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Complex Networks Measures and deterministic models

Complex Networks Measures and deterministic models. Philippe Giabbanelli. Intro. Measures (clustering, degree distribution). Main course. Deterministic models. ∙ clustering augmentation …. ∙ fractal graphs. Side dish. Generalizing fractal graphs.

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Complex Networks Measures and deterministic models

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  1. Complex Networks Measures and deterministic models Philippe Giabbanelli

  2. Intro Measures (clustering, degree distribution) Main course Deterministic models ∙ clustering augmentation ….∙ fractal graphs Side dish Generalizing fractal graphs Leftover Discussion

  3. Motifs – Clustering – Average distance – Degree distribution 2 1 0 1 3 0 a motif is a subgraph that appears at a ‘very’ different frequence in G than in S. 2 0 Given a graph G… and a set S of random graphs of the same size and average degree, Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Complex networks

  4. Motifs – Clustering – Average distance – Degree distribution Milo et al., Science, 303, 2004 Milo et al., Science, 298, 2002 Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Complex networks

  5. Motifs– Clustering – Average distance – Degree distribution For a given node i , we denote its neighborhood by Ni. The clustering coefficient Ci of i is the edge density of its neighborhood. Here, there are two edges between nodes in Ni. Ci = 2.2/(5.4) = 0.2 At most, it’d be a complete graph with Ni.(Ni-1) edges. Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Complex networks

  6. Motifs– Clustering – Average distance – Degree distribution For a given node i , we denote its neighborhood by Ni. The clustering coefficient Ci of i is the edge density of its neighborhood. If a graph has high clustering coefficient, then there are communities (i.e., cliques) in this graph. People tend to form communities so it is common in social networks. Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Complex networks

  7. Motifs– Clustering – Average distance – Degree distribution The average distance l is: ∙ small if l∝ln(n) Average distance: average length of shortest path between all pairs of nodes ∙ ultrasmall if l∝ln(ln(n)) ← M.E.J Newman, The structure and function of complex networks Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Complex networks

  8. Motifs– Clustering – Average distance – Degree distribution Many measured phenomena are centered around a particular value. Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Complex networks

  9. Motifs– Clustering – Average distance – Degree distribution Many measured phenomena are centered around a particular value. There also exists numerous phenomena with a heavy-tailed distribution. lets plot it on a log-log scale Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Complex networks

  10. Motifs– Clustering – Average distance – Degree distribution We say that this distribution follows a power-law, with exponent α. There also exists numerous phenomena with a heavy-tailed distribution. The equation of a line is p(x) = -αx + c. Here we have a line on a log-log scale: ln p(x) = -α ln x + c apply exponent e c -α p(x) = ecx Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Complex networks

  11. Motifs– Clustering – Average distance – Degree distribution We say that this distribution follows a power-law, with exponent α. computer files people’s incomes Keep in mind that this is quite common. visits on web pages moon craters Ref.: P. Giabbanelli, J. Peters, « Complex networks and epidemics », TSI, 2010 Complex networks

  12. Motifs– Clustering – Average distance – Degree distribution A network with high clustering and low average distance is small-world. fast communications locally and globally There are other definitions (e.g., network that you can navigate easily). A network with power-law degree distribution is scale-free. (Luckily, there aren’t other definitions, we’re already messy enough.) See: Efficient measurement of complex networks using link queries (Tarissan, NetSciCom’09), aaaaReverse centrality queries in complex networks (Nielsen, MSc Thesis SFU dec. ’09) Complex networks

  13. Small world models(Watts Strogatz, Comellas et al., Giabbanelli et al.) 0 Almost all examples you will find use a simplified version. 16 1 15 2 14 Get n nodes labelled from 0 to n. 3 13 A node i is connected to (i+1, i+2, …, i Δ/2) mod n. Lets use Δ = 4. 4 This scheme yields ‘good’ values for 0.01 < p 0.1 5 12 6 Change one endpoint for an edge with probability p 11 Small average distance 7 10 Large clustering coefficient 8 9 Ref.: Watts & Strogatz, « Collective dynamics of ‘small-world’ networks », Nature 393, 1998 Complex networks

  14. Small world models(Watts Strogatz, Comellas et al., Giabbanelli et al.) h=6, a=1, b=2 Get n nodes labelled from 0 to n. 0 0 16 1 A node i is connected to (i+1, i+2, …, i Δ/2) mod n. Lets use Δ = 6. 5 15 2 1 14 3 13 4 A double step graph C(h; a,b) has h nodes, and i is connected to i a (mod h), i b (mod h) 5 12 4 Select h equidistant nodes, and connect them as C(h;a,b). 6 2 11 7 10 Then, some deterministic fiddling to keep the degree unchanged… 8 3 9 Ref.: Comellas, Ozon, Peters « Deterministic small-world communication networks », 2000 Complex networks

  15. Small world models(Watts Strogatz, Comellas et al., Giabbanelli et al.) Intuition 0 Consider that we start with the cycle Cn. 16 1 15 2 The added edges should provide a good coverage of distances. 14 3 When we connect i to i 1,…,i (Δ/2), we create lots of short-range edges 13 4 5 12 Adding edges from a double-step graph mainly provides medium-range edges 6 11 7 10 8 9 Ref.: Giabbanelli & Peters, submitted to AlgoTel’10 Complex networks

