y 16 y o 44 l y 16 v y 44 w 1 16 y s 44 y o 16 o x 43 v y 16 s 3 39 c 1 16 2 v 35 f y 16 y a 31 y j 16 y j 31 y g 16 w 1 31 1 3 15 y g 31 w x 15 y f 31 1 k 15 n y 31 y t 15 l y 31 d 1 14 x v 29 1 i 14 k 1 29 1 m 14 x u 29 7 1 14 y t 29 y u 14 3 t 29 y r 14 2 3 29 1 6 14 1 6 27 b 1 13 y u 27 x o 13 7 1 27 1 5 13 x 9 26 e 1 13 1 e 25 8 1 12 3 9 23 3 s 12 x g 21 x l 11 x n 21 v 2 11 j x 21 x f 11 l x 21 t 3 11 f x 21 v x 11 q o 19 u x 11 a 3 19 n x 11 1 2 17 x g 11 c 1 16 j x 11 k 2 15 1 4 10 o s 15 a 3 10 i 1 14 9 x 10 d 1 14 1 2 9 1 m 14 2 3 9 r y 14 3 9 9 e 3 13 x 3 8 1 5 13 k 2 8 b 1 13 o q 7 8 1 12 3 e 7 u o 11 2 i 7 p s 11 m 2 7 4 3 11 e 2 6 e 2 11 o s 6 1 4 10 3 8 6 q w 9 4 3 6 4 2 9 p s 5 p w 9 w t 5 w t 9 2 8 5 3 x 8 o u 5 i 2 7 w p 5 m 2 7 2 4 5 8 3 6 q w 5 9 v 5 4 d 4 r u 5 b 6 3 4 e 5 v 9 3 6 b 5 u r 3 8 2 5 e 4 3 7 5 5 5 7 3 4 d 4 h 6 2 6 7 3 8 4 2 b 5 3 7 6 2 q p 3 b 5 2 4 8 2 q p 2 h 7 2 7 h 2 h 6 2 BtwnGN sort uniq k y 509 q o 494 t 3 406 1 w 395 1 9 360 2 v 341 s 3 321 9 y 303 o x 264 y w 239 k 1 219 3 a 215 o y 209 x u 199 w x 197 9 x 194 e 1 169 6 1 167 7 1 167 u y 135 y v 113 1 3 104 y e 99 2 k 99 q w 93 v x 90 9 3 89 c 1 88 1 d 86 h 7 84 1 i 84 1 m 84 6 h 84 o s 83 5 1 83 b 1 82 p w 81 3 2 74 p s 68 f x 62 g x 62 j x 62 l x 62 n x 62 e 2 59 y s 59 y a 59 y r 59 m 2 56 f y 55 g y 55 j y 55 l y 55 n y 55 i 2 54 t y 44 t w 37 4 e 29 4 3 29 d 4 26 2 1 25 8 3 25 8 1 24 3 x 23 9 v 21 u o 17 3 e 15 4 1 14 q p 11 u r 11 8 2 11 4 2 9 6 b 5 7 5 5 8 4 4 5 b 3 7 6 3 G7: Friendships in Zachery’s Karate Club Btwn1.5 f r a g d k v y u e 4 5 9 l 6 x h j 3 1 2 n 7 o G7: Friendships in Zachery’s Karate Club. Btwn2.5 f r t a b s 8 g m w q i d k v c p y u e 4 5 9 l 6 x h j 3 1 2 7 o t b s 8 m w q i n f r c G7: Friendships in Zachery’s Karate Club. BtwnGN p a g d k v y u e 4 5 9 l 6 x h j 3 1 2 7 o t b s 8 m w q i c p

  13. Recomputing Between-nesses after every delete? GN does this. Can it be done by just updating the existing btwn-nesses is some way? 2 2-Level Stride=4, Edge pTree 1 Level1 1 1 1 1 If 24 is deleted, it appears to be easy, namely, turn off bit 4 in 2 and bit 2 in 4, delete all three 2 hop SPs since they all three involve 24. So at this point it looks like it might just be a matter of deleting those SPs that involve the edge (either at the beginning, end or middle). However, if we delete 14 then 134 and 1342 are barnd new SPs which weren’t there before. It looks like one has to start fresh computing Between-ness??? 1 0 0 1 1 2 2 0 0 0 1 1 3 1 0 0 1 2 4 1 1 1 0 3 Edge Map Edges V1 V2 3 4 E 0 0 1 1_ 0 0 0 1_ 1 0 0 1_ 1 1 1 0 1,1 1,2 1,3 1,4_ 2,1 2,2 2,3 2,4_ 3,1 3,2 3,3 3,4_ 4,1 4,2 4,3 4,4 E2 0 0 0 1 E3 1 0 0 1 E4 1 1 1 0 E1 0 0 1 1 1 4 0 1 0 0 1 2 4 1 0 1 0 2 3 4 0 1 0 0 1 Btwn14=1 Btwn24=2 Btwn34=1 Ek is the map of edge endpoints from k (points adjacent to k). Shi is the map of endpoints of 2-hop Shortest Paths Through h then i. Of course, |Shi&Ek|=0 since there is a SP of length<2 to each vertex in Ek. Shij is the map of endpoints of 3-hop Shortest Paths Through h then i then j. Of course, |Shij&Ek|=|Shij&Shi|=0 since there is a SP of length<3 to each Ek and Shi. Etc. The Main Theorem: The full SP participation count of hk is a + b + ab where a is |Shk|+|Shki|+|Shkij|+… and b is |Skh|+|Skhi|+|Skhij|+… . Proof: Let s=c…dhke…f by any SP in which hk participates. If h=c then s is counted in |Shk|+|Shki|+|Shkij|+… If k=f then hk is counted in c then hk is counted in |Skh|+|Sikh|+|Sjikh|+… . If hc and kf then hk occurs in the middle of s and the “left half” of s (hk and left) is counted in a = |Skh|+|Sikh|+|Sjikh|+… and the “right half” (hk and right) is counted in b = |Shk|+|Shki|+|Shkij|+… therefore s is counted in ab. Thus Participation Count of hk a + b + ab. But every a + b + ab counts a SP, so we get =.

  14. Clique Existence Thm (CLQe)Let G=(V,E) and WV with |W|=k and EW{ {x,y}E | x,y W}, then the induced subgraph, (W,EW)CLQk (is a k-clique) iff every induced (k-1)vertex subgraph of (W,EW)CLQk-1. key 1,1 1,2 1,3 1,4 1,5 1,6 1,7 2,1 2,2 2,3 2,4 2,5 2,6 2,7 3,1 3,2 3,3 3,4 3,5 3,6 3,7 4,1 4,2 4,3 4,4 4,5 4,6 4,7 5,1 5,2 5,3 5,4 5,5 5,6 5,7 6,1 6,2 6,3 6,4 6,5 6,6 6,7 7,1 7,2 7,3 7,4 7,5 7,6 7,7 E 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 1 0 EU 0 1 1 1 0 1 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 C 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 CU 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 20 1 2 3 4 5 6 7 8 9 30 1 2 3 4 5 6 7 8 9 40 1 2 3 4 5 6 7 8 9 Clique Mining Thm (CLQm) finds all cliques using a closure property: Let Candk+1CliqueSet CCLQk+1. By the CLQe thm, CCLQk+1= all s of CLQk-pairs having k-1 common vertices. Let CCCLQk+1 be a union of two k-cliques with k-1 common vertices. Let v and w be their kth (non-common) vertices respectively, then CCLQk+1 iff Evw=1 (Just check a single bit in PE.) CLQ4: 1234 since PE 0 1 1 1 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 0 1 0 0 0 0 1 1 0 key 1,1 1,2 1,3 1,4 1,5 1,6 1,7 2,1 2,2 2,3 2,4 2,5 2,6 2,7 3,1 3,2 3,3 3,4 3,5 3,6 3,7 4,1 4,2 4,3 4,4 4,5 4,6 4,7 5,1 5,2 5,3 5,4 5,5 5,6 5,7 6,1 6,2 6,3 6,4 6,5 6,6 6,7 7,1 7,2 7,3 7,4 7,5 7,6 7,7 CLQ2: 2 vertices, 1 edge, so just E, which as a list is: 12 13 14 1623 24 34 56 67 CLQ3: 123 124 134 234 CLQ5=. 123,1241234. 123,1341234. 123,2341234. 124,1341234. 124, 2341234. 134,2341234. 8 Clique ExistenceThm edge count (CLQec): C={1,2,3,4}, CU=C&EU. ct(CU)=comb(4,2)=4!/2!2!=6 CCLQ4. Is there an edge count Clique Mining Thm? E1 0 1 1 1 0 1 0 E2 1 0 1 1 0 0 0 E3 1 1 0 1 0 0 0 E4 1 1 1 0 0 0 0 E6 1 0 0 0 1 0 1 E5 0 0 0 0 0 1 0 E7 0 0 0 0 1 1 0 k=4: 1234 k=5:  k=2: E=12 13 14 16 23 2434 56 57 67. PE(4,8)=1 2348CS4 PE(4,8)=1 1248CS4 PE(3,8)=1 1348CS4 k=3: 123 124 134 234 567 Edge counting requires counting 1’s in mask pTree of each Subgraph (or candidate Clique, if we take the time to generate the CCSs – but then clearly the fastest way to finish up is simply to lookup the single bit position in E, i.e., use EC). Again, CLQec is: |UCC| = (k+1)!/(k-1)!2! iffCCLQk. The SG Clique Mining Alg only needs to find those pairs of subgraphs in CLQk that share k-1 vertices) then check E to see if the two non-shared vertices form an edge in G. The search for such pairs is standare in the Apriori ARM alg and has therefore been optimized and engineered ad infinitum!) PE(2,6)=0 PE(2,6)=0 key 1,1 1,2 1,3 1,4 1,5 1,6 1,7 1,8 2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8 3,1 3,2 3,3 3,4 3,5 3,6 3,7 3,8 4,1 4,2 4,3 4,4 4,5 4,6 4,7 4,8 5,1 5,2 5,3 5,4 5,5 5,6 5,7 5,8 6,1 6,2 6,3 6,4 6,5 6,6 6,7 6,8 7,1 7,2 7,3 7,4 7,5 7,6 7,7 7,8 8.1 8,2 8,3 8,4 8,5 8,6 8,7 8.8 E 0 1 1 1 0 1 0 1 1 0 1 1 0 0 0 1 1 1 0 1 0 0 0 1 1 1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 1 1 1 1 0 0 0 0 6 G2 6 G4 6 G3 PE(3,4)=1 1234CS4 PE(6,7)=1 567CS3 PE(6,7)=1 567CS3 PE(2,4)=1 124CS3 5 5 5 7 7 7 PE(2,4)=1 124CS3 PE(1,5)=0 PE(1,5)=0 PE(2,3)=1 123CS3 PE(2,4)=1 1234CS4 PE(4,8)=1 148CS3 PE(2,3)=1 So 123CS3 already have 567 PE(1,7)=0 have PE(1,7)=0 PE(6,8)=0 PE(3,8)=1 238CS3 PE(2,8)=1 128CS3 PE(3,8)=1 138CS3 2 2 2 1 1 1 PE(2,3)=1 234CS3 PE(3,8)=1 1238CS4 PE(4,8)=1 248CS3 k=4: 1234 1238 1248 1348 2348 k=2: 12 13 14 16 23 2434 56 57 67 18 28 38 48. PE(4,8)=1 348CS3 PE(2,3)=1 234CS3 Have 123CS3 k=5: 12348. k=6:  have 124CS3 Have Have 1234 PE(1,4)=1 134CS3 PE(1,4)=1 134CS3 PE(4,8)=1 12348CS5 Have 4 4 3 3 4 3 k=3: 123 124 134 234 567 128 138 148 238 248 348

  15. G6 Clique Mining Thm (CLQm) a e d 9 CLQ2: 2 vertices, 1 edge, so just E, which as a list is: 12 15 17 24 36 38 57 68. So CLQ3: 157 368 and CLQ4 =  EG5 2-level str=8 f b c g 1 0 1 0 0 1 0 1 0 2 1 0 0 1 0 0 0 0 3 0 0 0 0 0 1 0 1 4 0 1 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 6 0 0 1 0 0 0 0 1 7 1 0 0 0 1 0 0 0 8 0 0 1 0 0 1 0 0 8 Calculating pairwise &s is unnecessary! The most efficient algorithm is to consider CCLQk+1 from lowest common vertex set to highest (i.e., start with the lowest k and work up always keeping the max of the shared sets as low as possible). For every found candidate pair from CCLQk+1 sharing k-1 vertices in which >1 unshared vertex is higher than said shared max, check for an edge connecting those unshared vertices. E 1 2 3 4 5 6 7 8 9 a b c 1 0 1 1 1 0 0 0 0 0 0 0 0 2 1 0 1 1 0 0 0 0 0 0 0 0 3 1 1 0 0 0 0 0 0 0 0 0 1 4 1 1 0 0 0 0 1 0 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 6 0 0 0 0 1 0 1 1 0 0 0 0 7 0 0 0 1 1 1 0 0 0 0 0 0 b 0 0 0 0 0 0 0 0 1 1 0 1 9 0 0 0 0 0 0 0 1 0 1 1 1 c 0 0 1 0 0 0 0 0 1 1 1 0 8 0 0 0 0 0 1 0 0 1 1 0 0 a 0 0 0 0 0 0 0 1 1 0 1 1 G5 CLQ2: 13 16 24 34 48 56 57 67 9c ac bc df dg fg 1 2 CCLQ3: 136 134 234 248 348 567 156 167 9ac abcdfg CLQ3: y y CCLQ4:  CLQ4:  6 3 8 E14=0 E36=0 368CS3 E57=1 157CS3 E25=0 5 E27=0 4 7 7 b 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 g 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 c 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 8 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 2 3 4 5 6 7 8 9 a b c d e f g f 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 4 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 6 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 7 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 5 6 2 1 1 5 4 2 6 7 3 c CLQ2: 12 13 14 23 24 3c 47 56 57 67 68 89 8a 9a 9b 9c ab ac bc 9 b 8 a CCLQ3: 123 124 134 234 13c 23c 147 247 567 678 689 68a 89a CLQ3: y y y y 9ab 9ac 9bc abc y y y y 4 3 CCLQ4: 1234 89ab 89ac 9abc CCLQ5= CLQ4: y G5.1

