Centre for Informatics and Applied Optimization

1CIAO

Optimal Optimal NetworksNetworks

Mirka MillerSchool of Information Technology and Mathematical Sciences University of Ballaratm.miller@ballarat.edu.au

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Outline of the talkOutline of the talk

Interconnection networksDegree/diameter problem –

directed and undirectedRecent new results concerning

graphs close to Moore boundThree extremal problemsOpen problems

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Interconnection Interconnection networksnetworks

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.

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Small-world networks Small-world networks

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WWW (2000)

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TopologyNodes and edges and their arrangement.

DiameterMaximum distance between any pair of nodes. Connectivity Number of neighbours of a given node:

d degree ClusteringAre neighbours of a node also neighbours among themselves?

Network parametersNetwork parameters

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Most “real life” Most “real life” networksnetworks

small-world scale-free

small diameter Milgram 1967clustered Watts & Strogatz 1998

degrees follow a power-law Barabási & Albert 1999

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Structured high clustering large diameter regular

Random low clustering small diameter

Small-world high clustering small diameter almost regular

|V| = 1000 = 10D = 100 C = 0.67

|V|=1000 = 8-13D = 14 C = 0.63

|V|=1000 = 5-18D = 5 C = 0.01

Network topology

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DefinitionsWe study graphs with respect to 3

parameters: diameter, degree and order.Diameter

The longest distance between any two vertices in the graph.

DegreeThe number of edges attached to a

vertex.Order

The number of vertices in the graph.

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Degree/diameter Degree/diameter problemproblem

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

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Degree/diameter Degree/diameter problemproblem

Directed version:

Determine the largest number of vertices n of a digraph G for given maximum out-degree d and diameter at most k.

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Degree/diameter Degree/diameter problemproblem

Approaches to attack the problemApproaches to attack the problem

Increase the lower bound: by construction

o Voltage assignmentso Line digraphso Computer searcho Etc.

Decrease the upper bound: by proving a graph with given parameters cannot exist

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Upper boundUpper boundA 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.

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This bound is called the Moore bound.A digraph attaining this bound is called a Moore digraph.

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Moore digraphsMoore digraphsPlesnik & Znam ’74, Bridges & Toueg ’80 :

Moore digraphs exist only for trivial cases, namely for d =1 (cycles of k+1 vertices) or k =1 (complete digraphs on d+1 vertices).

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Moore digraphsMoore digraphs

Outline of the PROOF that (d,k)-digraphs do not exist when

d > 1 and k > 1 :

I+A+…+Ak=J where A is adjacency matrix of G, J is unit matrix, I is identity

matrix

J has eigenvalues n (once) and 0 (n-1 times)A has eigenvalues d (once) and n-1 roots of the

characteristic equation 1+ x+x2+…xk = 0 x = 0 and roots of xk+1 -1 =0Since tr(Aj)=0 for 1 jk, we obtain –d=-dk and so d=1 or k=1 are the only solutions.

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Diregularity of AMDsDiregularity of AMDsIs 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.

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Repeat of a vertexRepeat of a vertexLet 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.

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Diameter 2Diameter 2Fiol, 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 linedigraph is the only (d,2)-digraph if d 3.

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Diameter 2, degree 2Diameter 2, degree 2

There are exactly three (2,2)-digraphs.

1 2

5 6

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1

2

3

4

5

6

1 2

3

4

5

6(1)(2)(3)(4)(5)(6) (123)(456)

(12)(3456)All selfrepeats All order 3

Two order 2; four order 4

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Degree 2 and degree 3Degree 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?

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Existence of Existence of (d,k)-digraphsA (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

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Structure of the orders Structure of the orders of verticesof vertices

A (d,k)-digraph contains either k selfrepeats or none. [Baskoro, Miller, Plesnik, ’98]

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The orders of verticesThe 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]

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The orders of verticesThe orders of verticesExample. 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.

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Open problemsOpen 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?

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Degree/diameter Degree/diameter problemproblem

Undirected version:

Determine the largest number of vertices n of a graph G for given maximum degree d and diameter at most k.

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Upper boundUpper boundA 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.

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Moore graphsMoore 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.

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Almost Moore graphsAlmost Moore graphs

Is Md,k–1 attainable for all d>2 and k>1?

Erdos, Fajtlowicz and Hoffman, ’80:

k =2: almost Moore graph exists only for d =2 (4-cycle).

Bannai and Ito; Kurosawa and Tsujii, ’81:

k >3: almost Moore graphs exist only for d =2 (2k-gons).

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Graphs with defect > Graphs with defect > 11Defect 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).

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Almost almost Moore Almost almost Moore graphsgraphs 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]

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Repeat of a vertexRepeat of a vertexDefine (d,k,)-graph to be a graph of

degree d, diameter k and defect

Vertex y is a maximal repeat of x if y appears in R(x) times (x has no other repeats).

Theorem 7. For d >1, the number of maximal repeats in a (d,2,)-graph is 0 or 2 or 6.

[Nguyen and Miller, ’04]

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Structure of a Structure of a (d,2,2)-graph

Possible repeat configurations in a (d,2,2)-graph:

u u

r(u)

u

u

u r2(u)r1(u)

r1(u)

r1(u) r1(u)

r2(u)

r2(u) r2(u)

Define n0,n1,n2a,n2b,n2c.

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Structure of a Structure of a (d,2,2)-graph

Theorem 7. A (d,2,2)-graph contains

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) = (9,6,9,4,d2-25-4) or n2b = d2 – 1.

[Nguyen and Miller, ’04]

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Open problemsOpen 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 for k 4?

Are there any (4,k,3)-graphs for k 3?

.

..

Is there a Moore graph with diameter 2 and degree 57?

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Open problemsOpen 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.

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Example. K(10,3) = K(8,3) = 2

n cannot be more than 10

n = 8 n = 10

but K(9,3) = 3!

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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?

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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)

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K(n,d) problem

line digraph construction

Vertex deletion schemedigraph L(G) of degree d, order dn and diameter k+1

digraph G of degree d, diameter k and order n

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K(n,d) problem

Theorem [Slamin & Miller,’00]

If L(G) G(dn,d,k) is a line digraph of a diregular digraph GG(n,d,k-1) then there exists digraph L(G) G(dn-r,d,k’), k’k, for every 1r (d-1)n-1

n nd

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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.

K(n,d) problem

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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?