  16. Small world models(Watts Strogatz, Comellas et al., Giabbanelli et al.) Intuition 0 Consider that we start with the cycle Cn. 16 1 +1 15 2 The added edges should provide a good coverage of distances. +2 14 3 As long as d(i)≠Δ, connect i to i 2 , …, i 2 13 0 k 4 +4 5 12 +8 6 11 7 10 8 9 Ref.: Giabbanelli & Peters, submitted to AlgoTel’10 Complex networks

  17. Small world models(Watts Strogatz, Comellas et al., Giabbanelli et al.) Intuition 0 16 1 We want high clustering coefficient. 15 2 What’s the graph with the highest clustering coefficient? 14 3 → complete graph 13 4 If a node has degree Δ-1, we add to it a K 5 12 Δ 6 11 Pretty artificial… but has the values required for small-world. 7 10 8 9 Ref.: Giabbanelli & Peters, submitted to AlgoTel’10 Complex networks

  18. Fractal graphs(Graph grammar, Zhang et al., Perspectives) We’ll just introduce as much as we need. Starting graph Pattern graph A dotted edge is said to be active. At each time step, all dotted edges get replaced by a pattern graph. Complex networks

  19. Fractal graphs(Graph grammar, Zhang et al., Perspectives) We’ll just introduce as much as we need. Starting graph Pattern graph A dotted edge is said to be active. At each time step, all dotted edges get replaced by a pattern graph. Complex networks

  20. Fractal graphs(Graph grammar, Zhang et al., Perspectives) Here’s the definition of ZRG using our graph grammar. t This generates a (planar) small-world graph. It also has a simple labelling scheme. 0L L 1L Starting graph Pattern graph Ref.: Zhang, Rong, Guo, Physica A: Stat. Mech. And Appl., 363, 2006 Giabbanelli, Mazauric, Pérennes, submitted to AlgoTel’10 Complex networks

  21. Fractal graphs(Graph grammar, Zhang et al., Perspectives) Here’s the definition of M using our graph grammar. d,t There is no triangle so the clustering coefficient is 0. The result is scale-free, planar, with small average distance. d Starting graph Pattern graph Example for d = 2 Ref.: Miralles, Comellas, Chen, Zhang, Physica A, 389, 2010 For d=1: Comellas, Mirales, Physica A, 388, 2009 Complex networks

  22. Fractal graphs(Graph grammar, Zhang et al., Perspectives) We’re not limited to active edges. For example, lets have active cycles. Given the active cycle and the pattern, how do we know which edge of the cycle gets replaced by which edge of the pattern? We use a function that maps the active cycle in the pattern (= morphism) a a d b d b c c Starting graph Pattern graph Ref.: Comellas, Zhang, Chen, J. Physc. A., 42, 2009 Complex networks

  23. Fractal graphs(Graph grammar, Zhang et al., Perspectives) We’re not limited to active edges. For example, lets have active cycles. How many active cycles are there in the pattern graph? Starting graph Pattern graph Ref.: Comellas, Zhang, Chen, J. Physc. A., 42, 2009 Complex networks

  24. Fractal graphs(Graph grammar, Zhang et al., Perspectives) We’re not limited to active edges. For example, lets have active cycles. How many active cycles are there in the pattern graph? 4 This is NOT an active cycle. Starting graph Pattern graph Ref.: Comellas, Zhang, Chen, J. Physc. A., 42, 2009 Complex networks

  25. Fractal graphs(Graph grammar, Zhang et al., Perspectives) We’re not limited to active edges. For example, lets have active cycles. How many active cycles are there in the pattern graph? 4 The result is scale-free, planar, with small average distance. Starting graph Pattern graph Ref.: Comellas, Zhang, Chen, J. Physc. A., 42, 2009 Complex networks

  26. Fractal graphs(Graph grammar, Zhang et al., Perspectives) We can define tons of patterns (people actually did and published them). So maybe we could discuss their properties a bit more generally. Lets have a look at active edges. Complex networks

  27. Fractal graphs(Graph grammar, Zhang et al., Perspectives) We start from a triangle with a pattern having two active edges. Step t = 1 For each box, we add Np-2 nodes. We start with 3 nodes, add 3 boxes: 3(Np-2)+3 nodes Diameter at most Dp 2 Black box Diameter D N nodes P P Pattern graph Complex networks

  28. Fractal graphs(Graph grammar, Zhang et al., Perspectives) We start from a triangle with a pattern having two active edges. The number of added nodes doubles at each step: we now add 3.2(Np-2) nodes. Black box Diameter D N nodes The longest path is through 2t boxes P Diameter at most 2tDp P Pattern graph Complex networks

  29. Fractal graphs(Graph grammar, Zhang et al., Perspectives) We start from a triangle with a pattern having two active edges. The average distance is small regardless of the pattern you choose. The same conclusion holds for a pattern graph with at least two active edges, and any starting graph. Black box Diameter D N nodes P P. Giabbanelli, Properties of fractal network models, submitted to Physica A P Pattern graph Complex networks

  30. Complex networks

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