  16. E 1 2 3 4 5 6 7 8 9 a b c d e f g h I j k l m n o p q r s t u v w x y Clique Mining Thm (CLQm) E 1 2 3 4 5 6 7 8 9 a b c d e f g h I j k l m n o p q r s t u v w x y A early exist for stealth programmers: (W,EW)CLQk iff every induced (k-1)vertex subgraph of (W,EW)CLQk-1. This tells us that 12348CLQ5. We know it is max containing {8}, since if there were other vertices in a bigger clique they would have shown up here. Can we now delete 12348??? Are their other early exits? Other execution time issues? e 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 l 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 m 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 a 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 b 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 c 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 d 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 h 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 j 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 k 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 5 p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 3 q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 3 r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 2 s 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 4 t 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 3 u 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 4 v 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 w 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 6 x 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 b y 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 g 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 5 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 g 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 9 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 a 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 i 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1111111111222222222233333 1234567890123456789012345678901234 123456789abcdefghijklmnopqrstuvwxy 1 with 23 24 25 26 27 28 29 2b 2c 2d 2e 2i 2k 2m 2w 34 35 36 37 38 39 3b 3c 3d 3e 3i 3k 3m 3w 45 46 47 48 49 4b 4c 4d 4e 4i 4k 4m 4w 56 57 58 59 5b 5c 5d 5e 5i 5k 5m 5w 67 68 69 6b 6c y yyyyyyyyyyyyyyyyy 2 with 13 14 18 1e 1i 1k 1m 1v 34 38 3e 3i 3k 3m 3v 48 4e 4i 4k 4m 4v 8e 8i 8k 8m 8v eiekemevikim iv km kv mv y yyyyyyyyyyy 3 with 12 14 18 19 1a 1e 1s 1t 1x 24 28 29 2a 2e 2s 2t 2x 48 49 4a 4e 4s 4t 4x 89 8a 8e 8s 8t 8x 9a 9e 9s 9t 9x ae as at ax es et ex stsxtx y yyyyyyyyyy 8 with 12 13 14 23 24 34 y yyyyy 4 with 12 13 18 1d 1e 23 28 2d 2e 38 3d 3e 8d 8e ed y yyyyyyyyy 5 with 17 1b 7b y y 6 with 17 1b 1h 7b 7h bh y yy 7 with 15 16 1h 56 5h 6h y yy 9 with 13 1v 1x 1y 3v 3x 3y vxvyxy y yy a with 3y b with 15 16 56 y y c with e with 12 13 14 1y 23 24 2y 34 3y 4y y yyyyy f with xy d with 14 y g with xy h with 67 y i with 12 y j with xy k with 12 1y 2y y l with xy p with qsqwsw y m with 12 y n with xy o with qsquqxqysusxsyuxuyxy y n q with op ow pw y s with 3o 3p 3y op oy py y t with 3w 3y wy y u with or ox oy rxryxy y yy v with 29 2x 2y 9x 9y xy y y r with uy y f r a W: 1p 1q 1t 1x 1y pqptpxpyqtqxqytx ty xy y y g d k v x: 39 3f 3g 3j 3l 3n 3o 3u 3v 3w 9f 9g 9j 9l 9n 9o 9u 9v 9w gjglgn go gugvgwjljn jo jujvjw ln lo lu lv lw no nu nvnwouov ow uvuwvw y yy y u e 4 5 9 y: 9a 9e 9f 9g 9j 9k 9l 9n 9o 9r 9s 9t 9u 9v 9w ae af ag ajak an aoar as at au av aw efegejekeneoeres et euevewfg fj fkfnfofr fs ftfufvfwgjgkgn go gsgtgugvgw y l 6 x h j y: jkjn jo jrjsjtjujvjwknkokrksktkukv kw no nr ns nt nu nvnw or osotouov ow rsrtrurvrwstsusvswtutvtwuvuwvw y yyy 3 1 2 7 o We already know 12348 is a MCLQ5. What other CLQ3s? t b s 12e 12i 12k 12m 139 13e 14d 14e 157 15b 167 23e 34e 39x 24e 348 34e 16b 67h 39x 3vy osypqwruytvwosuouxouyruy 9vx 9vy pqwtwy 39x 9vx oux 9vy 8 m w q i c p

  17. 1 with 23 24 25 26 27 28 29 2b 2c 2d 2e 2i 2k 2m 2w 34 35 36 37 38 39 3b 3c 3d 3e 3i 3k 3m 3w 45 46 47 48 49 4b 4c 4d 4e 4i 4k 4m 4w 56 57 58 59 5b 5c 5d 5e 5i 5k 5m 5w 67 68 69 6b 6c y yyyyyyyyyyyyyyyyy 2 with 13 14 18 1e 1i 1k 1m 1v 34 38 3e 3i 3k 3m 3v 48 4e 4i 4k 4m 4v 8e 8i 8k 8m 8v eiekemevikim iv km kv mv y yyyyyyyyyyy Clique Mining Thm (CLQm) on G7 UCLQ3s p1 p2 p3 b 1 5 b 1 6 d 1 4 e 1 4 e 1 2 e 1 3 e 2 3 e 2 4 e 3 4 h 6 7 i 1 2 k 1 2 m 1 2 o s y o u x o u y p q w r u y t w y v x 9 v y 9 x 3 9 1 2 8 1 2 3 1 2 4 1 3 9 1 3 4 1 3 8 1 4 8 1 5 7 1 6 7 2 3 4 2 3 8 2 4 8 3 4 8 3 with 12 14 18 19 1a 1e 1s 1t 1x 24 28 29 2a 2e 2s 2t 2x 48 49 4a 4e 4s 4t 4x 89 8a 8e 8s 8t 8x 9a 9e 9s 9t 9x ae as at ax es et ex stsxtx y yyyyyyyyyy 4 with 12 13 18 1d 1e 23 28 2d 2e 38 3d 3e 8d 8e ed y yyyyyyyyy 5 with 17 1b 7b y y 6 with 17 1b 1h 7b 7h bh y yy 8 with 12 13 14 23 24 34 y yyyyy 7 with 15 16 1h 56 5h 6h y yy 9 with 13 1v 1x 1y 3v 3x 3y vxvyxy y yy a with 3y b with 15 16 56 y y c with e with 12 13 14 1y 23 24 2y 34 3y 4y y yyyyy f with xy h with 67 y i with 12 y j with xy k with 12 1y 2y y l with xy d with 14 y g with xy q with op ow pw y s with 3o 3p 3y op oy py y p with qsqwsw y r with uy y m with 12 y n with xy o with qsquqxqysusxsyuxuyxy y t with 3w 3y wy y u with or ox oy rxryxy y yy v with 29 2x 2y 9x 9y xy y y W: 1p 1q 1t 1x 1y pqptpxpyqtqxqytx ty xy y y x: 39 3f 3g 3j 3l 3n 3o 3u 3v 3w 9f 9g 9j 9l 9n 9o 9u 9v 9w gjglgn go gugvgwjljn jo jujvjw ln lo lu lv lw no nu nvnwouov ow uvuwvw y yy y: 9a 9e 9f 9g 9j 9k 9l 9n 9o 9r 9s 9t 9u 9v 9w ae af ag ajak an aoar as at au av aw efegejekeneoeres et euevewfg fj fkfnfofr fs ftfufvfwgjgkgn y y: go gsgtgugvgwjkjn jo jrjsjtjujvjwknkokrksktkukv kw no nr ns nt nu nvnw or osotouov ow rsrtrurvrwstsusvswtutvtwuvuwvw y yyy CCLQ4: b1 with 56 e1 with 23 24 34 ou with xy e2 with 34 12 with 34 38 48 13 with 48 49 89 23 with 48 oy with su v9 with xy e4 with 12 13 23 e3 with 12 28 with 34 14 with 23 28 38 18 with 23 24 34 17 with 14 with de 67 with 1h 12 with ik im km uy with ro 39 with 1x 38 with 12 48 with 12 13 23 UCLQ4: 123e 124e 134e e234 1234 1238 1248 1348 2348 CCLQ5: 123 with 48 4e 8e UMCLQ5: 12348 1234e UMCLQ3 b 1 5 b 1 6 d 1 4 h 6 7 i 1 2 k 1 2 m 1 2 o s y o u x o u y p q w r u y t w y v x 9 v y 9 x 3 9 1 3 9 1 5 7 1 6 7 n f r a g d k v y u e 4 5 9 l 6 x h j 3 1 2 7 o t b s 8 m w q i c p

  18. Clique Mining on G10.At each step, we branch (in parallel?) to each of the lowest degree vertices. UCLQ3pTrees: for Max Ct=26 vertex=91. All & with 91 have Ct=0 so 91 is part of no 3cliques 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 40 46 76 81 86 88 89 90 92 93 94 97 98 99 a0 a1 a2 a4 b1 b4 c6 c7 d9 e0 h8 h9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 UCLQ3pTrees: for Ct=24 vertex=D2. All & with D2 have Ct=0 so D2 is part of no 3cliques 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 91 40 46 76 81 86 88 89 90 92 93 94 97 98 99 a0 a1 a2 a4 b1 b4 c6 c7 d9 e0 h8 h9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G10 for Ct=14 vertex=52 & Ct=0 so 52 part of no 3clique 5252525252525252525252525252 4653545556575859606162636494 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G10 UCLQ3pTrees: for Ct=23 vertex=38. All & with 38 have Ct=0 so 38 is part of no 3cliques 3838383838383838383838383838383838383838383838 2122242526272829303132333435364041424345464980 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G10: Web graph of pages of a website and hyperlinks. Communities by color (Girvan Newman Algorithm). |V|=180 (1-i0) and |E|=478. We have unPTrees (undirected graph). inPTrees (showing all incoming edges and where they come in from) and outPTrees. b7 G10 for Ct=13 vertex=174 is part of 3cliques H0 H2 H4 and H3 H4 I0 h4h4h4h4h4h4h4h4h4h4h4h4h4 4681d0g6h0h1h2h3h5h6h7h8i0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 G10 Ct(B2)=9 part of 3clique, 45 76 B2 b2b2b2b2b2b2b2b2b2 4572737476b1b3c0h1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 G10 Ct(45)=9 &cts=0 a3 b8 G10 Ct(78)=9 &cts=0 G10 Ct(49)=8 all 0s b6 b5 G10 Ct(81)=8 all 0s b9 a5 c3 a4 G10 Ct(C4)=7 all 0s G10 Ct(A7)=5 all 0s c2 G10 Ct(H9)=5 all 0s c4 2 1 3 4 a7 5 a6 a8 G10 for Ct=13 vertex=46 & Ct=0 so 46 part of no 3clique 46464646464646464646464646 3845474849505152787991d2h4 0 0 0 0 0 0 0 0 0 0 0 0 0 a9 b4 6 b0 c1 c0 7 19 20 8 18 22 97 98 99 21 23 a1 90 24 9 35 a0 84 a2 25 b1 85 36 40 89 10 37 26 92 76 83 27 86 b3 28 93 38 91 11 82 75 70 94 42 29 87 41 30 12 69 80 88 39 b2 57 81 43 31 32 58 77 95 13 There are only three 3Cliques: {H0 H2 H4} {H3 H4 I0} {45 76 B2} (I quickly checked the rest). No two share and edge so there are no 4Cliques. 56 h9 34 59 68 c5 44 79 33 96 14 78 60 55 15 52 51 67 16 45 61 50 h8 17 46 i0 62 63 54 49 The fact there are so few cliques may be a characteristic of web page link graphs. Was it worthwhile doing the Clique analysis? Yes! The 8 vertices involved in the three 3Cliques (and the three cliques themselves) are outliers! We can examine each to try to determine what’s unique about them. What does it mean that the three vertices {H3 H4 I0} are a 3Clique in the undirected graph of page references. In this case, after close examination, we see that they form a cycle (in the directed graph sense). Should there ever be circular references like that in web pages? The 3Clique {45 76 B2} appears to be a mistake (no edge from 45 to 76). The clique {H0 H3 H4} does not appear to be a cycle. 66 h7 65 53 64 48 h3 47 h5 h6 h4 h2 c7 c6 h0 71 h1 74 g9 72 g2 73 g3 g6 g8 c8 d4 g4 d3 g5 c9 g7 d2 g1 e6 d1 e7 d0 e5 g0 e2 e4 d7 d5 f9 f7 e3 f8 e1 f6 d6 e8 e0 d8 d9 e9 f0 f1 f5 f2 f4 f3

