Replicator Equations, Maximal Cliques, and Graph Isomorphismpelillo/papers/NeuralComputation 1999.pdf · Replicator Equations, Maximal Cliques, and Graph Isomorphism 1935 just the

LETTER Communicated by Steven Zucker

Replicator Equations Maximal Cliques and GraphIsomorphism

Marcello PelilloDipartimento di Informatica Universita Carsquo Foscari di Venezia 30172 Venezia MestreItaly

We present a new energy-minimization framework for the graph isomor-phism problem that is based on an equivalent maximum clique formu-lation The approach is centered around a fundamental result proved byMotzkin and Straus in the mid-1960s and recently expanded in variousways which allows us to formulate the maximum clique problem in termsof a standard quadratic program The attractive feature of this formulationis that a clear one-to-one correspondence exists between the solutions ofthe quadratic program and those in the original combinatorial problemTo solve the program we use the so-called replicator equationsmdasha classof straightforward continuous- and discrete-time dynamical systems de-veloped in various branches of theoretical biology We show how despitetheir inherent inability to escape from local solutions they neverthelessprovide experimental results that are competitive with those obtainedusing more elaborate mean-eld annealing heuristics

1 Introduction

The graph isomorphism problem is one of those few combinatorial opti-mization problems that still resist any computational complexity character-ization (Garey amp Johnson 1979 Johnson 1988) Despite decades of activeresearch no polynomial-time algorithm for it has yet been found At thesame time while clearly belonging to NP no proof has been provided thatit is NP-complete Indeed there is strong evidence that this cannot be thecase for otherwise the polynomial hierarchy would collapse(Boppana Has-tad amp Zachos 1987 Schoning 1988) The current belief is that the problemlies strictly between the P and NP-complete classes

Because of its theoretical and practical importance the problem has at-tracted much attention in the neural network community and various pow-erful heuristics have been developed (Kree amp Zippelius 1988 Gold amp Ran-garajan 1996Mjolsness Gindiamp Anandan 1989Rangarajan Goldamp Mjol-sness 1996 Rangarajan amp Mjolsness 1996 Simi Acircc 1991) Following Hopeldand Tankrsquos (1985) seminal work the customary approach has been to derivea (continuous) energy function in such a way that solutions of the originaldiscrete problem map onto minimizers of the function in a continuous do-

Neural Computation 11 1933ndash1955 (1999) cdeg 1999 Massachusetts Institute of Technology

1934 Marcello Pelillo

main For graph isomorphism the continuous domain usually correspondsto the unit hypercube and the energy function is quadratic The energy isthen minimized using an appropriate dynamical system and after conver-gence a solution to the discrete problem is recovered from the minimizerthus found Almost invariably the minimization algorithms developed sofar incorporate techniques borrowed from statistical mechanics in particu-lar mean eld theory which allow one to escape from poor local solutions

Early formulations suffer from the lack of a precise characterization ofthe local and global minimizers of the continuous energy function in termsof the solutions of the discrete problem which are usually in the form ofa permutation matrix In other words while the solutions of the originalproblem correspond (by construction) to solutions of its continuous coun-terpart the inverse is not necessarily true Recently however Yuille andKosowsky (1994) showed that by adding a certain term to the quadraticobjective minimizers in the unit hypercube can lie only at the verticesthereby overcoming this drawback Their formulation has been success-fully employed in conjunction with double normalization and Lagrangiandecomposition methods (Rangarajan et al 1996 Rangarajan amp Mjolsness1996) An additional remark on standard neural network models for graphisomorphism is that it is not clear how to interpret the solutions of the con-tinuous problem when the graphs being matched are not isomorphic Inthis case in fact there is no permutation matrix that solves the problemand yet there will be minima in continuous space since the domain is com-pact and the function being minimized is continuous Although this issueis more closely related to the subgraph isomorphism problem (which isknown to be computationally intractable) it would be desirable for a graphisomorphism algorithm always to return ldquomeaningfulrdquo solutions

In this article we develop a new energy-minimization framework forgraph isomorphism based on the idea of reducing it to the maximum cliqueproblemanother well-known combinatorial optimizationproblem (BomzeBudinich Pardalos amp Pelillo 1999) Central to our approach is a powerfulresult originallyprovedby Motzkin and Straus (1965) and recently extendedin various ways (Bomze 1997 Gibbons Hearn amp Pardalos 1996 GibbonsHearn Pardalos amp Ramana 1997 Pelillo amp Jagota 1995) which allows us toformulate the maximum clique problem in terms of an indenite quadraticprogram In the proposed formulation an elegant one-to-one correspon-dence exists between the solutions of the quadratic program and those ofthe original problem We also present a class of straightforward continuous-and discrete-time dynamical systems known in mathematical biology asreplicator equations and show how owing to their properties they providea natural and useful heuristic for solving the Motzkin-Straus program andhence the graph isomorphism problem

It may be argued that trying to solve the graph isomorphism problem byreducing it to the maximum clique problem is an altogether inappropriatechoice In contrast to graph isomorphism in fact the problem of nding

Replicator Equations Maximal Cliques and Graph Isomorphism 1935

just the cardinality of the maximum clique in a graph is known to be NP-complete and according to recent theoretical results so is the problem ofapproximating it within a certain tolerance (Arora Lund Motwani Su-dan amp Szegedy 1992 Bellare Goldwasser amp Sudan 1995 Hastad 1996)1

The experimental results presented in this article however seem to con-tradict this claim By using simple relaxation equations that are inherentlyunable to avoid local optima we get results that compare favorably withthose obtained using state-of-the-art sophisticated deterministic annealingalgorithms that by contrast are explicitly designed to escape from localsolutions This suggests that the proposed Motzkin-Straus formulation isa promising framework within which to develop powerful graph isomor-phism heuristics

The outline of the article is as follows Section 2 presents the quadraticprogramming formulation for graph isomorphism derived from theMotzkin-Straus theorem In section 3 we introduce the replicator equationsdiscuss their fundamental dynamical properties and present the experi-mental results obtained over hundreds of 100-vertex graphs of various con-nectivities In section 4 an exponential replicator dynamics is presentedthat turns out to be dramatically faster and more accurate than the classicalmodel Finally section 5 concludes the article

