Optimal Networks

1. Optimal Networks Mirka Miller School of Information Technology and Mathematical Sciences University of Ballarat m.miller@ballarat.edu.au Centre for Informatics and Applied Optimization

2. Outline of the talk • Interconnection networks • Degree/diameter problem – directed and undirected • Recent new results concerning graphs close to Moore bound • Three extremal problems • Open problems

3. Interconnection networks Examples: • Communications • Transportation • Computer • Social Networks can be modeled as graphs. The structure (topology) of the graph is useful when designing algorithms (for communication, broadcasting etc.) and for analyzing the performance of a network.

4. Small-world networks

5. WWW (2000)

6. Network parameters TopologyNodes and edges and their arrangement.DiameterMaximum distance between any pair of nodes.ConnectivityNumber of neighbours of a given node: d degreeClusteringAre neighbours of a node also neighbours among themselves?

7. Most “real life” networks • small-world scale-free small diameter Milgram 1967 clustered Watts & Strogatz 1998 degrees follow a power-law Barabási & Albert 1999

8. Network topology Structured • high clustering • large diameter • regular Small-world • high clustering • small diameter • almost regular Random • low clustering • small diameter |V| = 1000 d= 10 D = 100 C = 0.67 |V|=1000 d= 8-13 D = 14 C = 0.63 |V|=1000 d= 5-18 D = 5 C = 0.01

9. Definitions We study graphs with respect to 3 parameters: diameter, degree and order. • Diameter • The longest distance between any two vertices in the graph. • Degree • The number of edges attached to a vertex. • Order • The number of vertices in the graph.

10. Degree/diameter problem Determine the largest number of vertices n of a graph G for given maximum degree d and diameter at most k. • Survey by Miller and Siran “Moore bound and beyond: A survey of the degree/diameter problem” • Research supported by a grant from Australian Research Council (ARC) CI’s: McKay and Miller

11. Degree/diameter problem Directed version: Determine the largest number of vertices n of a digraph G for given maximum out-degree d and diameter at most k.

12. Degree/diameter problem Approaches to attack the problem Increase the lower bound: by construction • Voltage assignments • Line digraphs • Computer search • Etc. Decrease the upper bound: by proving a graph with given parameters cannot exist

13. Upper bound v A natural upper bound on the number of vertices n of digraph G with given maximum out-degree d and diameter at most k is: n  Md,k = 1+d +d 2 + … + d k. This bound is called the Moore bound. A digraph attaining this bound is called a Moore digraph.

14. Moore digraphs Plesnik & Znam ’74, Bridges & Toueg ’80 : Moore digraphs exist only for trivialcases, namely for d =1 (cycles of k+1 vertices) or k =1 (complete digraphs on d+1 vertices).

15. Moore digraphs Outline of the PROOF that (d,k)-digraphsdo not exist when d> 1 and k > 1 : I+A+…+Ak=J where A is adjacency matrix of G, J isunit matrix, I isidentity matrix J has eigenvalues n (once)and 0 (n-1 times) A has eigenvaluesd (once)and n-1 roots of the characteristic equation 1+ x+x2+…xk = 0 x = 0 and roots of xk+1-1 =0 Since tr(Aj)=0 for 1  jk, we obtain –d=-dk and so d=1 or k=1 are the only solutions.

16. Diregularity of AMDs Is Md,k–1 attainable for all d>1 and k>1? Notation. (d,k)-digraph is a digraph of maximum out-degree d, diameter k and order n = Md,k – 1. (We also call such a digraph almost Moore digraph). Miller, Gimbert, Siran & Slamin, ‘00: The (d,k)-digraphs are diregular of degree d. Note that: to show the regularity of out-degree is easy (by a counting argument). However, to show the regularity of in-degree is not easy.