  19. G6 Topdown k-plex Mining Algorithm: If G isn’t a k-plex, Let H1 be an ISG of G which is simply G with a vertex of least degree removed. If H1 still isn’t a k-plex, let H2 be an ISG of H1 with a vertex of least degree (in H1) removed, etc., until we find Hj is a k-plex. Remove Hj and restart the algorithm until all vertexes are removed. Note, we know Hj exists since an edge is a 0-plex. Letting H be an ISG and |VH|=h, |EH|=H, H=h(h-1)/2. H is a k-plex iff H–Hk. Downward Closure: If H a k-plex and F is an ISG of H, then F is a k-plex (If F is missing an edge, H is missing it too. So, F can’t be missing more edges than H). |E{123}|  #edges_in_induced_subgraph_123 = 3 so 123 is a 0-plex(a clique). Edges are 0-plexes. G=12*11/2=66 and G=19 so G is a kplex for k  66-19 = 47. H1=ISG{12346789abc} (degG5=2). H1=11*10/2=55, H1=17. H1 is a kplex for k  38. |E{124}| = 3 so 124 is a -0plex (clique) E 1 2 3 4 5 6 7 8 9 a b c 1 0 1 1 1 0 0 0 0 0 0 0 0 3 2 1 0 1 1 0 0 0 0 0 0 0 0 3 3 1 1 0 0 0 0 0 0 0 0 0 1 3 4 1 1 0 0 0 0 1 0 0 0 0 0 3 5 0 0 0 0 0 1 1 0 0 0 0 0 2 6 0 0 0 0 1 0 1 1 0 0 0 0 3 7 0 0 0 1 1 1 0 0 0 0 0 0 3 b 0 0 0 0 0 0 0 0 1 1 0 1 3 9 0 0 0 0 0 0 0 1 0 1 1 1 4 c 0 0 1 0 0 0 0 0 1 1 1 0 4 8 0 0 0 0 0 1 0 0 1 1 0 0 3 a 0 0 0 0 0 0 0 1 1 0 1 1 4 H2=ISG{1234789abc} (degH16=2). H2=10*9/2=45, H2=15. H2 is a kplex for k  30. (Must we AND all Fx&E5’ x5 to get the degH1(x)s? No! We already retrieved E5={6,7} so we just decrement the 1Counts (of 6 and 7) by 1 each (to 2 and 2) ). H3=ISG{123489abc} (degH27=1). H3=9*8/2=36, H3=14. H3 is a kplex for k  22. H4=ISG{12389abc} (degH34=2). H4=8*7/2=28, H4=12. H4 is a kplex for k  16. H5=ISG{1239abc} (degH48=2). H5=7*6/2=21, H5=10. H5 is a kplex for k  11. H6=ISG{239abc} (degH51=2). H6=6*5/2=15, H6=8. H6 is a kplex for k  7. H7=ISG{39abc} (degH62=1). H7=5*4/2=10, H7=7. H7 is a kplex for k  3. H8=ISG{9abc} (degH73=1). H8=4*3/2=6, H8=6. H8 is a kplex for k  0. So take {9abc} out of G (call it G1) and start over. G1={12345678} G1=8*7/2=28. G1=10 G1 is a kplex for k  18 33322331=deg H1=ISG{1234567} (degG18=1). H1=7*6/2=21, H1=9. H1 is a kplex for k  12. deg=2223223 H2=ISG{234567} (degH11=2). H2=6*5/2=15, H2=6. H2 is a kplex for k  9. deg=112223 H3=ISG{34567} (degH22=1). H3=5*4/2=10, H3=4. H3 is a kplex for k  6. deg=01222 H4=ISG{4567} (degH33=0). H4=4*3/2=6, H4=4. H4 is a kplex for k  2. deg=1222 H5=ISG{567} (degH44=1). H5=3*2/2=3, H5=3. H5 is a kplex for k  0. deg=222 So take {567} out of G1 (call it G2) and start over. G2={12348} G2=5*4/2=10. G2=5 G2 is a kplex for k  5. 33220=deg 1 5 4 2 6 7 H1=ISG{1234} (degG28=0). H1=4*3/2=6, H1=5. H1 is a kplex for k  1. deg=3322 This is what we want ! 1234 is a 1-plex (missing 1 edge). 124 was determined to be a clique (0-plex) It’d have been great if 123 was revealed as a clique and if 89abc was detected as a 1plex before 9abc was detected as a clique and removed. Can we modify the algorithm to do that? We’ll try by returning to remove all degree ties before moving on (on the next slide). NOTE: We only used E, and never used SP2, SP3, SP4 and that’s significant because those structures are hard to generate! 3 c 9 b H2=ISG{124} (degH13=2). H2=3*2/2=3, H2=3. H2 is a kplex for k  0. deg=222 8 a Miscellany: S(V,E) is a k-plex iff C(|V|,2)–|E|=|V|(|V|-1)/2-|E|  k iff |V|2/2–|V|/2–|E|–k  0. Adding |V| to both sides iff |V|2/2+|V|/2–|E|–k  |V|. If S’(V’,E’) adds 1 vertex, x, to S and adds only odx new edges out from x, S’ a k-plex iff (|V|+1)|V|/2–(|E’|+|E|)  k iff |V|2/2+|V|/2 -|E|–k  odx. odxv so can say only that S=k-plexS’=kplex if odx=v (obvious). If S is a k-plex missing h edges (so the slack is k-h more edges can be missing), and S’ is as above, S’ is a k-plex iff k-hv-odx or odxv-k+h. And odx=Ct(ES’,x). So a bottom up approach (larger and larger SuperGraphs) might use this fact??

  20. G6 Topdown k-plex k-core Mining Alg If G isn’t a k-plex, Let H1 be an ISG of G with a vertex of least degree removed. In parallel remove all degree ties before moving on. If H1 still isn’t a k-plex, let H2 be an ISG of H1 with a vertex of least degree (in H1) removed, etc., until we find Hj is a k-plex (usually our interest is in k=0). Remove Hj and restart the alg until all vertexes are removed. Note, we know Hj exists since an edge is a 0-plex. H an ISG and |VH|=h, |EH|=H, H=h(h-1)/2. H is a k-plex iff H–Hk. Downward Closure: If H a k-plex and F is an ISG of H, then F is a k-plex (If F is missing an edge, H is missing it too. So, F can’t be missing more edges than H). A k-core is a Subgraph containing  k edges. A COMBO(|V|,2)-core is a clique. Upward Closure of k-cores: If H is a kcore and H is an ISG of F, then F is a kcore. E 1 2 3 4 5 6 7 8 9 a b c 1 0 1 1 1 0 0 0 0 0 0 0 0 3 2 1 0 1 1 0 0 0 0 0 0 0 0 3 3 1 1 0 0 0 0 0 0 0 0 0 1 3 4 1 1 0 0 0 0 1 0 0 0 0 0 3 5 0 0 0 0 0 1 1 0 0 0 0 0 2 6 0 0 0 0 1 0 1 1 0 0 0 0 3 7 0 0 0 1 1 1 0 0 0 0 0 0 3 b 0 0 0 0 0 0 0 0 1 1 0 1 3 9 0 0 0 0 0 0 0 1 0 1 1 1 4 c 0 0 1 0 0 0 0 0 1 1 1 0 4 8 0 0 0 0 0 1 0 0 1 1 0 0 3 a 0 0 0 0 0 0 0 1 1 0 1 1 4 {123456789abc} 47plex19core 333323334434 Topdown Mining all kplexes and kcores. At each step, we [potentially] branch to each of the lowest degree vertices. {1234 6789abc} 37plex 17core 3333 2234434 {1234 6 89abc} 30plex 15core 3332 1 34434 {1234 789abc}30plex 15core 3333 124434 {1234 89abc} 22plex 14core 3332 24434 same asstop {1234 89abc}22plex 14core 3332 24434 {123 89abc}16plex 12core 223 24434 {1234 9abc}16plex 12core 3332 3334 {123 9abc}11plex 10core 223 3334 same asstop { 23 89abc}11plex 10core 12 24434 {1 3 89abc}11plex 10core 1 2 24434 {123 9abc}11plex 10core 223 3334 { 3 89abc}6plex 9core 1 24434 same as  stop { 23 9abc} 7plex 8core 12 3334 {1 3 9abc}7plex 8core 1 2 3334 { 3 89abc}6plex 9core 1 24434 { 3 9abc}3plex 7core 1 3334 { 3 9abc}3plex 7core 1 3334 same as stop { 89abc}2plex 8core 24433 1 5 { 9abc}0plex 6core 3333 { 9abc}0plex 6core 3333 4 2 6 7 3 c So take {9abc} and start over. 9 b 8 a {12345678} 18plex 10core 33232331 {12345 7} 8plex 7core 33231 2 {1234567} 12plex 9core 3323223 {1234 67} 8plex 7core 3323 12 {12 4567} 12plex 9core 22 3223 { 2 4567} 5plex 5core 1 2223 {1 4567} 5plex 5core 1 2223 {12 4 67} 5plex 5core 22 312 {12 45 7} 5plex 5core 22 312 {1234 7} 4plex 6core 3323 1 {1234 7} 4plex 6core 3323 1 same as stop { 4567} 2plex 4core 1223 { 4567} 2plex 4core 1223 same as stop {12 4 7} 2plex 4core 22 31 {12 4 7} 2plex 4core 22 3 1 same as stop {1234 } 1plex 5core 3322 { 567} 0plex 3core 222 {12 4 } 0plex 3core 22 2 {12 4 } 0plex 3core 222 same as stop {123 } 0plex 3core 222

  21. Topdown kplex/kcore mining On G7. Delete lowest degree vertices. 1111111111222222222233333 1234567890123456789012345678901234 123456789abcdefghijklmnopqrstuvwxy E 1 2 3 4 5 6 7 8 9 a b c d e f g h I j k l m n o p q r s t u v w x y e 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 l 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 m 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 a 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 b 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 c 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 d 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 f 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 h 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 j 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 k 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 5 p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 3 q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 3 r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 2 s 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 4 t 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 3 u 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 4 v 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 w 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 6 x 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 b y 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 g 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 5 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 g 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 9 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 a 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 i 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 4 cliques not revealed (Son’t combine steps (e.g., del 0,1,2,3 at 1 time)? 1111111111222222222233333 1234567890123456789012345678901234 g9a63444523125222223222533243446bg 122222233333 12346789445678901234 96952245553324344669 12233333 12348944801234 76854554334367 67b2cor 211 1plx 57b2cor 211 1plx 56b2cor 112 1plx 5672cor 112 1plx 12333 12348944134 66754552444 1333 1234894134 6675455433 1111 11111222222222233333 567 90123 56789012345678901234 233 31200 222021202533232435af 1111112222222233 56790156790135678912 23310200200003201113 13 12348941 66654342 222233 9568912 1321113 1111122222222233333 5679 1567901345678901234 233322222122533232435ae 111111 2222222233333 5679015679 1345678901234 233312222222533232435ae 11223 56717562 23322221 1 1234894 6565424 2233core 562 0plex 222 Del 25,26,32 restart G7 11111 2222222233333 5679 15679 1345678901234 23332222222533232435ad 11111 2222222233333 5679 15679 1345678901234 23332222222533232435ad same 1122 5671756 2332211 1 14core 123484 1plex 555544 223 9891 1001 3 91 11 111112222222233333 5679156791345678901234 23332222222533232435ad 11 6core 56717 4plex 23322 1234810core 44444 0plex 14cor 5671 2plx 2222 2333339core 401234 7plex 332222 14cor 5677 2plx 1232 11 4cor 6717 2plx 3212 1234e10core 44444 0plex Del 12348e restart 23335core 4034 1plex 3322 Del 5,6,7,11,17. restart. 67h3cor 222 0plx 67h 232 2333cor 404 0plx 222 2333cor 403 0plx 222 Del 24,30,33,34. Restart. Here we stick with just one count deleted at a time, fully in parallel for each lowest count value. 1111111111222222222233333 1234567890123456789012345678901234 g9a63444523125222223222533243446bg 11 1111111222222222233333 12345678901 3456789012345678901234 f9a63444523 25222223222533243446bg 11111111222222222233333 12345678913456789012345678901234 f99634445325222223222533243446bf 11111111222222222233333 12345678901456789012345678901234 e9a534445235222223222533243446bg 11111111222222222233333 12345678901346789012345678901234 f9a634445232522223222533243446af 11111111222222222233333 12345678901345789012345678901234 f9a634445232522223222533243446af 11111111122222222233333 12345678901345678902345678901234 f9a634445232522222322533243446af 11111111222222222233333 12345678901345679012345678901234 e8a634445232522223222533243446bg 11111111222222222233333 12345678901345678012345678901234 f9a634445232522223222533243446af 11111111222222222233333 12345678901345689012345678901234 f9a633345232522223222533243446bg This is one del at a time. Oof! Let’s try one count at a time on next slide 11111111122222222233333 12345678901345678901345678901234 e8a634445232522222322533243446bg 11111111122222222233333 12345678901345678901245678901234 f9a634445232522222322533243446af 11111111122222222233333 12345678901345678901234568901234 f9a634445232522222322253343346bf

