Landscapes and Their Correlation FunctionsPeter F. Stadlera;b;�aTheoretische Biochemie, Institut f�ur Theoretische ChemieUniversit�at Wien, Vienna, AustriabSanta Fe Institute, Santa Fe, NM�Mailing Address:Peter F Stadler, Inst. f. Theoretische Chemie, Univ. WienW�ahringerstr. 17, A-1090 Wien, AustriaPhone: [43] 1 40480 665 Fax: [43] 1 40480 660E-Mail: studla@tbi.univie.ac.atKeywordsCoherent Con�guration| Combinatorial Optimization| Correlation Function| Fourier Series| Landscapes | Laplace Operator | Random Walk

P.F. Stadler: Landscapes and Their Correlation FunctionsAbstractFitness landscapes are an important concept in molecular evolution. Many important examples oflandscapes in physics and combinatorial optimization, which are widely used as model landscapesin simulations of molecular evolution and adaptation, are \elementary", i.e., they are (up to anadditive constant) eigenfunctions of a graph Laplacian. It is shown that elementary landscapesare characterized by their correlation functions. The correlation functions are in turn uniquelydetermined by the geometry of the underlying con�guration space and the nearest neighborcorrelation of the elementary landscape. Two types of correlation functions are investigatedhere: the correlation of a time series sampled along a random walk on the landscape and thecorrelation function with respect to a partition of the set of all vertex pairs.1. IntroductionSince Sewall Wright's seminal paper [1] the notion of a �tness landscape underly-ing the dynamics of evolutionary optimization has proved to be one of the mostpowerful concepts in evolutionary theory. Implicit in this idea is a collection ofgenotypes arranged in an abstract metric space, with each genotype next to thoseother genotypes which can be reached by a single mutation, as well as a valueassigned to each genotype. Such a construction is by no means restricted to bio-logical evolution; Hamiltonians of disordered systems, such as spin glasses [2, 3],and the cost functions of combinatorial optimization problems [4] have the samebasic structure. It has been known since Eigen's [5] pioneering work on the molec-ular quasispecies that the dynamics of evolutionary adaptation (optimization) ona landscape depends crucially on detailed structure of the landscapes itself. Exten-sive computer simulations, see, e.g., [6, 7] have made it very clear that a completeunderstanding of the dynamics is impossible without a thorough investigation ofthe underlying landscape [8].The landscapes of a number of well known combinatorial optimization problemssuch as the Traveling Salesman Problem TSP [9], the Graph Bipartitioning ProblemGBP [10], or the Graph Matching Problem GMP have been investigated in somedetail, see [11, 12, 13]. A detailed survey of a variety of model landscapes derivedfrom folding RNA molecules into their secondary structures has been performedrecently [14, 6, 7, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].{ 1 {

P.F. Stadler: Landscapes and Their Correlation FunctionsMost of the knowledge about landscapes has so far been derived using statisticalmethods, considering randommodels of landscapes rather than a single landscape.The distribution of local optima and the statistical characteristics of down-hillwalks have been computed for the uncorrelated landscape of the random energymodel [25, 26, 27], Furthermore, two one-parameter families of tunably ruggedlandscapes have been studied extensively: the Nk model and its variants [28, 29,30, 16] and the p-spin models [31, 32, 33, 34]. Local optima of 2-spin models areconsidered in [35, 36, 37]. While the statistical approach is the natural one, e.g., inthe physics of spin glasses, it seems to be rather contrived in evolutionary biologybecause it is by no means clear what a reasonable statistical model should looklike, even if there is a computational procedure to model the landscapes of, say,RNA free energies.A theory of landscapes is based on three ingredients: we are given a �nite, butvery large set V of \con�gurations" and a \�tness function" f : V ! IR. Thethird ingredient is a notion of neighborhood between the con�gurations, whichallows us to interpret V as the vertex set of a graph �. We will refer to � as thecon�guration space of the landscape f . Let us brie y discuss two examples here:Consider the set of RNA molecules of given chain length n. A particular moleculex can be represented as a string of length n taken from the alphabet fG, C, A,Ug; molecular biologists call this string the sequence of the RNA. The \�tness"function f is, for instance, the free energy of folding x into its secondary structure[15]. In silico the folding is done by an algorithm containing a large numberof experimentally determined parameters [38]. In nature as well as in in vitroexperiments variation is introduced by mutations, predominantly point-mutations.Neighboring sequences are thus those that di�er in only a single position. Theresulting graph is known as the sequence space [39, 40].A very di�erent example is the Travelling Salesman Problem. A salesman startsfrom his home city and visits exactly once each of the n cities on a given list, thenhe returns home. The con�gurations are the possible tours, i.e., all permutations ofthe cities on the salesman's list. The numerical value assigned to a particular tour{ 2 {