2 A Quadratic Programming Formulation for Graph Isomorphism

21 Graph Isomorphism as Clique Search Let G D (V E) be an undi-rected graph where V is the set of vertices and E microV poundV is the set of edgesThe order of G is the number of its vertices and its size is the number ofedges Two vertices i j 2 V are said to be adjacent if (i j) 2 E The adjacencymatrix of G is the n pound n symmetric matrix A D (aij) dened as follows

aij Draquo

1 if (i j) 2 E0 otherwise

The degree of a vertex i 2 V denoted by deg(i) is the number of verticesadjacent to it that is deg(i) D

Pj aij

Given two graphs G0 D (V0 E0 ) and G00 D (V00 E00 ) an isomorphism be-tween them is any bijection w V0 V00 such that (i j) 2 E0 (w (i) w (j)) 2E00 for all i j 2 V0 Two graphs are said to be isomorphic if there exists anisomorphism between them The graph isomorphism problem is thereforeto decide whether two graphs are isomorphic and in the afrmative tond an isomorphism The maximum common subgraph problem is moregeneral and difcult (Garey amp Johnson 1979) and includes the graph iso-

1 However these are worst-case results and there are certain classes of graphs forwhich the problem is solvable in polynomial time (Grotschel Lov Acircasz amp Schrijver 1988Bomze et al 1999)

1936 Marcello Pelillo

morphism problem as a special case It consists of nding the largest iso-morphic subgraphs of G0 and G00 A simpler version of this problem is tond a maximal common subgraphmdashan isomorphism between subgraphsthat is not included in any larger subgraph isomorphism

Barrow and Burstall (1976) and also Kozen (1978) introduced the notionof an association graph as a useful auxiliary graph structure for solvinggeneral graphsubgraph isomorphism problems

Denition 1 The association graph derived from graphs G0 D (V0 E0 ) andG00 D (V00 E00 ) is the undirected graph G D (V E) dened as follows

V D V0 pound V00

and

E Dcopy((i h) (j k)) 2 V pound V i 6D j h 6D k and (i j) 2 E0 (h k) 2 E00ordf

Given an arbitrary undirected graph G D (V E) a subset of vertices C iscalled a clique if all its vertices are mutually adjacent that is for all i j 2 Cwe have (i j) 2 E A clique is said to be maximal if it is not contained in anylarger clique and maximum if it is the largest clique in the graph The cliquenumber denoted by v(G) is dened as the cardinality of the maximumclique

The following result establishes an equivalence between the graph iso-morphism problem and the maximum clique problem

Theorem 1 Let G0 D (V0 E0 ) and G00 D (V00 E00 ) be two graphs of order n andlet G be the corresponding association graph Then G0 and G00 are isomorphic if andonly if v(G) D n In this case any maximum clique of G induces an isomorphismbetween G0 and G00 and vice versa In general maximal and maximum cliques in Gare in one-to-one correspondence with maximal and maximum common subgraphisomorphisms between G0 and G00 respectively

Proof Suppose that the two graphs are isomorphic and let w be an iso-morphism between them Then the subset of vertices of G dened as Cw Df(i w (i)) 8i 2 V0g is clearly a maximum clique of cardinality n Converselylet C be an n-vertex maximum clique of G and for each (i h) 2 C denew (i) D h Then because of the way the association graph is constructed it isclear that w is an isomorphism between G0 and G00 The proof for the generalcase is analogous

22 Continuous Formulation of the Maximum Clique Problem LetG D (V E) be an arbitrary undirected graph of order n and let Sn denotethe standard simplex of Rn

Sn D

(x 2 Rn xi cedil 0 for all i D 1 n and

nX

iD1

xi D 1

)

Replicator Equations Maximal Cliques and Graph Isomorphism 1937

Given a subset of vertices C of G we shall denote by xc its characteristicvector which is the point in Sn dened as

xci D

raquo1 |C| if i 2 C0 otherwise

where |C| denotes the cardinality of CNow consider the following quadratic function

f (x) D xTAx

DnX

iD1

nX

jD1

aijxixj (21)

where A D (aij) is the adjacency matrix of G and T denotes transpositionA point xcurren 2 Sn is said to be a global maximizer of f in Sn if f (xcurren) cedil f (x)for all x 2 Sn It is said to be a local maximizer if there exists an e gt 0 suchthat f (xcurren) cedil f (x) for all x 2 Sn whose distance from xcurren is less than e and iff (xcurren) D f (x) implies xcurren D x then xcurren is said to be a strict local maximizer

The Motzkin-Straus theorem (Motzkin amp Straus 1965) establishes a re-markable connection between global (local) maximizers of the function f inSn and maximum (maximal) cliques of G Specically it states that a subsetof vertices C of a graph G is a maximum clique if and only if its character-istic vector xc is a global maximizer of f on Sn A similar relationship holdsbetween (strict) local maximizers and maximal cliques (Gibbons et al 1997Pelillo amp Jagota 1995) This result has an intriguing computational signi-cance in that it allows us to shift from the discrete to the continuous domainin an elegant manner Such a reformulation is attractive for several reasonsIt not only allows us to exploit the full arsenal of continuous optimizationtechniques thereby leading to the development of new algorithms but mayalso reveal unexpected theoretical properties Additionally continuous op-timization methods are often described in terms of (ordinary) differentialequations and are therefore potentially implementable in analog circuitryThe Motzkin-Straus theorem has served as the basis of many clique-ndingprocedures (Bomze Pelillo amp Giacomini 1997 Bomze Budinich Pelillo ampRossi 1999 Gibbons et al 1996 Pardalos amp Phillips 1990 Pelillo 1995)and has also been used to determine theoretical bounds on the clique num-ber (Pardalos amp Phillips 1990 Wilf 1986)

One drawback associated with the original Motzkin-Straus formulationrelates to the existence of spurious solutionsmdashmaximizers of f that are not inthe form of characteristic vectors This was observed empirically by Parda-los and Phillips (1990) and has more recently been formalized by Pelillo andJagota (1995) In principlespurious solutions represent a problemalthoughthey provide information about the cardinality of the maximum clique theydo not allow us to extract its vertices easily Fortunately there is straight-forward solution to this problem which has recently been introduced and

1938 Marcello Pelillo

studied by Bomze (1997) Consider the following regularized version offunction f

Of (x) DnX

iD1

nX

jD1

aijxixj C12

nX

iD1

x2i (23)

which is obtained from equation 21 by substituting the adjacency matrix Aof G with