  22. Topdown kplex/kcore Mining on G7.At each step, we branch (in parallel?) to each of the lowest degree vertices. Here delete just one count value at a time, but we delete all occurrences of that value at one time. We use this to cluster. It does not separate the white from the light blue. 1 4 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 5 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 7 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 5 2 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 3 2 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 3 2 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 2 2 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 4 2 9 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 4 3 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 6 3 3 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 3 4 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 6 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 6 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 9 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 8 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 Mine kplexes/cores each step, del low cts. Then cluster based on lo plex, hi core affinity. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 1111111111222222222233333 1234567890123456789012345678901234 123456789abcdefghijklmnopqrstuvwxy We try recursing the process on blue only. 11111111222222222233333 567901356789012345678901234 2333120222021202533232435af 1111111111222222222233333 1234567890123456789012345678901234 g9a63444523125222223222533243446bg 11112222222233333 905691345678901234 3122222533232435ae 111112222222233333 5679156791345678901234 23332222222533232435ae 11 1111111222222222233333 12345678901 3456789012345678901234 f9a63444523 25222223222533243446bg 1112222222233333 95691345678901234 322222533232435ad G7 222233333 679456801234 113533333456 1122222233333 1234567891404568901234 c795333453535334334669 222233333 9456801234 3533333456 222233333 9456801234 19core 3533333456 26plex 1223333 1234894481234 7685455334356 2333 4234 No help! 2222 Does not separate white from light blue 2333 94234 6cor 22233 4plx 2333 4234 4core 2222 2plex 1333 1234894134 6675455433 233 423 2core 112 1plex 333 234 2core 211 1plex 233 434 2core 211 1plex 233 424 2core 112 1plex 13 12348941 66654342 Take our 0plexes, {12348 1234e 67h ox oy}, move up 1 step at a time stopping prior to overlap, we move up 1 step in thread 1 and 2 in threads 2 and 3. getting {1,2,3,4,8,14 24,32,33,34 5,6,7,11,17}. If we move up one more level in threads 1 and 2 we encounter overlap on 9. We want a low plex and a high core, so if we score threads with an overlapping vertex by increase in core minus increase in plex, 9 has thread1 score of (16-14)-(5-1)=-2 and thread2 score of (6-4)-(4-2)=0 so we put 9 in thread2. We can move thread 2 up one more level now and get: {1,2,3,4,8,14} {9,24,25,26,28,30,31,32,33,34} {5,6,7,11,17} 33 24 1core 11 0plex 23 43 1core 21 0plex 1 1234894 16cor 6565424 5plx 1111111122222222233 56790135678901235678901 23310200020000002111011 1 123484 14core 5555441plex 112 567175 233220 1 12344 10core 44444 0plex 12348 10core 44444 0plex 11 56717 6core 23322 4plex Now if we finish off by putting each remaining vertex with the core to which it is maximally connected (break ties with core size?), we get: {1,2,3,4,8,12,13,14,18,22} {9,10,15,16,19,21,23,24,25,26,27,28,29,30,31,32,33,34} {5,6,7,11,17} 11 6717 4cor 3212 2plx 1 5677 4cor 1232 2plx 1 5671 4cor 2222 2plx 1 677 3core 222 0plex

  23. Topdown Mining all kplexes and kcores on G10At each step, we branch to each of the lowest degree vertices. Here delete several count values at a time. 1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899a0a1a2a3a4a5a6a7a8a9 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 231 3 2 2 4 1 9 133 5 8 5 2 142 2 3 4 2 2 2 2 2 1 2 1 1 2 1 1 1 1 1 3 3 3 2 4 4 9 8 3 8 1 1 1 2 2 1 3 3 2 261 1 2 1 2 2 2 3 2 2 2 1 2 2 1 5 1 1 b0b1b2b3b4b5b6b7b8b9c0c1c2c3c4c5c6c7c8c9d0d1d2d3d4d5d6d7d8d9e0e1e2e3e4e5e6e7e8e9f0f1f2f3f4f5f6f7f8f9g0g1g2g3g4g5g6g7g8g9h0h1h2h3h4h5h6h7h8h9i0 If we treat B10 as undirected, these are unique listings of edges. Del 1,2 counts: 1 2 9 2 2 1 2 1 1 1 2 2 2 2 7 2 2 2 2 2 2 3 241 1 2 2 2 2 3 2 2 1 2 3 3 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 1 2 1 1 1 3 3 2 2 132 2 2 2 5 4 v Ct 29 1 38 7 40 3 43 3 45 7 46 12 47 3 48 3 49 7 50 3 52 3 55 3 56 2 72 3 73 3 74 3 76 4 77 1 78 6 79 3 80 2 81 5 88 3 89 1 91 9 99 3 a7 1 b2 6 c4 0 d1 3 d2 8 d9 2 e4 1 e5 1 h0 1 h1 2 h4 5 h9 5 i0 2 v Ct 38 5 40 3 43 3 45 7 46 12 47 3 48 3 49 7 50 2 52 2 55 3 72 3 73 3 74 3 76 3 78 6 79 3 81 4 88 3 91 7 99 2 b2 5 d1 3 d2 5 h4 2 h9 4 v Ct 38 5 40 3 43 3 45 7 46 9 47 3 48 3 49 7 55 1 72 3 73 3 74 3 76 3 78 6 79 3 81 3 88 2 91 6 b2 5 d1 3 d2 5 h9 4 Del Cts 1,2,3 v Ct 38 3 45 4 46 6 49 3 78 2 91 2 b2 1 d2 1 h9 2 12cor 24plx Del Cts 1,2 v Ct 38 3 45 3 46 3 49 3 6core 0plex Del and restart Del Cts 123 v Ct 50 1 52 1 56 1 77 0 78 2 79 2 81 4 91 3 b2 0 c4 0 d2 1 d9 2 h4 2 h9 4 i0 2 Del Cts 12 v Ct 78 2 79 2 81 4 91 3 d9 1 h4 2 h9 4 i0 2 10core 18plex DelCt 1 v Ct 81 1 91 2 h9 1 2core 1plex Not very effective but then, G10 is known to be without many cliques. b7 a3 b8 b6 b5 b9 a5 c3 a4 c2 c4 2 1 3 4 a7 5 a6 a8 a9 b4 6 b0 c1 c0 7 19 20 8 18 22 97 98 99 21 23 a1 90 24 9 35 a0 84 a2 25 b1 85 36 40 89 10 37 26 92 76 83 27 86 b3 28 93 38 91 11 82 75 70 94 42 29 87 41 30 12 69 80 88 39 b2 57 81 43 31 32 58 77 95 13 56 h9 34 59 68 c5 44 79 33 96 14 78 60 55 15 52 51 67 16 45 61 50 h8 17 46 i0 62 63 54 49 66 h7 65 53 64 48 h3 47 h5 h6 h4 h2 c7 c6 h0 71 h1 74 g9 72 g2 73 g3 g6 g8 c8 d4 g4 d3 g5 c9 g7 d2 g1 e6 d1 e7 d0 e5 g0 e2 e4 d7 d5 f9 f7 e3 f8 e1 f6 d6 e8 e0 d8 d9 e9 G10: Web graph of pages of a website and hyperlinks. Communities by color (Girvan Newman Algorithm). |V|=180 (1-i0) and |E|=266. Vertices with OutDeg=0 (leaves) do not have pTrees shown because pTrees display only OutEdges and thus those OD=1 have a pure0 pTree. f0 f1 f5 f2 f4 f3

  24. Topdown Mining all kplexes and kcores on G8At each step, delete lowest degree vertices as indicated. 1 5 4 2 D 1,2 v Ct 1 2 2 3 3 1 8 2 9 0 16 1 35 1 41 2 51 0 52 2 D 0-1 D 0-6 v Ct 6 6 7 7 9 5 12 10 13 6 14 7 16 5 17 5 19 5 20 5 23 7 24 7 25 8 27 7 29 4 31 7 39 3 40 4 42 3 43 5 44 5 51 4 54 11 3 At each step, delete 1 lowest degree vertex but we do several rounds of that before restarting alg. 41 46 42 8 45 47 Del 1 then 2 Then 2 again v Ct 13 1 14 3 16 2 19 3 20 2 21 3 Del 1 Then 1 again Then 1 sgain Then 1 again Then 1 again v Ct 4 2 8cor 5 2 7plx 13 4 46 3 47 2 48 3 7 Del 1 Then 1 again Then 1 sgain v Ct 29 1 8cor 33 2 7plx 51 1 D 29 33 51 Del 3 then 4 Then 5 Then 6 v Ct 7 2 12 3 23 4 24 6 25 5 27 4 31 5 54 5 Del 1 then 2 Then 3 v Ct 8 3 14 2 16 4 17 5 19 4 21 2 35 3 52 3 44 6 43 40 9 39 • 6 0 • 17 1 1cor • 19 1 0plx • 23 2 3cor • 25 2 0plx • 2 • D 17,19,23 25,31 38 53 48 12 52 10 G8 13 1 2 3cor 2 2 0plx 41 2 8 1 1cor 52 1 0plx D all 5 14 11 17 36 54 16 Del 2 13 1 14 3 16 2 19 3 20 2 21 3 Del 2 8 39cor 16 26plx 17 5 19 2 35 3 52 3 Del 0 1 2 2cor 2 2 1plx 3 1 D 1 2 3 All cts=0 done. 24 35 15 Del 3 23 4 13cor 24 5 2plx 25 5 27 4 31 5 54 3 Del 2 13 2 2cor 46 1 1plx 48 1 D 4 5 13 46 47 48 23 v Ct 5 1 9 2 11 1 14 1 16 2 22 2 30 1 38 1 43 1 46 1 48 1 51 2 D 1 22 37 21 49 D 345 6 4 7 5 12 6 13 5 14 4 23 5 24 6 25 5 27 4 31 5 54 5 19 Del 2 8 25cor 17 3 1plx 35 2 52 3 34 20 Del 3 23 4 10cor 24 4 0plx 25 4 27 4 31 4 D 23 24 25 27 31 27 18 Del 2 14 12cor 19 21plx 21 1 D 14 19 21 25 Overlapping communities in a network of word association. The groups, labeled by the colors, were detected with the Clique Percolation Method by Palla et al. 50 Del 2 17 1 0plx 52 1 1cor D 8 17 35 52 51 26 • D 1-6 • 6 0 • 39 2 • 40 2 3cor • 2 0plx • D 39,40,42 Del 1 then 2 Then 3 Then 4 Then 5 6 4 7 6 12 6 14 4 17 4 19 3 40 2 43 3 44 5 54 7 v Ct 22 11cor 51 10plx D 22 51 30 D 4 7 3 12 4 13 3 23 4 24 5 25 4 31 4 54 5 16core 12plex 29 28 33 31 32 D 0,1 5 1 9 2 11 1 16 2 35 1 38 1 43 1 46 1 53 2 D 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 6 1 2 0 0 0 1 0 1 1 1 0 1 1 0 1 1 0 1 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 7 1 3 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 9 1 4 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 7 1 5 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 1 1 0 2 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 1 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 6 2 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 2 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 2 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 1 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 6 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 5 3 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 1 1 4 3 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 5 3 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 4 3 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 6 3 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 5 3 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 3 8 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 4 3 9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 6 4 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 8 4 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 5 4 2 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 7 4 3 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 6 4 4 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 8 4 5 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 3 4 6 0 0 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 4 7 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 3 4 8 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 5 4 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 4 5 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 9 5 2 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 6 5 3 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 5 4 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 4 2 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 4 3 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 4 0 0 0 0 1 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 5 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 4 6 0 0 1 1 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 1 7 0 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 1 1 1 1 8 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 5 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 1 6 1 Scientist 2 Science 3 Astronomy 4 Earth 5 Space 6 Moon 7 Star 8 Ray 9 Intelligent 10 Golden 11 Glare 12 Sun 13 Sky 14 Moonlight 15 Eyes 16 Sunshine 17 Light 18 Lit 19 Dark 20 Brown 21 Tan 22 Orange 23 Blue 24 Yellow 25 Color 26 Gray 27 Black 28 Race 29 White 30 Green 31 Red 32 Crayon 33 Pink 34 Velvet 35 Flashlight 36 Glow 37 Dim 38 Gifted 39 Genius 40 Smart 41 Inventor 42 Einstein 43 Brilliant 44 Shine 45 Laser 46 Telescope 47 Horizon 48 Sunset 49 Ribbon 50 Violet 51 Purple 52 Beam 53 Night 54 Bright D 3,4 24 1 54 1 1core 0plex D 24,54 rest. Del 3 6 4 18cor 7 6 3plx 12 6 14 3 17 3 44 4 54 4 v Ct 9 2 3cor 38 1 0plx 43 1 5 1 1cor 46 1 0plx D all 5 D 1-9 6 2 7 3 12 3 17 3 19 1 23 2 25 2 31 2 Del 3 again 6 3 9cor 7 4 1plx 12 4 44 4 54 3 v Ct 11 1 1cor 15 1 0plx 30 1 1cor 32 1 0plx 35 1 1cor 37 1 0plx 47 1 1cor 48 1 0plx D these 8 leaves all 0 counts! Del 3s 1 at a time 6 3 6cor 7 3 0plx 12 3 44 3 D 12 v Ct 7 1 1cor C 1 0plx D 7,c Del 3s 1 at a time 7 3 6cor 12 3 0plx 44 3 54 3 Del 6 7 12 44 54 v Ct 14 2 16 1 2cor 53 1 1plx D 14,16,53 Del 1 then 2 Then 3 Then 4 Then 3 again Then 2 again 9 4 13cor 39 5 2plx 40 5 41 3 42 5 43 4 D 1-4 6 2 2cor 13 1 1plx 44 1 20 2 27 2 3cor 29 2 0plx 51 0 D all 6 Del 3 9 4 10cor 39 4 0plx 40 4 42 4 43 4 D 9 39 40 41 42 43