17. Repeat of a vertex Let G be a (d,k)-digraph. • For every vertex x of G there exists a vertex y, called the repeat of x, such that there are two walks of lengths  k from x to y. • If r(x) = x then vertex x is called a selfrepeat. • The function r : V(G)  V(G) is an automorphism on V(G); namely, (x,y)  E(G) iff (r (x), r (y))  E(G). • Let the order of vertex v be the smallest positive integer (v) so that r (v)(v)= v.

18. Diameter 2 Fiol, Allegre, and Yebra ’83: (d,2)-digraphs exist for any d  2. • Example: The line digraph of L(Kd+1) of the complete digraph Kd+1. • But Gimbert ’01 showed that this line digraph is the only (d,2)-digraph if d 3.

19. Diameter 2, degree 2 3 1 2 1 4 4 5 3 2 1 2 4 5 5 6 3 6 6 (123)(456) (1)(2)(3)(4)(5)(6) (12)(3456) All order 3 All selfrepeats Two order 2; four order 4 There are exactly three (2,2)-digraphs.

20. Degree 2 and degree 3 Miller and Fris ’92: There are no (2,k)-digraphs for any k  3. Baskoro, Miller, Siran & Sutton (in press): There are no (3,k)-digraphs for any k  3. The remaining cases are still open: Do there exist any (d,k)-digraphs, d  4, k  3?

21. Existence of (d,k)-digraphs A (d,k)-digraph, d  4, k  3 (if it exists) may contain a selfrepeat or no selfrepeat. Further study may focus on the existence of: • (d,k)-digraphs with selfrepeats • (d,k)-digraphs with no selfrepeats

22. Structure of the orders of vertices • A (d,k)-digraph contains either k selfrepeats or none. [Baskoro, Miller, Plesnik, ’98]

23. The orders of vertices • We can determine the structure of the orders of vertices in a (d,k)-digraph with selfrepeats,d 2, k 2. [Baskoro, Cholily, Miller, ’04]

24. The orders of vertices Example.k=2, d=6. Label vertices 0,1,2,…,41. Suppose the digraph G contains a selfrepeat and that the out-neighbourhood of a selfrepeat consists of vertices of orders 2 and 3 (as well as a selfrepeat). Then (up to isomorphism) the permutation cycles of repeats of G are (0) (1) (2,3)(4,5,6) (7,8)(9,10,11)(12,18)(13,19)(14,20) (15,21,16,22,17,23)(24,30,36)(25,31,37)(26,32,38)(27,33,39) (28,34,40)(29,35,41) Two cycles of length 1, five cycles of length 2, eight cycles of length 3, one cycle of length 6.

25. Open problems • Are there any (d,k)-digraphs, d 4, k 3, with selfrepeats? • Are there any (d,k)-digraphs, d 4, k 3, without selfrepeats? • For d= 3, k 3, are there any digraphs of order M3,k– 2? • For d= 2, k 3, are there any digraphs of order Md,k– 3? • Are almost almost Moore digraphs diregular?

26. Degree/diameter problem Undirected version: Determine the largest number of vertices n of a graph G for given maximum degree d and diameter at most k.

27. Upper bound v A natural upper bound on the number of vertices n of a graph G of given maximum degree d and diameter at most k is: n  Md,k=1+d+d(d-1)+…+d(d-1)k-1 • This bound is called the Moore bound. • A graph attaining this bound is called a Moore graph.

28. Moore graphs k =1: Moore graphs are complete graphs on d+1 vertices. Hoffman and Singleton, ’60: k =2: Moore graphs exist for d =2 (pentagon) or d =3 (Petersen graph) or d =7 (Hoffman-Singleton graph) or d =57? k =3: Moore graph exists for d =2 (7-gon). Damerell; Bannai and Ito, ’73: k >3: Moore graph are (2k+1)-gons.

29. Almost Moore graphs Is Md,k–1 attainable for all d>2 and k>1? Erdos, Fajtlowicz and Hoffman, ’80: k =2: almost Moore graphexists only for d =2 (4-cycle). Bannai and Ito; Kurosawa and Tsujii, ’81: k >3: almost Moore graphsexist only for d =2 (2k-gons).