OA D A C12

In

where In is the n pound n identity matrix The following is the spurious-freecounterpart of the original Motzkin-Straus theorem (see Bomze 1997 forproof)

Theorem 2 Let C be a subset of vertices of a graph G and let xc be its charac-teristic vector Then the following statements hold

1 C is a maximum clique of G if and only if xc is a global maximizer of thefunction Of over the simplex Sn In this case v(G) D 1 2(1 iexcl f (xc))

2 C is a maximal clique of G if and only if xc is a local maximizer of Of in Sn

3 All local (and hence global) maximizers of Of over Sn are strict

Unlike the Motzkin-Straus formulation the previous result guaranteesthat all maximizers of Of on Sn are strict and are characteristic vectors ofmaximal or maximum cliques in the graph In an exact sense therefore aone-to-one correspondence exists between maximal cliques and local max-imizers of Of in Sn on the one hand and maximum cliques and global max-imizers on the other hand This solves the spurious solution problem in adenitive manner

23 A Quadratic Program for Graph Isomorphism In the light of theabove discussion it is now a straightforward exercise to formulate the graphisomorphism problem in terms of a standard quadratic programming prob-lem Let G0 and G00 be two arbitrary graphs of order n and let A denote theadjacency matrix of the corresponding association graph whose order isN D n2 The graph isomorphism problem is equivalent to the followingprogram

maximize Of (x) D xT (A C 12 IN)x

subject to x 2 SN (23)

More precisely the following result holds which is a straightforward con-sequence of theorems 1 and 2

Replicator Equations Maximal Cliques and Graph Isomorphism 1939

Theorem 3 Let G0 D (V0 E0 ) and G00 D (V00 E00 ) be two graphs of order nand let xcurren be a global solution of program 23 where A is the adjacency matrix ofthe association graph of G0 and G00 Then G0 and G00 are isomorphic if and only ifOf (xcurren) D 1 iexcl 1 2n In this case any global solution to 23 induces an isomorphism

between G0 and G00 and vice versa In general local and global solutions to 23are in one-to-one correspondence with maximal and maximum common subgraphisomorphisms between G0 and G00 respectively

Note that the adjacency matrix A D (aih jk) of the association graph canbe explicitly written as follows

aih jk Draquo

1 iexcl (a0ij iexcl a00

hk)2 if i 6D j and h 6D k

0 otherwise

where A0 D (a0ij) and A00 D (a00

hk) are the adjacency matrices of G0 and G00

respectively The regularized Motzkin-Straus objective function Of thereforebecomes

Of (x) DX

ih

X

j 6Di

X

k 6Dh

a0ija

00hkxihxjk

CX

ih

X

j 6Di

X

k 6Dh

(1 iexcl a0ij)(1 iexcl a00

hk)xihxjk C12

X

ih

x2ih (24)

Many interesting observations about the previous objective function canbe made It consists of three terms The rst is identical to the one usedin Mjolsness et al (1989) Gold and Rangarajan (1996) Rangarajan et al(1996) and Rangarajan and Mjolsness (1996) which derives from the so-called rectangle rule Intuitively by restricting ourselves to binary variablesxih 2 f0 1g it simply counts the number of consistent ldquorectanglesrdquo betweenG0 and G00 that are induced by the tentative solution x The second termis new and by analogy with the rectangle rule can be derived from whatcan be called the antirectangle rule in case of binary variables it countsthe number of rectangles between the complements of the original graphs2

Finally the third term in equation 24 which has been added to avoid spu-rious solutions in the Motzkin-Straus program is just the self-amplicationterm introduced in a different context by Yuille and Kosowsky (1994) forthe related purpose of ensuring that the minimizers of a generic quadraticfunction in the unit hypercube lie at the vertices The self-amplication termhas also been employed recently in Rangarajan et al (1996) and Rangarajanand Mjolsness (1996) Like ours the self-amplication term has the formc

Pih x2

ih but the parameter c depends on the structure of the quadratic

2 The complement of a graph G D (V E) is the graph G D (V E) such that (i j) 2 E (i j) 2 E

1940 Marcello Pelillo

program matrix In our case c D 12 and it can easily be proved that theo-

rem 2 holds true for all c 2 (0 1) Its choice is therefore independent of thestructure of the matrix A and only affects the basins of attraction aroundlocal optima3

3 Replicator Equations and Graph Isomorphism

31 The Model and Its Properties Replicator equations have been de-veloped and studied in the context of evolutionary game theory a disciplinepioneered by J Maynard Smith (1982) that aims to model the evolution ofanimal behavior using the principlesand tools of game theory In this sectionwe discuss the basic intuition behind replicator equations and present a fewtheoretical properties that will be instrumental in the subsequent develop-ment of our graph isomorphism algorithm For a more systematic treatmentsee Hofbauer and Sigmund (1988) and Weibull (1995)

Consider a large population of individuals belonging to the same speciesthat compete for a particular limited resource such as food or territory Thiskind of conict is modeled as a game the players being pairs of randomlyselected population members In contrast to traditional application elds ofgame theory such as economics or sociology (Luce amp Raiffa 1957) playershere do not behave rationally but act instead according to a preprogrammedbehavior pattern or pure strategy Reproduction is assumed to be asexualwhich means that apart from mutation offspring will inherit the samegenetic material and hence behavioral phenotype as their parents Let J Df1 ng be the set of pure strategies and for all i 2 J let xi(t) be the relativefrequency of population members playing strategy i at time t The state ofthe system at time t is simply the vector x(t) D (x1(t) xn(t))T

One advantage of applying game theory to biology is that the notionof utility is much simpler and clearer than in human contexts Here aplayer rsquos utility can be measured in terms of Darwinian tness or repro-ductive successmdashthe player rsquos expected number of offspring Let W D (wij)be the n poundn payoff (or tness) matrix Specically for each pair of strategiesi j 2 J wij represents the payoff of an individual playing strategy i againstan opponent playing strategy j Without loss of generality we shall assumethat the payoff matrix is nonnegative that is wij cedil 0 for all i j 2 J At timet the average payoff of strategy i is given by

p i(t) DnX

jD1

wijxj(t)

while the mean payoff over the entire population isPn

iD1 xi (t)p i(t)

3 The effects of allowing c to take on negative values and of varying it during theoptimization process are studied in Bomze Budinich Pelillo and Rossi (1999)