  25. Breadth-First Inductive Clique Search Alm: Let CLQK be the set of all Kcliques, 1st find CLQ3 using CS0 (Common Siblings=0) or using CCLQ3. Breadth-1st Clique Alg: Find CLQ3. Induction theorem: A Kclique and 3clique that share an edge form a (K+1)clique iff all K-2 edges from the non-shared Kclique vertices to the non-shared 3clique vertex exist. Next find CLQ4, then CLQ5, … G7 on G7 ( List Version): 1 2 3 18,20,22CLQ4 since 3:18,20,22E3 E 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 8 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 7 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 9 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 2 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 4 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 3 3 3 2 4 2 4 Note checkback. Is it required? (No: if 132 in 4CLQ it’d show up already). Already in CLQ4 1 1 1 2 2 2 3 4 3 4 since 34E3 1 2 3 8 1 2 3 14 22 20 18 14 1 8 12 4 3 2 33 23 19 21 34 17 30 15 27 16 31 10 32 26 29 28 24 25 11 6 5 7 9 13 1 3 4 8 1 2 4 8 1 2 4 14 UCLQ4 done. pTree version faster? 1 3 4 14 2 3 4 8 2 3 4 14 UCLQ3 Unique 3cliques as lists 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 8 14 18 20 22 4 8 9 14 1 1 1 1 1 1 1 2 2 2 4 4 4 5 5 6 6 3 3 3 8 13 14 7 11 7 11 4 8 14 2 2 3 3 3 6 9 9 4 4 4 4 9 7 31 31 8 14 8 14 33 17 33 34 24 24 24 25 27 29 28 30 30 26 30 32 34 33 34 32 34 34 UCLQ5 1 3 9 1 4 13 1 6 7 1 6 11 1 2 20 1 2 22 1 5 7 1 5 11 MUCLQs 1 2 18 3 9 33 1 2 3 4 8 1 2 3 4 14 1 2 3 4 8 1 2 3 4 14 9 31 33 6 7 17 9 31 34 24 28 34 24 30 33 24 30 34 25 26 32 27 30 34 29 32 34 16 9 10 6 3 4 4 4 5 2 3 1 2 5 2 2 2 2 2 3 2 2 2 5 3 3 2 4 3 4 4 6 11 16 CLQ3 (as pTrees) Remaining edges after CS0 (removal of PURE0 edge endpoint pair ANDs). 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 5 6 6 6 7 9 9 9 24242424 2525 26 2727 28 2929 3030 3131 32 2 3 4 5 6 7 8 9 111314182022 3 4 8 14182022 4 8 9 1433 8 1314 7 11 7 1117 17 313334 28303334 2632 32 3034 34 3234 3334 3334 34 7 5 5 2 2 2 3 1 2 1 3 1 1 1 4 4 3 3 1 1 1 4 3 2 3 1 3 1 3 1 1 2 1 1 1 2 2 1 1 2 1 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 1 1 0 0 0 1 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 0 1 0 0 0 1 1 0 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 1 0 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 7 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 1 1 1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 18 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 20 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 22 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 24 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 26 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 27 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 28 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 29 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 30 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 Is there a pTree Version of this Algorithm ? Is it faster?

  26. n-clique (n-clan, n-club) Community Search on G6 n-clique = subgraph s.t. pairs  ageodesic path of length n (all pairs are n-connected). Downward Closures: SubGraph, SGDC (Every subgraph of an n-clique is an n-clique.) Path Length, PLDC (If C is an n-clique then C is an (n-1)-clique.) Apriori, ADC (If C,D are n-cliques of size=k sharing k-1 vertices, then CD is (n+1)-clique) n-clan = n-clique with diameter n. n-club = subgraph of diameter =n. G6 8 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 1 4 0 0 0 0 0 0 0 0 4 0 0 1 0 0 0 0 0 4 0 0 1 0 0 0 0 0 4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 5 1 1 0 0 0 0 0 5 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 5 1 1 0 0 0 0 0 5 0 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 0 0 0 0 0 0 6 1 1 0 0 0 0 6 1 1 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 7 0 0 0 0 0 SP3 1 2 3 4 5 6 7 8 9 a b c SP4 1 2 3 4 5 6 7 8 9 a b c SP 1 2 3 4 5 6 7 8 9 a b c SP1 1 2 3 4 5 6 7 8 9 a b c 9 1 1 1 9 0 0 0 9 1 1 1 9 0 0 0 9 0 0 0 8 0 0 0 0 8 0 0 0 0 8 1 1 0 0 8 1 1 0 0 a 0 0 a 0 0 a 1 1 a 1 1 a 0 0 b 0 b 0 b 1 b 0 b 1 1-clique=clique We already have many clique search algorithms, but the pattern we are going to use for n=2… may give us another one?: 1 is 1conn to 234. 123 1clique? (23 edge?). Yes! 124 1clique? (24 edge?) Yes! 134 1clique? (34 edge?) No! 1234 not a clique by SGDC. SP2 1 2 3 4 5 6 7 8 9 a b c 2 0 0 0 0 1 0 0 0 0 1 3 1 0 0 0 0 1 1 1 0 4 1 1 0 0 0 0 0 0 5 0 0 1 0 0 0 0 7 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 6 0 0 1 1 0 0 8 0 0 1 1 2 is 1conn to 34. 234 1clique? No! 3 is 1conn to just c. 4 is 1conn to just 7. 5 is 1conn to 67. 567 1clique? (67 edge?) Yes! 6 is 1conn to 78. 578 1clique? (78 edge?) No! 8 is 1conn to 9a. 89a 1clique? (9a edge?) Yes! 9 is 1conn to abc. 9ab 1clique? Yes! 9ac 1clique? Yes! 9bc 1clique? Yes! So 9abc is 1clique by ASGDC and abc is 1clique by SGDC. 1 is 2conn to 7c. 17c 2clique? 7c turned on in SP1 or SP2? No! 2 is 2conn to 7c. 27c 2clique? 7c turned on in SP1 or SP2? No! 2 0 0 1 1 0 0 1 1 1 0 3 0 0 0 1 1 0 0 0 0 3 0 0 0 1 0 0 0 0 0 4 0 0 0 1 0 0 0 1 5 0 0 0 1 1 0 0 6 0 0 0 0 1 1 7 0 1 1 0 0 1 0 0 0 1 1 0 0 1 1 1 0 3 2con to 49abc. 349 2clique? 49 on in SP1|2? N! 34a? 4a on SP1|2? N! 34b? 4b on SP1|2? N! 34c? 4c on SP1|2? N! 39a 2clique?Y! (9a on in SP1). 39b 2clique? Y! (9b on SP1). 39c 2clique? Y! (9c on SP1). 3ab 2clique? Y! (ab on in SP1). 3ac 2clique? Y! (ac on in SP1). 3bc 2clique? Y! (bc on in SP1). 39ab, 39ac, 3abc, are 2cliques by ASGDC. Research questions: • Did I miss anything by using the lower triangular matricies? • What about n-clans and n-clubs? • Would it be better to start with SP4, then SP3, then SP2, then SP1 by virtue of PLDC or some other closure or property? • Keeping in mind that it is a big task to creatSPk’s for large graphs, is there a better way? 4 2con to 56. 456 2clique?Y! (56 on in SP1). 5 2con to 8 only. 6 2con to 9a. 69a 2clique?Y! (9a on in SP1). 7 2con to 8 only. 1 0 0 0 0 0 0 1 0 0 0 0 2 0 0 0 0 0 1 0 0 0 0 3 0 1 1 0 0 0 0 0 0 4 0 0 0 0 1 1 1 0 5 0 0 0 0 0 1 1 7 0 0 0 1 1 5 0 0 0 0 0 0 0 7 0 0 0 0 0 8 2con to bc. 8bc 2clique?Y! (bc on in SP1). 9 a b are not 2con to anybody new. 1 5 4 2 6 7 3 c Upward Closures: Apriori, ADC (If C,D are n-cliques of size=k sharing k-1 vertices, then CD is (n+1)-clique) All SubGraphs, ASGDC (C a subgraph size=k and all subgraphs of C of size (k-1) are n-cliques then C is a n-clique?? n-clan = n-clique with diameter n. n-club = subgraph of diameter =n. 9 b 8 a 1 1 1 1 3 3 2 4 3 3 3 2 1 1 1 1 3 3 2 0 3 3 3 2 2 1 1 3 3 2 4 3 3 3 2 2 1 1 3 3 2 0 3 3 3 2 3 2 0 0 0 0 2 2 2 1 1 1 1 1 0 0 2 0 0 0 0 2 2 1 1 0 0 2 0 0 0 0 2 3 2 0 0 3 3 2 2 2 1 3 2 0 0 3 0 2 2 2 1 4 2 2 1 0 0 0 0 0 5 1 1 2 0 0 0 0 6 1 1 2 2 0 0 7 2 0 0 0 0 3 2 4 4 3 3 2 2 2 1 5 1 1 2 3 3 4 4 4 2 2 1 3 0 0 0 3 5 1 1 2 3 3 0 0 6 1 1 2 2 3 3 7 2 3 3 0 0 7 2 3 3 4 4 8 1 1 2 2 4 2 2 1 3 4 4 4 3 1 is 4conn to 8 so 18 is a 4-clique. 2 is 4conn to 8 so 28 is a 4-clique. 3 is 4conn to 56, 5 is 1conn to 6 so 356 is a 4-clique. 4 is 4conn to 9ab, 9a 9b ab are edges so 49ab is a 4-clique. 5 is 4conn to bc, bc is an edge so 5bc is a 4-clique. 7 is 4conn to bc bc, bc is an edge so 7bc is a 4-clique. 1 is 3conn to 569ab checking it out, 169ab is a 3-clique.

  27. G6 E 1 2 3 4 5 6 7 8 9 a b c d e f g h I j k l m n o p q r s t u v w x y E 1 2 3 4 5 6 7 8 9 a b c d e f g h I j k l m n o p q r s t u v w x y E 1 2 3 4 5 6 7 8 9 a b c d e f g h I j k l m n o p q r s t u v w x y E 1 2 3 4 5 6 7 8 9 a b c d e f g h I j k l m n o p q r s t u v w x y 1 c 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 g 1 c 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 b 1 c 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 6 1 c 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 7 e 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 e 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 e 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 e 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 l 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 m 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 1 0 1 0 0 1 1 5 p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 3 p 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 3 q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 3 q 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 3 r 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 2 s 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 3 s 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 4 s 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 4 t 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 3 t 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 3 u 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 4 u 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 3 v 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 v 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 v 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 v 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 w 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 3 w 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 6 w 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 6 x 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 5 x 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 3 x 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 6 x 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 b y 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 6 y 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 3 y 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 9 y 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 g 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 5 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 5 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 5 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 5 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 6 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 9 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 6 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 7 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 9 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 7 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 8 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 a 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 i 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 G6 has 4 max cliques: 123 124 567 9abc. 8 is the only point not in an MCLQ and 8 is maximally connected to 9abc. So the G6 Clique Aura Partition of: 123 124 567 89abc E 1 2 3 4 5 6 7 8 9 a b c 1 0 1 1 1 0 0 0 0 0 0 0 0 3 2 1 0 1 1 0 0 0 0 0 0 0 0 3 3 1 1 0 0 0 0 0 0 0 0 0 1 3 4 1 1 0 0 0 0 1 0 0 0 0 0 3 5 0 0 0 0 0 1 1 0 0 0 0 0 2 6 0 0 0 0 1 0 1 1 0 0 0 0 3 7 0 0 0 1 1 1 0 0 0 0 0 0 3 b 0 0 0 0 0 0 0 0 1 1 0 1 3 9 0 0 0 0 0 0 0 1 0 1 1 1 4 c 0 0 1 0 0 0 0 0 1 1 1 0 4 8 0 0 0 0 0 1 0 0 1 1 0 0 3 a 0 0 0 0 0 0 0 1 1 0 1 1 4 G7 1 5 4 2 6 7 3 c 9 b 8 a 11111111 12345678901234567 123456789abcdefgh 11222222222233333 89012345678901234 ijklmnopqrstuvwxy