30. Graphs with defect > 1 Defect 2: d =2: (2k-1)-gons. d >2: only 5 such graphs are known so far: (d,k) = (3,2) (two); (4,2); (5,2); (3,3) (unique).

31. Almost almost Moore graphs • There are no almost almost Moore graphs of degree 3 and diameter k>3. [Jorgensen, ’92] • Theorem 6. For k>2,there are no almost almost Moore graphs of degree 4. [Miller and Simanjuntak, ’04]

32. Repeat of a vertex Define (d,k,d)-graph to be a graph of degree d, diameter k and defect d. Vertex y is a maximal repeat of x if y appears in R(x)d times (x has no other repeats). Theorem 7. For d >1,the number of maximal repeats in a(d,2,d)-graphis 0 or 2 or 6. [Nguyen and Miller, ’04]

33. Structure of a (d,2,2)-graph Possible repeat configurations in a (d,2,2)-graph: u u u r1(u) r2(u) r(u) r1(u) r2(u) u u r1(u) r2(u) r1(u) r2(u) Define n0,n1,n2a,n2b,n2c.

34. Structure of a (d,2,2)-graph Theorem 7. A (d,2,2)-graphcontains • if d is even then n0 = 3 and n2b = d2 – 4 • if d = 3 then (n0,n1,n2c) = (3,2,3) • if d is odd then (n0,n1,n2c,n2a,n2b) = (9,6,9,4a,d2-25-4a) or n2b = d2 – 1. [Nguyen and Miller, ’04]

35. Open problems • Are there any (d,2,2)-graphs for d 6? • Are there any (d,k,2)-graphs for d 5 and k 3? • Are there any (3,k,3)-graphs fork 4? • Are there any (4,k,3)-graphs fork 3? . . . • Is there a Moore graph with diameter 2 and degree 57?

36. Open problems • We know that for directed graphs N(d,k)is monotonic in both d and k. Let K(n,d)be the smallest possible diameter of a digraph on n vertices and maximum out-degree d. Is K(n,d)monotonic in n? For undirected graphs we know that the corresponding K(n,d) is not monotonic in n.

37. Example. K(10,3) = K(8,3) = 2 n = 8 n = 10 but K(9,3) = 3! ncannot be more than 10

38. Three optimisation problems • N(d,k) = max{n: G(n,d,k)} • D(n,k) = min{d: G(n,d,k)} • K(n,d) = min{k: G(n,d,k)} • G(n,d,k) denotes the set of all directed graphs of order n, degree d, and diameter k. Question: are these three problems equivalent to each other?

39. Known monotonic relationships • d1<d2 implies N(d1,k)< N(d2,k) • k1<k2 implies N(d,k1)< N(d,k2) • d1<d2 implies K(n,d1) K(n,d2) ?n1<n2 implies K(n1,d)  K(n2,d) ?k1<k2 implies D(n,k1)  D(n,k2) ?n1<n2 implies D(n1,k)  D(n2,k)

40. K(n,d) problem digraph G of degree d, diameter k and order n line digraph construction digraph L(G) of degree d, order dn and diameter k+1 Vertex deletion scheme

41. K(n,d) problem Theorem [Slamin & Miller,’00] If L(G) G(dn,d,k) is a line digraph of a diregular digraph GG(n,d,k-1) then there exists digraph L(G) G(dn-r,d,k’), k’k, for every 1r (d-1)n-1 nd n

42. K(n,d) problem The largest n such that G(n,d,k-1) is not empty for given d. G(nd,d,k) The largest n’ such that G(n’,d,k) is not empty for given d.

43. K(n,d) problem • K(n,d) is monotonic in n in some intervals of d values • K(n,d) problem is equivalent to N(d,k) in those intervals • N(d,k)  K(n,d)  D(n,k) For directed graphs, are these three problems in fact all equivalent?