Replicator Equations Maximal Cliques and Graph Isomorphism 1941

In evolutionary game theory the assumption is made that the game isplayed over and over generation after generation and that the action of nat-ural selection will result in the evolution of the ttest strategies If successivegenerations blend into each other the evolution of behavioral phenotypescan be described by the following set of differential equations (Taylor ampJonker 1978)

Pxi (t) D xi(t)

0

p i(t) iexclnX

jD1

xj(t)pj(t)

1

A i D 1 n (31)

where a dot signies derivative with respect to time The basic idea behindthis model is that the average rate of increase Pxi(t) xi(t) equals the differencebetween the average tness of strategy i and the mean tness over the entirepopulation It is straightforward to show that the simplex Sn is invariantunder equation 31 or in other words any trajectory starting in Sn willremain in Sn To see this simply note that d

dtP

i xi(t) DP

i Pxi(t) D 0 whichmeans that the interior of Sn (the set dened by xi gt 0 for all i D 1 n)is invariant The additional observation that the boundary too is invariantcompletes the proof

Similar arguments provide a rationale for the following discrete-timeversion of the replicator dynamics assuming nonoverlapping generations

xi (t C 1) Dxi(t)p i(t)PnjD1 xj(t)pj (t)

i D 1 n (32)

Because of the nonnegativity of the tness matrix W and the normalizationfactor this system too makes the simplex Sn invariant as its continuouscounterpart

A point x D x(t) is said to be a stationary (or equilibrium) point for ourdynamical systems if Pxi(t) D 0 in the continuous-time case and xi (t C 1) Dxi(t) in the discrete-time case (i D 1 n) Moreover a stationary point issaid to be asymptotically stable if any trajectory starting in its vicinity willconverge to it as t 1 It turns out that both the continuous-time anddiscrete-time replicator dynamics have the same set of stationary pointsthat is all the points in Sn satisfying the condition

xi (t)

0

p i(t) iexclnX

jD1

xj(t)p j(t)

1

A D 0 i D 1 n (33)

or equivalently p i(t) DPn

jD1 xj(t)p j(t) whenever xi gt 0Equations 31 and 32 arise independently in different branches of theo-

retical biology (Hofbauer amp Sigmund 1988) In population ecology for ex-ample the famous Lotka-Volterra equations for predator-prey systems turn

1942 Marcello Pelillo

out to be equivalent to the continuous-time dynamics (see equation 31)under a simple barycentric transformation and a change in velocity In pop-ulation genetics they are known as selection equations (Crow amp Kimura1970) In this case each xi represents the frequency of the ith allele Ai andthe payoff wij is the ldquotnessrdquo of genotype AiAj Here the tness matrix Wis always symmetric The discrete-time dynamical equations turn out tobe a special case of a general class of dynamical systems introduced byBaum and Eagon (1967) and studied by Baum and Sell (1968) in the contextof Markov chain theory They also represent an instance of the so-calledrelaxation labeling processes a class of parallel distributed algorithms de-veloped in computer vision to solve (continuous) constraint satisfactionproblems (Rosenfeld Hummel amp Zucker 1976 Hummel amp Zucker 1983)An independent connection between dynamical systems such as relaxationlabeling and Hopeld-style networks and game theory has recently beendescribed by Miller and Zucker (1991 1992)

We are now interested in studying the dynamical properties of replicatorsystems it is these properties that will allow us to employ them for solvingthe graph isomorphism problem The following theorem states that underreplicator dynamics the populationrsquos average tness always increases pro-vided that the payoff matrix is symmetric (in game theory terminology thissituation is referred to as a doubly symmetric game)

Theorem 4 Suppose that the (nonnegative) payoff matrix W is symmetric (iewij D wji for all i j D 1 n) The quadratic polynomial F dened as

F(x) DnX

iD1

nX

jD1

wijxixj (34)

is strictly increasing along any nonconstant trajectory of both continuous-time(see equation 31) and discrete-time (see equation 32) replicator equations In otherwords for all t cedil 0 we have

ddt

F(x(t)) gt 0

for system 31 and

F(x(t C 1)) gt F(x(t))

for system 32 unless x(t) is a stationary point Furthermore any such trajectoryconverges to a (unique) stationary point

The previous result isknown in mathematical biology as the fundamentaltheorem of natural selection (Crow amp Kimura 1970 Hofbauer amp Sigmund

Replicator Equations Maximal Cliques and Graph Isomorphism 1943

1988 Weibull 1995) and in its original form traces back to Fisher (1930) Asfar as the discrete-time model is concerned it can be regarded as a straight-forward implication of the Baum-Eagon theorem (Baum amp Eagon 1967Baum amp Sell 1968) which is valid for general polynomial functions overproduct of simplices Waugh and Westervelt (1993) also proved a similar re-sult for a related class of continuous- and discrete-time dynamical systemsIn the discrete-time case however they put bounds on the eigenvalues ofW in order to achieve convergence to xed points

The fact that all trajectories of the replicator dynamics converge to a sta-tionary point has been proved more recently (Losert amp Akin 1983 LyubichMaistrowskii amp Olrsquokhovskii 1980) However in general not all stationarypoints are local maximizers of F on Sn The vertices of Sn for example areall stationary points for equations 31 and 32 whatever the landscape of FMoreover there may exist trajectories that starting from the interior of Sneventually approach a saddle point of F However a result recently provedby Bomze (1997) asserts that all asymptotically stable stationary points ofreplicator dynamics correspond to (strict) local maximizers of Of on Sn andvice versa

32 Application to Graph Isomorphism Problems The properties dis-cussed in the preceding subsection naturally suggest using replicator equa-tions as a useful heuristic for the graph isomorphism problem Let G0 D(V0 E0 ) and G00 D (V00 E00 ) be two graphs of order n and let A denote theadjacency matrix of the corresponding N-vertex association graph G Byletting

W D A C12

IN

we know that the replicator dynamical systems starting from an arbitraryinitial state will iteratively maximize the function Of (x) D xT (A C 1

2 IN)x inSN and eventually converge to a strict local maximizer that by virtue oftheorem 2 will then correspond to the characteristic vector of a maximalclique in the association graph4 We know from theorem 3 that this willin turn induce an isomorphism between two subgraphs of G0 and G00 thatis maximal in the sense that there is no other isomorphism between sub-graphs of G0 and G00 that includes the one found Clearly in theory there is noguarantee that the converged solution will be a global maximizer of Of andtherefore that it will induce a maximum isomorphism between the two orig-inal graphs However previous experimental work done on the maximumclique problem (Bomze Pelillo amp Giacomini 1997 Pelillo 1995) and also