  28. K-Degree-Difference Community Search: H SG s.t. ddHIntDegH-ExtDegHk. Thm: If hH, ddH-h = ddH – (2idh - edh). So want to remove h s.t. (2idh – edh) is min. G6 H=G={123456789abc} ddH=38 id= 333323334434 ddH/|VH|=38/12=3.16 ed= 000000000000 Remove 5 Very Simple Weighted SP1 and SP2 K-plex Search Weighting: 0,1path nbrs of x times 3; 2path nbrs of x times 2; Until all degrees are weighted, then back to actual subgraph degrees H={123456789abcH=3 H=3 0plex deg222623338861 x=3 after cut 1 (actual SG degrees) H= {12346789abc} ddH=34 id= 33333334434 ddH/|VH|=34/11=3.09 ed= 00001100000 2id-ed=66665568868 Remove 6,7 UNWEIGHTED Degrees H={123456789abc deg333323334434 E 1 2 3 4 5 6 7 8 9 a b c 1 0 1 1 1 0 0 0 0 0 0 0 0 3 2 1 0 1 1 0 0 0 0 0 0 0 0 3 3 1 1 0 0 0 0 0 0 0 0 0 1 3 4 1 1 0 0 0 0 1 0 0 0 0 0 3 5 0 0 0 0 0 1 1 0 0 0 0 0 2 6 0 0 0 0 1 0 1 1 0 0 0 0 3 7 0 0 0 1 1 1 0 0 0 0 0 0 3 b 0 0 0 0 0 0 0 0 1 1 0 1 3 9 0 0 0 0 0 0 0 1 0 1 1 1 4 c 0 0 1 0 0 0 0 0 1 1 1 0 4 8 0 0 0 0 0 1 0 0 1 1 0 0 3 a 0 0 0 0 0 0 0 1 1 0 1 1 4 H= {123489abc} ddH=26 id= 333224434 ddH/|VH|=26/9=2.88 ed= 000110000 2id-ed=666338868 Remove 4,8 H={123456789abc deg999923634438 x=1 H={1234578c H=15 H=7 kplex k8 deg99992638 x=1 after cutting 234 H={12345H=6 H=5 kplex k1 deg99992x=1, after cut 23468 H={123456789abcH=3 H=3 0plex deg333123314434x=5 after cut 1 from SG degs H= {1239abc} ddH=16 id= 2233334 ddH/|VH|=16/7=2.28 ed= 1101100 2id-ed=3365568 Remove 1,2 H={1234578c H=15 H=7 kplex k8 deg99992638 x=2 after cutting 234 H={12345H=6 H=5 kplex k1 deg99992x=2, after cut 23468 H={123456789abc deg999923634438 x=2 H={123456789abcH=3 H=2 1plex deg33312333223 x=6 after cut 12 SG degs211 H={123456789abc deg99962333886c x=3 H={123cH=6 H=4 2plex deg99962x=3, after cut 2368 H= {39abc} ddH=10 id= 13334 ddH/|VH|=10/5= 2 ed= 21100 2id-ed=05568 Remove 3 H={123456789abcH=3 H=3 0plex deg333122232234x=7 after cut 1 SG degs H={123456789abc deg996946334434 x=4 H={123456789abcH=3 H=3 0plex deg996946334434 x=4 after cut 2346 H= {9abc} ddH=9 id= 3333 ddH/|VH|=9/4=2.25 ed= 1101 2id-ed=5565 CLQ. Start over w 12345678 H={123456789abcH=10 H=5 5plex deg333669964434 x=5 after cut 34 H={123456789abc deg333669964434 x=5 H={123456789abc 2plex deg333342134433 x=8 after cut12 SG degs H= {12345678} ddH=17 id= 33232331 ddH/|VH|=17/8=2.13 ed= 00100002 2id-ed=66563660 Remove 8 H={123456789abc deg333669998834 x=6 H={123456789abc deg333669998834 x=6 after cut 34 H= {1234567} ddH=17 id= 3323223 ddH/|VH|=16/7=2.28 ed= 0010010 2id-ed=6636436 Remove 3,6 H={123456789abc deg333969998834x=7 after cut 34 H={123456789abc deg333969934434 x=7 H= {12457} ddH=6 id= 22312 ddH/|VH|=6/5=1.2 ed= 11011 2id-ed=33613 Remove 5 H={123456789abc deg33334969cc68 x=8 H={123456789abc deg33334969cc68 x=8 after cut 34 H= {1247} ddH=4 id= 2231 ddH/|VH|=4/4=1 ed= 1102 2id-ed=3360 Remove 7 H={123456789abcH=10 H=8 H a kplex k 2 deg33632639cc9c x=9 after Cutting 2,3,6 H={123456789abc deg33632639cc9c x=9 1 5 4 2 6 7 3 H= {124} ddH=3 id= 222 ddH/|VH|=3/3=1 ed= 111 2id-ed=333 CLQ. Start over w 35678 c H={123456789abcH=10 H=8 H a kplex k 2 deg33632639cc9c x=a after cut 2,3,6 H={123456789abc deg33632639cc9c x=a 9 b 8 a H={123456789abcH=6 H=6 H a kplex k 0 deg33632639cc9c x=b after cut 2,3,6 H={123456789abc deg33632336cc9c x=b H={35678} id=02321 ed=30012 ddH=2 2id-ed=-34630 ddH/|VH|=2/5=.4 Remove 3 H={123456789abcH=6 H=6 H a kplex k 0 deg66932336cc9c x=c after cut 2,3,6 H={123456789abc deg66932336ccpc x=c H={5678} id=2321 ed=0012 ddH=5 2id-ed= 4630 ddH/|VH|=5/4=1.2 Remove 8 By weighting the initial round we have gotten nearly perfect information for this example (G6). Weightings, 3 and 2, were arbitrarily chosen but worked here. In general, one should devise a formula to determine them. Also we could weight SP3 and etc. as well? If we have paid the price of constructing SPk k>1, this is a much simpler way to do it, as compared to the Clique Percolation method of Palla. H= {567} id=222 ed=011 ddH=4 2id-ed= 433 ddH/|VH|=4/3=1.33 Clique. remove 567. Start over w 38 (but it has 0 id)

  29. Very Simple Weighted SP1 k-plex Search on G7 Weighting: 0,1path nbrs of x times 1; 2path nbrs of x times 0; 1 4 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 5 1 8 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 1 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 1 7 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 3 2 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 2 2 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 1 5 2 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 3 2 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 3 2 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 2 2 8 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 4 2 9 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 3 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 4 3 1 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 4 3 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 1 6 3 3 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 3 4 0 0 0 0 0 0 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 1 1 1 1 1 1 0 0 1 6 SP1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 6 2 1 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 9 3 1 1 0 1 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 4 1 1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 5 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 6 1 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 7 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 9 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 5 8 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 1 2 3 H=1234567890123456789012345678901234 H=561 H=77 kplx k484 D g9a63444523125222223222533243446bg kcore k77 Cut 123: 1 2 3 H=1234567890123456789012345678901234 H=120 H=38 kplx k82 D 9685322452322522222322243323334367 kcore k38 Cut 23: 1 2 3 H=1234567890123456789012345678901234 H=55 H=26 kplx k24 D 6675322452322522222322223323334344 kcore k26 Cut 24: 1 2 3 H=1234567890123456789012345678901234H=15 H=12 kplx k3 D 5454322422322422222322223323334344kcore k12 G7 Cut 2: 1 2 3 H=1234567890123456789012345678901234H=10 H=10 kplx k0 D 4444322422322422222322223323334344kcore k10 {1,2,3,4, 14} is a clique. {1,2,3,4,9,14} is a 3plex. 2 3 4 4 4 5 5 6 6 6 6 7 8 9 10 11 12 13 15 5 5 1 4 8 2 6 1 3 6 8 0 5 7 9 1 3 5 8 0 2 4 9 2 5 7 1 4 8 2 8 9 5 Cut0: 1 2 3 H=5678901235678901235678901 H=21 H=4 kplx k17 D 2330102000020000002111011 kcore k4 Cut 1 leaves 25 only. 1 2 3 H=56789012356789012345678901234 D 232031200222021202533232435af 1 2 3 H=89023568901235678901 H=19 H=4 kplex k15 D 01000000000002010011 kcore k4 Cut012:1 2 3 H=56789012356789012345678901234 H=55 H=19 kplx k36 D 20203120022202120253323233456 kcore k19 Cut03: 1 2 3 H=56789012356789012345678901234 H=6 H=4 kplx k2 D 20203120022202120223323233222 kcore k6 {24,32,33,34} is a 2plex Cut0: 2 3 H=89023568901235678901 H=19 H=4 kplex k15 D 01000000000002010001 kcore k4 Cut 0 leaves {9,31} as a 0plex 1 2 3 H=5678901235678901235678901 D 2330102000020000002111011 1 2 3 H=89023568901235678901H=17 H=2 kplex k15 D 01000000000002010011 kcore k2 Cut 0 leaves {27,30} as a 0plex Cut01: 1 2 3 H=5678901235678901235678901 H=15 H=6 kplx k9 D 2330102000020000000111011kcore k6 1 2 3 H=89023568901235678901H=14 H=0 kplex k14 D 0100000000000201001kcore k0 no edges left Cut0: 1 2 3 H=5678901235678901235678901 H=10 H=6 kplx k4 D 2330102000020000000111011kcore k6 {5,6,7,11,17} is a 4plex 1 2 3 H=89023568901235678901 D 01000000000002111011 The expected communities are mostly not detected as kplexes or kcores. Cut0: 1 2 3 H=5678901235678901235678901 H=21 H=4 kplx k17 D 2330102000020000002111011 kcore k4 1 2 3 4 5 6 01234567890123456789012345678901234567890123456789012345678901234 0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ@#$ (Symbols for base 65 )

  30. Simple Weighted SP1, SP2 K-plex Search on G8 1 5 4 2 Weighting 444 0,1path neighbors (12012) times 5 334 2 path nbrs (39893) times 3 3 41 46 42 8 45 47 7 44 6 43 40 9 39 38 53 48 12 52 10 13 14 11 17 36 This gives C0={1,2,9,39,40,41,42,43} which is exactly the Intelligence Class except that v=38 (gifted) is missing. It is a kplex k8 (not that strong of a community!) 54 16 24 35 15 23 22 37 21 49 19 34 Within the Intelligence Class this is the 1plex, C1={1, 2,40,41,42} ( only edge missing is (2,40) ) with C1-degrees: 4 3 3 4 4 Thus if we cut next using C1-degrees (cut 2,40) leaves the clique (0plex) C2={1,41,42} Cutting C0 and starting over: 20 27 25 18 50 11 1 1 1 1 1 1 44444105645697461218645954938634545423675767353534965 12345678901234567890123456789012345678901234567890123 51 26 121663 23954136 3231353 2 212 1114 113 131 44242600640642266634620404734864545223675782958094865 cut<30 G8 30 29 28 Weighting 0,1path neighbors (367) times 5 1111445 2 path nbrs (452347483) times 3 33 31 32 This gives C2={3,4,5,6,7, 12,13,14,15,17,23,25,31,44, 48, 53} Whereas, Astronomy is 3,4,5,6,7,8,10,11,12,13,14,16,17, 44,45,46,47,48,52,53 so, not a good fit! Weighting 0,1 SP nbrs times 6 With replacement but using as starting vertex, the remaining vertex of highest degree (first, v=12). Weighting 0,1 SP nbrs times 5 2 SP nbrs times 3 221 11 1 1 1 1 1 1 1 13231 00244105845697461218645954938634545429855587353534965 next cut<18 11 1 1 1 1 1 1 44444105645697461218645954938634545423675767353534965 11 1 1 1 1 1 1 44444105645697461218645954938634545423675767353534965 12155222143135323144122131113112231 44202505605655205634025554734894545823675785955594705 cut<20 221 11 1 1 1 1 1 1 1 13231 00244105845697461218645954938634545429855587353534965 instead cut<19 221 11 1 1 1 1 1 1 1 13231 00244105845697461218645954938634545429855587353534965 11 1 2 1 1439512514372325 44444105645697461218640454488684045423675767353534465 12155222143135323144122131113112231 44202505605655205634025554734894545823675785955594705 cut<20 21155 14223 1 1 1 1 111 44522505645887163218645954938634545421675768353534965 next cut<10 21155 142231 1 1 1 111 44522505645887163218645954938634545421675768353534965 next cut<12 11 1 1 1 1 1 1 44544105645697461218645954938634545421675766353534965 G-C0 degs 12345678901234567890123456789012345678901234567890123 x=1 12345678901234567890123456789012345678901234567890123 x=1 12345678901234567890123456789012345678901234567890123 x=1 12345678901234567890123456789012345678901234567890123 x=1 12345678901234567890123456789012345678901234567890123 12345678901234567890123456789012345678901234567890123 x=25 12345678901234567890123456789012345678901234567890123 x=12 12345678901234567890123456789012345678901234567890123 x=3 12345678901234567890123456789012345678901234567890123 x=3 12345678901234567890123456789012345678901234567890123 x=3 12345678901234567890123456789012345678901234567890123 x=12 11111 11 44444 55 Astronomy is 345678 01234 67 45678 23 Weighting 0,1 SP nbrs times 6 2 SP nbrs times 3 Astronomy is 345678 01234 67 45678 23 1234567890123456789012345678901234567890123456789012 5 astronomy vertices missing (3,5,45,46,53} and 2 non-astronomy included {21,24} Weighting 0,1 SP nbrs times 6 2 SP nbrs times 1 Colors is 5 012345678901234 901 4 colors missing but zero non-colors included. 44444ba5645g9746b2b864f9f49386d4545423675767353534965