4 Because of the presence of saddle points the algorithm occasionally may convergetoward one such points However since the set of saddle points is of measure zero thishappens with probability tending to zero

1944 Marcello Pelillo

the results presented in this article suggest that the basins of attraction ofglobal maximizers are quite large and frequently the algorithm convergesto one of them Without any heuristic information about the optimal solu-tion it is customary to start out the replicator process from the barycenterof the simplexmdashthat is the vector ( 1

N 1N )T This choice ensures that no

particular solution is favoredThe emergent matching strategy of our replicator model is identical to

the one adopted by Simi Acirccrsquos algorithm (Simi Acircc 1991) which is speculatedto be similar to that employed by humans in solving matching problemsSpecically it seems that the algorithm rst tries to match what Simi Acircc calledthe notable verticesmdashthat is those vertices having highest or lowest con-nectivity To illustrate consider two vertices i 2 V0 and h 2 V00 and assumefor simplicity that they have the same degree deg(i) D deg(h) It is easy toshow that the corresponding vertex in the association graph has at most de-gree deg(i h) D

iexcldeg(i)2

centC

iexclniexcl1iexcldeg(i)2

centD deg2(i) iexcl (niexcl1) deg(i) C

iexclniexcl12

cent which

attains its minimum value when deg(i) D niexcl12 and maximum value when

deg(i) equals 0 or n iexcl 1 It follows that pairs of notable vertices give rise tovertices in the association graph having the largest degree Now considerwhat happens at the very rst iterations of our clique-nding relaxationprocess assuming as is customary that it is started from the barycenter ofSN At t D 0 the average payoff of a vertex (i h) in the association graphis p ih(0) D 1

NP

jk aih jk C 12N D 1

2N (2 deg(i h) C 1) Because of the payoffmonotonicity property of replicator dynamics (cf equations 42 and 44 inthe next section) this implies that at the very beginning of the relaxation pro-cess the components corresponding to pairs of notable vertices will growat a higher rate thereby imposing a sort of partial ordering over the set ofpossible assignments Clearly this simplied picture is no longer valid afterthe rst few iterations when local information begins to propagate

We illustrate this behavior with the aid of a simple example Consider thetwo isomorphic graphs in Figure 1 Our matching strategy would suggestrst matching vertex 1 to vertex A then 2 to B and nally either 3 to Cand 4 to D or 3 to D and 4 to C These are the only possible isomorphismsbetween the two graphs As shown in Figure 2 this is exactly what our al-gorithm accomplishes The gure plots the evolution of each component ofthe state vector x(t) a 16-dimensional vector under the replicator dynamics(see equation 32) Observe how after rapidly trying to match 1 to A and 2to B it converges to a saddle point which indeed incorporates the informa-tion regarding the two possible isomorphisms After a slight perturbationat around the seventy-fth step the process makes a choice and quicklyconverges to one of the two correct solutions

33 Experimental Results In order to assess the effectiveness of the pro-posed approach extensive simulations were performedover randomly gen-erated graphs of various connectivities Random graphs represent a useful

Replicator Equations Maximal Cliques and Graph Isomorphism 1945

1

2

3 4

A

B

C D

Figure 1 A pair of isomorphic graphs

benchmark not only because they are not constrained to any particular ap-plication but also because it is simple to replicate experiments and henceto make comparisons with other algorithms Before going into the detailsof the experiments however we need to enter a preliminary caveat

It is often said that random graph isomorphism is trivial Essentially thisclaim is based on a result due to Babai Erdos and Selkow (1980) whichshows that a straightforward linear-time graph isomorphism algorithm

Iterations

Co

mp

on

ents

of

stat

e ve

cto

r

0

005

0 1

015

0 2

025

0 3

035

0 4

045

0 5 0 1 00 1 50 20 0

(1 A)

(2B) (3C) and (4D)

(3D) and (4C)

Figure 2 Evolution of the components of the state vector x(t) for the graphs inFigure 1 using the replicator dynamics (see equation 32)

1946 Marcello Pelillo

does work for almost all random graphs5 It should be pointed out how-ever that there are various probability models for random graphs (Palmer1985) The one adopted by Babai et al (1980) considers random graphs asuniformly distributed random variables they assume that the probabilityof generating any n-vertex graph equals 2iexcl(n

2) By contrast the customaryway in which random graphs are generated leads to a distribution that isuniform only in a special case Specically given a parameter p (0 lt p lt 1)that represents the expected connectivity a graph of order n is generatedby randomly entering edges between the vertices with probability p Notethat p is related to the expected size of the resulting graph which indeedis

iexcln2

centp It is straightforward to see that in so doing the probability that a

graph of order n and size s be generated is given by ps(1 iexcl p)(n2)iexcls which

only in the case p D 12 equals Babairsquos uniform distribution The results pre-

sented by Babai et al (1980) are based on the observation that by using auniform probability model the degrees of the vertices have large variabilityand this is in fact the key to their algorithm In the nonuniform probabilitymodel the degree random variable has variance (n iexcl 1)p(1 iexcl p) and it isno accident that it attains its largest value exactly at p D 1

2 However as pmoves away from 1

2 the variance becomes smaller and smaller tending to0 as p approaches 0 or 1 As a result Babai et alrsquos arguments are no longerapplicable It therefore seems that using the customary graph generationmodel random graph isomorphism is not as trivial as is generally believedespecially for very sparse and very dense graphs In fact the experiencereported by Rangarajan et al (1996) Rangarajan and Mjolsness (1996) andSimi Acircc (1991) and also the results presented below provide support to thisclaim

In the experiments reported here the algorithm was started from thebarycenter of the simplex and stopped when either a maximal clique (alocal maximizer of Of on Sn) was found or the distance between two suc-cessive points was smaller than a xed threshold which was set to 10iexcl17In the latter case the converged vector was randomly perturbed and thealgorithm restarted from the perturbed point Because of the one-to-onecorrespondence between local maximizers and maximal cliques this situa-tion corresponds to convergence to a saddle point All the experiments wererun on a Sparc20