  31. 3 3 6 4 8 8 a e c 5 4 6 3 3 AN pTree Ct H a k-plex and F is a ISG, F is a kplex G=(V,E) a k-plex iff |V|(|V|-1)/2 – |E| k K-plex search AN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 10 11 12 13 14 TS Women123456789abcdefghi 18*17/2=153 degs=hfhfbffghhgghhhgcc |Edges| =139 kplex k14 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 0 0 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 k-plex=SG missing k edges. Women123456789abcdefgh degs=gfgfbfffggffgggfc |14 Women123456789abcdefg1 degs=ffffbffeffeefffe |Edg 1 0 0 1 1 1 1 0 1 0 0 1 1 0 Dow? Women12346789abcdefg 15*14/2=105 degs=eeeeeeeeeeeeeee |Edges| =105 15kplex k0 15Clique So take out {12346789abcdefg} and start over. ANalystTickerSymbolRelationship with labels Women5hi 3*2/2=3 degs=011 |Edges| =1 kplex k2 Womenhi 2*1/2=1 degs=11 |Edges| =1 kplex k0 Clique H B B SB B SS S S H H B B B SB Buy-Hold-Sell No info from kplex search alg to WSP2. Avoid the work? Notice the very high 1-density of the pTrees? (only 28 zeros)? 1 3 1 3 3 1 2 3 1 1 2 2 3 3 TS SA TS pTree Ct 8 7 8 7 4 4 4 3 4 4 4 6 7 8 5 2 2 2 AN SA 3 3 2 1 3 2 3 1 3 2 2 1 1 1 3 1 2 3 Sal 7 5 7 6 2 1 6 4 5 7 4 5 6 3 4 1 6 1 C 2 3 0 1 0 1 0 0 0 0 0 0 2 0 0 0 0 3 Events123456789abcde 14*13/2=91 degs=88888dddd88888 |Edge|=66 kplex k25 Events23456789abcde Not calculating k degs=7777cccc88888 til it gets lower 1 0 0 1 1 1 1 0 1 0 0 1 1 0 Dow? Events456789abcde degs=55aaaa88888 Events3456789abcde degs=666bbbb88888 0 0 0 0 1 1 1 1 1 0 0 0 0 0 C3 0 0 1 1 0 0 0 1 1 1 1 1 0 0 C2 1 1 1 0 0 0 1 1 0 0 0 1 1 1 C1 1 1 0 0 0 0 0 0 0 1 0 0 1 1 C0 Events6789abcde 9*8/2=36 A 9Clique! degs=888888888 |Edges|=36 kplex k0 Events56789abcde1 degs=499998888825 0 0 0 0 0 1 0 0 0 0 0 0 0 0 SS 0 0 0 0 0 0 1 1 0 0 0 0 0 0 S 1 0 0 0 0 0 0 0 1 1 0 0 0 0 H So take out {6789abcde} and start over. 0 1 1 0 1 0 0 0 0 0 1 1 1 0 B 0 0 0 1 0 0 0 0 0 0 0 0 0 1 SB 0 1 0 1 1 0 1 1 0 0 1 1 1 1 SA1 1 1 1 1 1 1 0 0 1 1 0 0 0 1 SA0 Events12345 5*4/2=10 |Edges|=10 kplex k 0 A 5clique! degs: 44444 S A 1 1 1 0 1 1 0 1 1 1 0 0 1 1 1 1 1 0 1 S A 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 C 3 1 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 C 2 0 1 0 1 1 1 1 0 1 1 1 1 1 0 1 0 0 0 C 1 0 1 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 1 C 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 S 2 1 1 1 1 0 0 1 1 1 1 1 1 1 0 1 0 1 0 S 1 1 0 1 1 1 0 1 0 0 1 0 0 1 1 0 0 1 0 F 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 AN 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 S 0 1 1 1 0 0 1 0 0 1 1 0 1 0 0 0 1 0 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 TS Community structure in multipartite networks. This bipartite graph refers to the Southern Women Event Participation data set. Women are represented as open symbols with black labels, events as filled symbols with white labels. The illustrated vertex partition has been obtained by maximizing a modified version of the modularity by NewmanGirvan, tailored on bipartite graphs (Barber07). Note that most bipartite graphs involve a subject part and an object part such as women-events, Investors-stocks, Subjects-Objects in a access restriction system. 10 01 00 10 01 01 01 00 00 00 00 00 00 00 00 00 00 00 11 11 10 01 00 00 00 00 00 00 00 00 00 00 00 00 00 00 10 11 11 10 10 11 00 00 00 00 00 00 00 00 00 00 00 00 11 01 11 11 11 01 00 00 00 00 00 00 00 00 00 00 00 00 10 10 10 10 10 10 10 00 10 00 00 00 00 00 00 00 00 00 01 01 01 01 00 01 01 01 00 00 00 00 00 10 00 00 00 01 01 11 11 11 10 01 10 01 10 10 00 00 01 01 01 01 01 01 10 10 10 01 00 10 01 10 01 10 10 10 01 00 10 10 00 00 10 01 11 01 00 00 00 10 10 10 01 10 01 10 00 01 11 11 10 10 01 10 10 00 00 00 01 10 11 11 11 11 11 01 10 01 00 00 00 00 00 00 00 00 00 00 00 00 00 10 01 00 10 01 00 00 00 00 00 01 01 01 10 11 11 11 11 11 11 01 01 01 00 00 00 00 00 00 00 00 00 00 00 10 10 10 00 00 00 00 00 00 00 00 00 01 01 10 01 10 01 11 11 11 01 01 01 01 G9 If we had used the full algorithm which pursues each minimum degree tie path, one of them would start by eliminating 14 instead of 1. That will result in the 9Clique 123456789 and the 5Clique abcde. All the other 8 ties would result in one of these two situations. How can we know that ahead of time and avoid all those unproductive minimum degree tie paths? Every ISG of a Clique is a Clique so 6789 and 789 are Cliques (which seems to be the authors intent?) If the goal is to find all maximal Cliques, how do we know that CA=123456789 is maximal? If it weren’t then there would be at least one of abcde which when added to CA=123456789 would results in a 10Clique. Checking a: PCA&Pa would have to have count=9 (It doesn’t! It has count=5) and PCA(a) would have to be 1 (It isn’t. It’s 0). The same is true for bcde. The same type of analysis shows 6789abcde is maximal. I think one can prove that any Clique obtained by our algorithm would be maximal (without the above expensive check), since we start with the whole vertex set and throw out one at a time until we get a clique, so it has to be maximal? The Women associated strongly with the blue EventClique, abgde are {12 13 14 15 16} and associated but loosely are {10 11 17 18} associated strongly with the green EventClique, 12345 are {1 2 3 4 5} and associated but loosely are {6 7 9}

  32. Most bipartite graphs involve a subject part and an object part, e.g., women-events, Investors-Stocks, Customers-Items, Subject-Accessibility in system access . Most tripartite hypergraphs involve a subject, object and circumstance part, eg, Investor-Stock-Day. Most quadrapartite hypergraphs involve a subject, object and 2 circumstance parts, e.g., Customer-Item-Store-Day. Most multi-partite hypergraphs can be realized as a subject-object bipartite graph with circumstances as edge labels. A biclique is a complete subgraph of a bipartite graph (having every edge it is allowed to have and at least 1 (rules out the trivial case of no edges)). Every induced subgraph of a biclique is a biclique and thus the downward closure: If two Kbicliques overlap in K-2 verticies and if the other 2 vertices form an edge, then the union of the two k-biciques is a (k+1)biclique. B B A A B So we could start with two 3-bicliques sharing 2 points, x,y and not sharing two points, u,v (unless u and v are from different parts, there is nothing to check), except that the downward closure gives us a good way to get the set of 3bicliques, 3bCLQ. 1 1 1 2 2 3BCLQs 1AB 1AC 1AD 1AE 1AF 1AH 1AI 1BC 1BD 1BE 1BF 1BH 1BI 1CD 1CE 1CF 1CH 1CI 1DE 1DF 1DH 1DI 1EF 1EH 1EI 1FH 1FI 1HI 2AB 2AC 2AE 2AF 2AG 2AH 2BC 2BE 2BF 2BG 2BH 2CE 2CF 2CG 2CH 2EF 2EG 2EH 2FG 2FH 2GH 3BC 3BD 3BE 3FB 3BG 3BH 3BI 3CD 3CE 3CF 3CG 3CH 3CI 3DE 3DF 3DG 3DH 3DI 3EF 3EG 3EH 3EI 3FG 3FH 3FI 3GH 3GI 3HI 4AC 4AD 4AE 4AF 4AG 4AH 4CD 4CE 4CF 4CG 4CH 4DE 4DF 4DG 4DH 4EF 4EG 4EH 4FG 4FH 4GH Note, 2 2-bicliques (edges) that share a point form a 3biclique 6CE 6CF 6CH 6EF 6EH 6FH 8FH 8FI 8HI 5CD 5CE 5CG 5DE 5DG 5EG 7DE 7DF 7DG 7EF 7EG 7FG 9EG 9EH 9EI 9GH 9GI 9HI bHI bHJ bHL bIJ bIL bJL aGH aGI aGL aHI aHL aIL cHI cHJ cHL cHM cHN cIJ cIL cIM cIN cJL cJM cJN cLM cLN cMN dGH dGI dGJ dGL dGM dGN dHI dHJ dHL dHM dHN dIJ dIL dIM dIN dJL dJM dJN dLM dLN dMN eFG eFH eFI eFJ eFL eFM eFN eGH eGI eGJ eGL eGM eGN eHI eHJ eHL eHM eHN eIJ eIL eIM eIN eJL eJM eJN eLM eLN eMN hIK gHI iIK fGH fGJ fFK fFL fHJ fHK fHL fJK fJL fKL The union of each of these 17 sets (e.g., 1AB..1HI) is a 1HUB-kSPOKE (k+1)biclique The  of each pair is a 2HUB-kSPOKE (k+2)biclique (nothing to check). 12AB 12AC 12AE 12AF 12AH 12BC 12BE 12BF 12BH 12CE 12CF 12CH 12EF 12EH 12FH 13BC 13BD 13BE 13BF 13BH 13BI 13CD 13CE 13CF 13CH 13CI 13DE 13DF 13DH 13DI 13EF 13EH 13EI 13FH 13FI 13HI 38FH 38FI 38HI 15CD 15CE 15DE 16CE 16CF 16CH 16EF 16EH 16FH 14AC 14AD 14AE 14AF 14AH 14CD 14CE 14CF 14CH 14DE 14DF 14DH 14EF 14EH 14FH I 1dHI 1gHI 1aHI 1bHI 1cHI 29EG 29EH 29GH 2eFG 2eFH 2eGH 2aGH 2dGH H 17DE 17DF 17EF 18FH 18FI 18HI 19EH 19EI 19HI 1eFH 1eFI 1eHI 16 g 27EF 27EG 27FG 26CE 26CF 26CH 26EF 26EH 26FH 23BC 23BE 23BF 23BG 23BH 23CE 23CF 23CG 23CH 23EF 23EG 23EH 23FG 23FH 23GH 28FH 2fGH 35CD 35CE 35CG 35DE 35DG 35EG 24AC 24AE 24AF 24AG 24AH 24CE 24CF 24CG 24CH 24EF 24EG 24EH 24FG 24FH 24GH 25CE 25CG 25EG 37DE 37DF 37DG 37EF 37EG 37FG 36CE 36CF 36CH 36EF 36EH 36FH 34CD 34CE 34CF 34CG 34CH 34DE 34DF 34DG 34DH 34EF 34EG 34EH 34FG 34FH 34GH 3gHI 4eFG 4eFH 4eGH 39EG 39EH 39EI 39GH 39GI 39HI 3aGH 3aGI 3aHI 3bHI 3cHI 3dGH 3dGI 3dHI 3eFG 3eFH 3eFI 3eGH 3eGI 3eHI 3fGH 4aGH 45CD 45CE 45CG 45DE 45DG 45EG 46CE 46CF 46CH 46EF 46EH 46FH 39EG 39EH 39GH 47DE 47DF 47DG 47EF 47EG 47FG 48FH 4dGH 4fGH 8 59EG 69EH 56CE 57DE 57DG 57EG 67EF 68FH 6eFH 79EG 7eFG 89HI 8aHI 9aGI 9aHI 9bHI 8bHI 8cHI 8dHI 8gHI 9cHI 8eFH 8eFI 8eHI 9dGH 9dGI 9dHI 9fGH 9gHI abHI abIL acHI acHL acIL adGH adGI adGL adHI adIL afGH 9eGH 9eGI 9eHI aeGH aeGI aeGL aeHI aeHL aeIL dgHI bcHI bcHJ bcHL bcIJ bcIL bcJL bdHI bdHJ bdHL bdIJ bdIL bdJL beHI beHJ beHL beIJ beIL beJL bfHJ bfJL bgHI cdHI cdHJ cdHL cdHM cdHN cdIJ cdIL cdIM cdIN cdJL cdJM cdJN cdLM cdLN cdMN cgHI cfHJ cfHL cfJK cfJL dfGH dfGJ dfHJ dfHL dfJL ceHI ceHJ ceHL ceHM ceHN ceIJ ceIL ceIM ceIN ceJL ceJM ceJN ceLM ceLN ceMN efGH efGJ efFL efHJ efHL efJL egHI deGH deGI deGJ deGL deGM deGN deHI deHJ deHL deHM deHN deIJ deIL deIM deIN deJL deJM deJN deLM deLN deMN 11 b We need to start with a smaller bipartite graph to get a feel for efficiencies and shortcuts. We will come back to G9 after that. 18 i 9 1 6 10 a 17 h F G9 E 12 c 7 C 3 1 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 a b c d e f g h i A 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 B 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 C 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 6 D 1 0 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 5 E 1 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 8 F 1 1 1 1 0 1 1 1 0 0 0 0 0 1 0 0 0 0 8 G 0 1 1 1 1 0 1 0 1 1 0 0 1 1 1 0 0 0 10 H 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 1 0 0 13 I 1 0 1 0 0 0 0 1 1 1 1 1 1 1 0 1 1 1 12 J 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 5 K 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 4 L 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 6 M 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 N 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 3 8 7 8 7 4 4 4 3 4 4 4 6 7 8 5 2 2 2 A L M A B C D E F G H I J K L M N 2 D K 15 f J B 4 13 d N G 5 G9: Bipartite graph of the Southern Women Event Participation. Women are numbers (18), events are letters (14) (89 edges) Or Investors are numbers, stocks are letters in a recommends graph 14 e

  33. Clique Mining Thm (CLQm) Use AN ARM-like downward closure alg:bCCLQk =  pairs from bCLQk-1 that share k-2 vertices. Each such bCLQk iff the non-shared points E (THROW OUT  IF NON-SHARED PAIR COULD FORM AN EDGE BUT DOESN’T) Each such bCLQk iff the non-shared points are in the same part (do not qualify to be an edge). G6 6 8 D F G E 7 H A B C D 3 1 2 2 1 1 0 1 0 2 2 1 0 1 0 2 3 0 1 0 1 2 4 1 0 0 1 2 bCLQ31AC 2AC 3BD 4AD A12 A14 A24 C12 D34 5 bCLQ31AC 4AC 5BC 6DE 7DE 8FG 8FH 8GH bCCLQ412AC 12AC 14AC 12AC 12AC 24AC 34BD 24AD 34AD 124A 12AC bCCLQ414AC 67DE 8FGH = bCLQ4 bCLQ412AC 124A bCCLQ5=  bCCLQ512A4C bCLQ31AD 1AE 1DE 2AD 2AE 2DE 3EF 3EG 3FG 4CF 4CG 4FG 5AB 5AC 5BC 8 0 0 0 0 0 1 1 1 3 1 1 0 1 0 0 0 0 0 2 2 0 1 0 0 0 0 0 0 1 3 1 0 0 0 0 0 0 0 1 4 1 0 1 0 0 0 0 0 2 5 0 1 1 0 0 0 0 0 2 6 0 0 0 1 1 0 0 0 2 7 0 0 0 1 1 0 0 0 2 A B C D E F G bCCLQ41ADE 12AD 12AE 12DE 2ADE 3EFG 34FG 4CFG 5ABC = CLQ4 1 1 0 0 1 1 0 0 3 2 1 0 0 1 1 0 0 3 3 0 0 0 0 1 1 1 3 4 0 0 1 0 0 1 1 3 5 1 1 1 0 0 0 0 3 F E D C B A 1EF 1EF 2EF 3EF 6EF 7EF A B C D E F G H 3 2 3 2 2 1 1 1 1DF 1DF 3DF 3 1 2 2 3 2 2 bCCLQ512ADE 34EFG 34CFG G5.1b 1CF 1CF 2CF 3CF 6CF G5b 1BF 2BF 3BF 1AF 1AF 2AF F E D C B 1 2 1DE 5DE 1DE 3DE 1CE 2CE 3CE 5CE 1CE 6CE A 1BE 2BE 3BE 3 D 4AE 1AE 2AE 4 4 C 3 5CD 4CD 1CD 3CD 1BD 2BD 3BD A B 4AD 1AD Bclq3 1BC 2BC 3BC 2 4AC 1AC 2AC 1 G B 4 F 5 C 1AB 2AB 1 2 3 4 5 6 7 3 E 2 D 1 2 3 4 5 6 7 A 1 1 0 1 0 0 0 3 B 1 1 1 0 0 0 0 3 C 1 1 1 1 1 1 0 6 D 1 0 1 1 1 0 0 4 E 1 1 1 1 1 1 1 7 F 1 1 1 1 0 1 1 6 6 5 5 5 3 3 2 1 6 A 1 A B C D E F 1AB 1AC 1AD 1AE 1AF 1BC 1BD 1BE 1BF 1CD 1CE 1CF 1DE 1DF 1EF 2AB 2AC 2AE 2AF 2BC 2BE 2BF 2CE 2CF 2EF 3BC 3BD 3BE 3BF 3CD 3CE 3CF 3DE 3DF 3EF 4AC 4AD 4AE 4AF 4CD 4CE 4CF 4DE 4DF 4EF 5CD 5CE 5DE 6CE 6CF 6EF 7EF F E 7 C 3 B A C 2 D Bclq312A 14A 24A 12B 13B 23B 12C 13C 14C 15C 16C 23C 24C 25C 26C 34C 35C 36C 45C 46C 56C 13D 14D 15D 34D 35D 45D 12E 13E 14E 15E 16E 17E 23E 24E 25E 26E 27E 34E 35E 36E 37E 45E 46E 47E 56E 57E 67E 12F 13F 14F 16F 17F 23F 24F 36F 37F 34F 36F 37F 46F 47F 67F B 4 G9B1 5

  34. F E D C B A 1EF 1EF 2EF 3EF 6EF 7EF 1DF 1DF 3DF 1CF 1CF 2CF 3CF 6CF 1BF 2BF 3BF 1AF 1AF 2AF F E D C B 1DE 5DE 1DE 3DE 1CE 2CE 3CE 5CE 1CE 6CE 1BE 2BE 3BE 4AE 1AE 2AE 5CD 4CD 1CD 3CD 1BD 2BD 3BD 1 2 3 4 5 6 7 A 1 1 0 1 0 0 0 3 B 1 1 1 0 0 0 0 3 C 1 1 1 1 1 1 0 6 D 1 0 1 1 1 0 0 4 E 1 1 1 1 1 1 1 7 F 1 1 1 1 0 1 1 6 6 5 5 5 3 3 2 4AD 1AD A B C D E F 1BC 2BC 3BC 4AC 1AC 2AC 1AB 2AB 1 2 3 4 5 6 7 67F 37F 47F 27F 7 6 5 4 3 2 17F 12A 14A 24A 12B 13B 23B 12C 13C 14C 15C 16C 23C 24C 25C 26C 34C 35C 36C 45C 46C 56C 13D 14D 15D 34D 35D 45D 12E 13E 14E 15E 16E 17E 23E 24E 25E 26E 27E 34E 35E 36E 37E 45E 46E 47E 56E 57E 67E 12F 13F 14F 16F 17F 23F 24F 36F 37F 34F 36F 37F 46F 47F 67F 36F 46F 26F 16F 34F 24F 14F 23F 13F Bclq3 1AB 1AC 1AD 1AE 1AF 1BC 1BD 1BE 1BF 1CD 1CE 1CF 1DE 1DF 1EF 2AB 2AC 2AE 2AF 2BC 2BE 2BF 2CE 2CF 2EF 3BC 3BD 3BE 3BF 3CD 3CE 3CF 3DE 3DF 3EF 4AC 4AD 4AE 4AF 4CD 4CE 4CF 4DE 4DF 4EF 5CD 5CE 5DE 6CE 6CF 6EF 7EF 12F F E D C B A 67E 37E 47E 57E 27E 17E 36E 46E 56E 26E 16E 35E 45E 25E 15E 34E 24E 14E 23E 13E 12E 35D 45D 15D 34D 14D 13D 36C 46C 56C 26C 16C 35C 45C 25C 15C 34C 24C 14C 1 6 23C 13C 12C F E 7 C 3 A 23B 13B 2 D 12B 7 6 5 4 3 2 B Let’s take and even smaller bipartite graph to get a feel for how datacube technology might help us catalog bicliques, since there are clearly going to be a large number of them! 4 G9B1 5 14A 24A 12A 1 2 3 4 5 6 7

  35. Remember the Sales Data Cube? Each cell contains a sales measurement, e.g., the number of sales (may contain many other measurements of product-date-country instances) We will attempt to apply this technology to the task of finding bicliques later, after reviewing the technology. Date 2Qtr 1Qtr 3Qtr 4Qtr TV Product U.S.A PC VCR Canada Country Mexico

  36. Total of all product sales by country and quarter Total sales by country and dateRollup (aggregate under +) along product (e.g., using the aggregate, sum) Date 2Qtr 1Qtr 3Qtr 4Qtr TV Product U.S.A PC VCR Canada Country Mexico

  37. Total annual sales by country and product Rollup along date (e.g., using the aggregate, sum) Date 2Qtr 1Qtr 3Qtr 4Qtr TV Product U.S.A PC VCR Canada Country Mexico

  38. Total of all product sales by product and date Total of all product sales by product and date Rollup along country (e.g., using the aggregate, sum) Date 2Qtr 1Qtr 3Qtr 4Qtr TV Product U.S.A PC VCR Canada Country Mexico

  39. sales by product, country sales by country sales by country sales by country, date sales by product sales by product sales by product, country Total sales Total sales Total sales sales by date sales by date All rollups (e.g., using the aggregate, sum) Date 2Qtr 1Qtr 3Qtr 4Qtr TV Product U.S.A PC sales by product, country and quarter VCR Canada Country Mexico

  40. TV VCR PC Partial Rollup: climbing up a concept hierarchy(instead of eliminating Product altogether by summing over all products, rollup partially on Product, from (VCR, PC, TV) to computer (includes PC only) and non-computer (includes VCR + TV) Date 2Qtr 1Qtr 3Qtr 4Qtr Product U.S.A non-comp comp Canada Country Mexico

  41. TV VCR PC SLICE e.g., slice off PC Date 2Qtr 1Qtr 3Qtr 4Qtr Product U.S.A Canada Country Mexico

  42. 3Qtr 4Qtr PC Mexico DICE (e.g. dice off PC, the last two quarters, the country Mexico) Date 2Qtr 1Qtr Product TV U.S.A VCR Canada Country

  43. secondary Country Mexico Canada U.S.A Date 2Qtr 1Qtr 3Qtr 4Qtr 4Qtr TV Product U.S.A PC VCR 3Qtr Date Canada Country 2Qtr Mexico 1Qtr TV PC VCR Product tertiary Pivot/Rotate primary

  44. bCLQ3s centered on numbers. 1AC 2AB 2AC 2BC A B C 2 2 2 Now let’s apply this technology to finding all bicliques. 1 1 0 1 2 2 1 1 1 3 3 0 1 0 1 1 2 3 G5b1 A A B 2AB 2AB C 1AC 2AC 1 2 B 3 2BC C A B C

  45. 2ABC RollUp along the front-to-back dimension using the hub intersection and spoke union gives the expanded hub-and-spoke biclique, hub={2}, spokes={A,B,C} or hub={2A}, spokes={B,C} or the hub-union (of hubs {B},{C}), spoke-intersection (of spokes {2,A}). Rather than view it as an intersection-union of hubs and spokes, I think it suffices to just take the union??? bCLQ3s centered on numbers. 1AC 2AB 2AC 2BC A B C 2 2 2 1 1 0 1 2 2 1 1 1 3 3 0 1 0 1 1 2 3 G5b1 A A B 2AB C 1AC 2AC 1AC 2AC 1 2 B 3 2BC C A B C

  46. 12AC RollUp along the left-right dimension using the hub intersection and the spoke union gives the one expanded biclique, (hub={AC}, spokes={1,2} bCLQ3s centered on numbers. 1AC 2AB 2AC 2BC A B C 2 2 2 1 1 0 1 2 2 1 1 1 3 3 0 1 0 1 1 2 3 G5b1 A B 2AB C 1AC 2AC 1 2 A 3 2BC C A B B C

  47. 1AC 2AC 2BC 2ABC 2ABC RollUp along the top-bottom dim using hub intersection and spoke union gives the expanded hub-and-spoke biclique, (hub={2}, spokes={A,B,C} RollUp along the top-bottom dim using hub intersection and spoke union gives the expanded hub-and-spoke biclique, (hub={2}, spokes={A,B,C} bCLQ3s centered on numbers. 1AC 2AB 2AC 2BC A B C 2 2 2 1 1 0 1 2 2 1 1 1 3 3 0 1 0 1 1 2 3 G5b1 A B 2AB C 1 2 A 3 C A B B C

  48. 12 AC 2ABC 2ABC bCLQ3s centered on numbers. 1AC 2AB 2AC 2BC A B C 2 2 2 1 1 0 1 2 2 1 1 1 3 3 0 1 0 1 1 2 3 G5b1 A A B 2AB C 1AC 2AC 1 2 B 3 2BC C A B C

  49. bCLQ3s centered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 3AB 3AC 3BC A B C 3 3 3 1 1 1 1 3 2 1 1 1 3 3 1 1 1 3 1 2 3 G5b2 A A B 1AB 2AB 3AB 2AB C 1AC 2AC 3AC 1 2 B 3 1BC 2BC 3BC C A B C

  50. bCLQ3s centered on numbers. 1AB 1AC 1BC 2AB 2AC 2BC 3AB 3AC 3BC A B C 3 3 3 1 1 1 1 3 2 1 1 1 3 3 1 1 1 3 1 2 3 G5b2 A B 1AB 2AB 3AB 2AB C 1AC 2AC 3AC 1 2 A 3 1BC 2BC 3BC C A B B C

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