Undirected 100-vertex random graphs were generated with expectedconnectivities ranging from 1 to 99 Specically the values of the edge-probability p were as follows 001 003 005 095 097 099 and from 01 to09 in steps of 01 For each connectivity value 100 graphs were producedand each had its vertices randomly permuted so as to obtain a pair of iso-morphic graphs Overall 1500 pairs of isomorphic graphs were generated

5 A property is said to hold for almost all graphs if the probability that the propertyholds tends to 1 as the order of the graph approaches innity

Replicator Equations Maximal Cliques and Graph Isomorphism 1947

To keep the order of the association graph as low as possible its vertexset was constructed as follows

V Dcopy(i h) 2 V0 pound V00 deg(i) D deg(h)

ordf

the edge set E being dened as in denition 1 It is straightforward to seethat when the graphs are isomorphic theorem 1 continues to hold sinceisomorphisms preserve the degree property of the vertices This simpleheuristic may signicantly reduce the dimensionality of the search space

Each pair of isomorphic graphs was given as input to the replicatormodel after convergence a success was recorded when the cardinality ofthe returned clique was equal to the order of the graphs given as input (thatis 100)6 Because of the stopping criterion employed this guarantees thata maximum clique and therefore a correct isomorphism was found Fig-ure 3a plots the proportion of successes as a function of p and Figure 3bshows the average CPU time (in logarithmic scale) taken by the algorithmto converge

These results are signicantly superior to those Simi Acircc (1991) reportedpoor results at connectivities less than 40 even on smaller graphs (up to75 vertices) They also compare favorably with the results obtained morerecently by Rangarajan et al (1996) on 100-vertex random graphs for con-nectivities up to 50 Specically at 1 and 3 connectivities they reporta percentage of correct isomorphisms of about 0 and 30 respectivelyUsing our approach we obtained on the same kind of graphs a percentageof success of 10 and 56 respectively Rangarajan and Mjolsness (1996)also ran experiments on 100-vertex random graphs with various connec-tivities using a powerful Lagrangian relaxation network Except for a fewinstances they always obtained a correct solution The computational timerequired by their model however turns out to exceed ours greatly As an ex-ample the average time their algorithm took to match two 100-vertex 50-connectivity graphs was about 30 minutes on an SGI workstation As shownin Figure 3b we obtained identical results in about 3 seconds However forvery sparse and very dense graphs our algorithm becomes extremely slowIn the next section we present an exponential version of our replicator dy-namics which turns out to be dramatically faster and even more accuratethan the classical model 32

All of the algorithms mentioned above do incorporate sophisticated an-nealing mechanisms to escape from poor local minima By contrast in thepresented work no attempt was made to prevent the algorithm from con-verging to such solutions It seems that as far as the graph isomorphismproblem is concerned global maximizers of the Motzkin-Straus objective

6 Due to the high computational time required in the p D 001 and p D 099 cases thealgorithm was tested on only 10 pairs instead of 100

1948 Marcello Pelillo

Expected connectivity

Per

cen

tag

e o

f co

rrec

t is

om

orp

his

m

0

2 5

5 0

7 5

1 00

001 003 005 01 02 03 04 05 06 07 08 09 095 097 099

(a)

Expected connectivity

Ave

rag

e C

PU

tim

e (s

ecs)

1

1 0

100

1000

10000

100000

001 0 03 005 01 0 2 03 04 05 06 07 08 09 095 097 099

(plusmn215818)

(plusmn2030)

(plusmn4382)

(plusmn196)

(plusmn107)

(plusmn065)

(plusmn069)

(plusmn058)

(plusmn094)

(plusmn226)

(plusmn4569)

(plusmn29486)

(plusmn190717)

(plusmn1310701) (plusmn1707976)

(b)

Figure 3 Results obtained over 100-vertex graphs of various connectivities us-ing dynamics 32 (a) Percentage of correct isomorphisms (b) Average computa-tional time taken by the replicator equations The vertical axis is in logarithmicscale and the numbes in parentheses represent the standard deviation

have large basins of attraction A similar observation was also made inconnection to earlier experiments concerning the maximum clique problem(Bomze et al 1997 Pelillo 1995)

Replicator Equations Maximal Cliques and Graph Isomorphism 1949

4 Faster Replicator Dynamics

Recently there has been much interest in evolutionary game theory aroundthe following exponential version of replicator equations which arises asa model of evolution guided by imitation (Hofbauer 1995 Hofbauer ampWeibull 1996 Weibull 1994 1995)

Pxi (t) D xi(t)

sup3ekp i(t)

PnjD1 xj(t)ekpj (t)

iexcl 1

i D 1 n (41)

where k is a positive constant As k tends to 0 the orbits of this dynamicsapproach those of the standard ldquorst-orderrdquo replicator model 31 slowedby the factor k moreover for large values ofk the model approximates theso-called best-reply dynamics (Hofbauer amp Weibull 1996) It is readily seenthat dynamics 41 is payoff monotonic (Weibull 1995) which means that

Pxi (t)xi (t)

gtPxj(t)

xj(t) p i (t) gt pj(t) (42)

for i j D 1 n This amounts to stating that during the evolution processthe components corresponding to higher payoffs will increase at a higherrate Observe that the rst-order replicator model 31 also is payoff mono-tonic The class of payoff monotonic dynamics possesses several interestingproperties (Weibull 1995) In particular all have the same set of stationarypoints which are characterized by equation 33 Moreover when the tnessmatrix W is symmetric the average population payoff dened in equa-tion 34 is also strictly increasing as in the rst-order case (see Hofbauer1995 forproof)After discussing various propertiesof payoff monotonic dy-namics Hofbauer (1995) has recently concluded that they behave essentiallyin the same way as the standard replicator equations the only differencebeing the size of the basins of attraction around stable equilibria

A customary way of discretizing equation 41 is given by the followingdifference equations (Cabrales amp Sobel 1992 Gaunersdorfer amp Hofbauer1995) which is also similar to the ldquoself-annealingrdquo dynamics recently intro-duced by Rangarajan (1997)

xi (t C 1) Dxi (t)ekp i(t)

PnjD1 xj(t)ekpj (t)

i D 1 n (43)

As its continuous counterpart this dynamics is payoff monotonic that is

xi (t C 1) iexcl xi(t)xi(t)

gtxj(t C 1) iexcl xj (t)

xj(t) p i(t) gt p j(t) (44)

1950 Marcello Pelillo

for all i j D 1 n Observe that the standard discrete-time equations 32also possess this property

From our computational perspective exponential replicator dynamicsare particularly attractive because as demonstrated by the extensive nu-merical results reported below they seem to be considerably faster andeven more accurate than the standard rst-order model To illustrate inFigure 4 the behavior of the dynamics 43 in matching the simple graphs ofFigure 1 is shown for various choices of the parameter k Notice how thequalitative behavior of the algorithm is the same as the rst-order modelbut now convergence is dramatically faster (cf Figure 2) In this examplethe process becomes unstable when k D 5 suggesting as expected that thechoice of this parameter is a trade-off between speed and stability Unfor-tunately there is no theoretical principle to choose this parameter properly

To test the validity of this new model on a larger scale we conducted asecond series of experimentsover the same 1500 graphs generated for testingthe rst-order dynamics The discrete-time equations 43 were used andthe parameter k was heuristically set to 10 The process was started fromthe barycenter of the simplex and stopped using the same criterion used inthe previous set of experiments Figure 5 shows the percentage of successesobtained for the various connectivity values and the average CPU timetaken by the algorithm to converge (in logarithmic scale) It is evident fromthese results that the exponential replicator system 43 may be dramaticallyfaster than the rst-order model 32 and may also provide better results

5 Conclusion

In this article we have developed a new energy-minimization frameworkfor the graph isomorphism problem that is centered around an equivalentmaximum clique formulation and the Motzkin-Straus theorem a remark-able result that establishes an elegant connection between the maximumclique problem and a certain standard quadratic programThe attractive fea-ture of the proposed formulation is that a clear one-to-one correspondenceexists between the solutions of the quadratic program and those in the origi-nal discrete problemWehave then introduced the so-called replicatorequa-tions a class of continuous- and discrete-time dynamical systems developedin evolutionary game theory and various other branches of theoretical biol-ogy and have shown how they naturally lend themselves to approximatelysolving the Motzkin-Straus program The extensive experimental resultspresented show that despite their simplicity and their inherent inability toescape from local optima replicator dynamics are nevertheless able to pro-vide solutions that are competitive with more sophisticated deterministicannealing algorithms in terms of both quality of solutions and speed

Our framework is moregeneral than presented here and we are now em-ploying it for solving more general subgraph isomorphism and relationalstructure matching problems Preliminary experiments seem to indicate that

Replicator Equations Maximal Cliques and Graph Isomorphism 1951

Figure 4 Evolution of the components of the state vector x(t) for the graphsin Figure 1 using the exponential replicator model equation 43 for differentvalues of the parameter k

1952 Marcello Pelillo

Expected connectivity

Per

cen

tag

e o

f co

rrec

t is

om

orp

his

m

0

2 5

5 0

7 5

1 00

001 003 005 01 02 03 04 05 06 07 08 09 095 097 099

(a)

Expected connectivity

Ave

rag

e C

PU

tim

e (s

ecs)

1

1 0

100

1000

10000

001 003 005 0 1 02 0 3 04 05 06 07 0 8 09 0 95 097 099

(plusmn109148)

(plusmn11226)

(plusmn594)

(plusmn700)

(plusmn071)

(plusmn065)

(plusmn053)

(plusmn056)

(plusmn054)

(plusmn056)

(plusmn089)

(plusmn650)

(plusmn625)

(plusmn9878)

(plusmn120370)

(b)

Figure 5 Results obtained over 100-vertex graphs of various connectivitiesusing the exponential dynamics 43 (a) Percentage of correct isomorphisms(b) Average computational time taken by the replicator equations The verti-cal axis is in logarithmic scale and the numbers in parentheses represent thestandard deviation

local optima might represent a problem here especially in matching verysparse or dense graphs Escape procedures like those developed in Bomze(1997) and Bomze et al (1999) would be helpful in these cases to avoidthem Nevertheless local solutions in the continuous domain always have

Replicator Equations Maximal Cliques and Graph Isomorphism 1953

a meaningful interpretation in terms of maximal commonsubgraph isomor-phisms and this is one of the major advantages of the presented approachWe are currently conducting a thorough investigation and plan to presentthe results in a forthcoming article The approach is also being applied withsuccess to the problem of matching hierarchical structures with applicationto shape matching problems arising in computer vision (Pelillo Siddiqi ampZucker 1999)

Acknowledgments

This work was done while I was visiting the Department of ComputerScience at Yale University it was supported by Consiglio Nazionale delleRicerche Italy I thank I M Bomze A Rangarajan K Siddiqi andS W Zucker for many stimulating discussions and for providing commentson an earlier version of the article and the anonymous reviewers for con-structive criticism

References

Arora S Lund C Motwani R Sudan M amp Szegedy M (1992) Proof veri-cation and the hardness of approximation problems In Proc 33rd Ann SympFound Comput Sci (pp 14ndash23) Pittsburgh PA

Babai L Erdos P amp Selkow S M (1980) Random graph isomorphism SIAMJ Comput 9(3) 628ndash635

Barrow H G amp Burstall R M (1976) Subgraph isomorphism matching rela-tional structures and maximal cliques Inform Process Lett 4(4) 83ndash84

Baum L E amp Eagon J A (1967) An inequality with applications to statisticalestimation for probabilistic functions of Markov processes and to a modelfor ecology Bull Amer Math Soc 73 360ndash363

Baum L E amp Sell G R (1968) Growth transformations for functions on man-ifolds Pacic J Math 27(2) 211ndash227

Bellare M Goldwasser S amp Sudan M (1995) Free bits PCPs and non-approximabilitymdashTowards tight results In Proc 36thAnn Symp Found Com-put Sci (pp 422ndash431) Milwaukee WI

Bomze I M (1997) Evolution towards the maximum clique J Global Optim10 143ndash164

Bomze I M Budinich M Pardalos P M amp Pelillo M (1999) The maximumclique problem In D Z Du amp P M Pardalos (Eds) Handbook of combinatorialoptimization vol 4 Boston MA Kluwer

Bomze I M Budinich M Pelillo Mamp Rossi C (1999)Annealed replicationAnew heuristic for the maximum clique problem To appear in Discrete AppliedMathematics

Bomze I M Pelillo M amp Giacomini R (1997) Evolutionary approach tothe maximum clique problem Empirical evidence on a larger scale InI M Bomze T Csendes R Horst amp P M Pardalos (Eds) Developmentsin global optimization (pp 95ndash108) Dordrecht Netherlands Kluwer

1954 Marcello Pelillo

Boppana R B Hastad J amp Zachos S (1987)Does co-NP have short interactiveproofs Inform Process Lett 25 127ndash132

Cabrales A amp Sobel J (1992)On the limit points of discrete selection dynamicsJ Econ Theory 57 407ndash419

Crow J F amp Kimura M (1970)An introductionto populationgenetics theory NewYork Harper amp Row

Fisher R A (1930) The genetical theory of natural selection London Oxford Uni-versity Press

Garey M R amp Johnson D S (1979) Computers and intractability A guide to thetheory of NP-completeness San Francisco W H Freeman

Gaunersdorfer A amp Hofbauer J (1995) Fictitious play Shapley polygons andthe replicator equation Games Econ Behav 11 279ndash303

Gibbons L E Hearn D W amp Pardalos P M (1996) A continuous basedheuristic for the maximum clique problem In D S Johnson amp M Trick (Eds)Cliques coloring and satisabilitymdashSecond DIMACS implementation challenge(pp 103ndash124) Providence RI American Mathematical Society

Gibbons L E Hearn D W Pardalos P M amp Ramana M V (1997)Continuouscharacterizations of the maximum clique problem Math Oper Res 22(3)754ndash768

Gold S amp Rangarajan A (1996) A graduated assignment algorithm for graphmatching IEEE Trans Pattern Anal Machine Intell 18 377ndash388

Grotschel M Lov Acircasz L amp Schrijver A (1988) Geometric algorithms and combi-natorial optimization Berlin Springer-Verlag

Hastad J (1996) Clique is hard to approximate within n1iexcl2 In Proc 37th AnnSymp Found Comput Sci (pp 627ndash636) Burlington VT

Hofbauer J (1995) Imitation dynamics for games Unpublished manuscriptCollegium Budapest

Hofbauer J amp Sigmund K (1988) The theory of evolution and dynamical systemsCambridge Cambridge University Press

Hofbauer J amp Weibull J W (1996) Evolutionary selection against dominatedstrategies J Econ Theory 71 558ndash573

Hopeld J J amp Tank D W (1985) ldquoNeuralrdquo computation of decisions in opti-mization problems Biol Cybern 52 141ndash152

Hummel R A amp Zucker S W (1983)On the foundations of relaxation labelingprocesses IEEE Trans Pattern Anal Machine Intell 5 267ndash287

Johnson D S (1988) The NP-completeness column An ongoing guide J Algo-rithms 9 426ndash444

Kozen D (1978) A clique problem equivalent to graph isomorphism SIGACTNews pp 50ndash52

Kree R amp Zippelius A (1988) Recognition of topological features of graphsand images in neural networks J Phys A Math Gen 21 L813ndashL818

Losert V amp Akin E (1983) Dynamics of games and genes Discrete versuscontinuous time J Math Biol 17 241ndash251

Luce R D amp Raiffa H (1957) Games and decisions New York WileyLyubich Yu Maistrowskii G D amp Olrsquokhovskii Yu G (1980) Selection-

induced convergence to equilibrium in a single-locus autosomal populationProblems of Information Transmission 16 66ndash75

Replicator Equations Maximal Cliques and Graph Isomorphism 1955

Maynard Smith J (1982) Evolution and the theory of games Cambridge Cam-bridge University Press

Miller D A amp Zucker S W (1991) Copositive-plus Lemke algorithm solvespolymatrix games Oper Res Lett 10 285ndash290

Miller D A amp Zucker S W (1992)Efcient simplex-like methods for equilibriaof nonsymmetric analog networks Neural Computation 4 167ndash190

Mjolsness E Gindi G amp Anandan P (1989)Optimization in model matchingand perceptual organization Neural Computation 1 218ndash229

Motzkin T S amp Straus E G (1965) Maxima for graphs and a new proof of atheorem of Tur Acircan Canad J Math 17 533ndash540

Palmer E M (1985) Graphical evolution An introduction to the theory of randomgraphs New York Wiley

Pardalos P M amp Phillips A T (1990) A global optimization approach forsolving the maximum clique problem Int J Computer Math 33 209ndash216

Pelillo M (1995) Relaxation labeling networks for the maximum clique prob-lem J Artif Neural Networks 2 313ndash328

Pelillo M amp Jagota A (1995) Feasible and infeasible maxima in a quadraticprogram for maximum clique J Artif Neural Networks 2 411ndash420

Pelillo M Siddiqi K amp Zucker S W (1999) Matching hierarchical structuresusing association graphs To appear in IEEE TransPatternAnal Machine Intell

Rangarajan A (1997) Self-annealing Unifying deterministic annealing andrelaxation labeling In M Pelillo amp E R Hancock (Eds) Energy minimiza-tion methods in computer vision and pattern recognition (pp 229ndash244) BerlinSpringer-Verlag

Rangarajan A Gold S amp Mjolsness E (1996) A novel optimizing networkarchitecture with applications Neural Computation 8 1041ndash1060

Rangarajan A amp Mjolsness E (1996) A Lagrangian relaxation network forgraph matching IEEE Trans Neural Networks 7(6) 1365ndash1381

Rosenfeld A Hummel R A amp Zucker S W (1976) Scene labeling by relax-ation operations IEEE Trans Syst Man Cybern 6 420ndash433

Schoning U (1988)Graph isomorphism is in the low hierarchy J Comput SystSci 37 312ndash323

Simi Acircc P D (1991) Constrained nets for graph matching and other quadraticassignment problems Neural Computation 3 268ndash281

Taylor P amp Jonker L (1978)Evolutionarily stable strategies and game dynam-ics Math Biosci 40 145ndash156

Waugh F R amp Westervelt R M (1993) Analog neural networks with localcompetition I Dynamics and stability Phys Rev E 47(6) 4524ndash4536

Weibull J W (1994)The ldquoas ifrdquo approach to game theory Three positive resultsand four obstacles Europ Econ Rev 38 868ndash881

Weibull J W (1995) Evolutionary game theory Cambridge MA MIT PressWilf H S (1986) Spectral bounds for the clique and independence numbers of

graphs J Combin Theory Ser B 40 113ndash117Yuille ALamp Kosowsky J J (1994)Statisticalphysics algorithms that converge

Neural Computation 6 341ndash356

Received January 27 1998 accepted December 11 1998