P.F. Stadler: Landscapes and Their Correlation Functions� is its total length f(� ). The notion of neighborhood between di�erent tours ismuch less obvious here than in the biological example above. Usually one says thattwo tours are neighbors if they can be interconverted by a simple operation on thelist of cities, such as swapping two cities (transpositions), or inverting the order ofa contiguous part of the list. It turns out that the performance of an optimizationheuristic depends crucially on the choice of the neighborhood relation. We willreturn to this topic later in this contribution.The main theme of this paper is the relationship between the Fourier-expansion ofa given landscapes f and its autocorrelation functions. Landscapes that are (up toan additive constant) eigenvectors of the Laplacian operator of the con�gurationspace � take center stage in the theory presented here. We shall refer to them aselementary landscapes. Two types of correlation functions will be considered here:The autocorrelation function r(s) of a \time series" generated by a random walkon the con�guration space, and the correlation function �(X ) de�ned on suitablepartitions of the set of all pairs of con�gurations. We will show that a landscapeis elementary if and only if r(s) is exponential which is in turn equivalent to �(X )being a left eigenvector of what is called the \collapsed adjacency matrix" of thecon�guration space.In section 2 we consider the properties of r(s) in some detail. Section 3 is devotedto the Laplacian operator on graphs and the Fourier expansion of landscapes.We shall also prove the �rst part of the main theorem at this stage. Section 4discusses the known elementary landscapes and reviews some the implications ofelementarity. Correlation measures de�ned on partitions of the set of pairs ofcon�gurations are the subject of section 5. The second part of the main theoremwill be proved there. The results of this contribution are summarized in section 6and discussed in section 7. { 3 {

P.F. Stadler: Landscapes and Their Correlation Functions2. \Random Walk" Correlation Functions of LandscapesBecause of the extremely large number of con�gurations, 4n for RNA and n! forthe TSP for example, we need a condensed description of a landscape. Correlationmeasures relating the values of nearby con�gurations with each other seem to be anatural approach. We will show that the most useful de�nition of such a measuredepends on the symmetry properties of the graph � = (V;E), i.e., on the choiceof the neighborhood relations.De�nition. For each landscape f : V ! IR we de�nef def=== 1jV j Xx2V f(x) and �2f def=== 1jV j Xx2V �f(x) � f�2 = f2 � f2:A landscape with �2f = 0 is called at.Note that �2f = 0 if and only if f is constant, i.e., for at landscapes. The quantityf is the mean of the landscape, and �2f can be interpreted as the variance ofthe landscape. There is nothing \statistical" about f or �2f . Both quantitiesare well de�ned functionals of f : V ! IR, and they should not be mistaken forthe averages over di�erent instances which are commonly used in the analysis ofstatistical models of landscapes.Both f and �2f do not depend on the neighborhood structure implied by theedge set E of �. Eigen and co-workers [8] have introduced correlation functionsthat depend on the Hamming distance in sequence space as a measure of thelocal structure of �tness landscapes. Weinberger [41] proposed to use a simplerandom walk fx0; x1; : : :g on the vertex set in order to sample a \time series"ff(x0); f(x1); : : :g and to use the autocorrelation function of this \time series" asa characteristic of the landscape. A simple random walk [42] on a graph � hastransition matrix T def=== D�1A, where A is the adjacency matrix of � and D isthe diagonal matrix of vertex degrees with entriesAxy def=== � 1 if fx; yg 2 E0 otherwise and Dxy def=== �x;y Xz2V Axz = �x;y[A1]x ;{ 4 {

P.F. Stadler: Landscapes and Their Correlation Functionsrespectively. Here �x;y is the Kronecker symbol, and 1 def=== (1; 1; : : : ; 1) is the vectorwith all entries 1.The expected autocorrelation function of a \time series" sampled along a simplerandom walk on � is de�ned asr(s) def=== hf(xt)f(xt+s)ix0;t � hf(xt)ix0;t hf(xt+s)ix0;tq� hf(xt)2ix0;t � hf(xt)i2x0;t� �hf(xt+s)2ix0;t � hf(xt+s)i2x0;t �where the notation h : ix0 ;t emphasizes that the expectation is taken over all \times"t and all initial conditions x0. We will refer to r(s) as the \random walk" correla-tion function of the landscape f on �. Since the averages are taken over all initialconditions with uniform weights, the de�nition of r(s) simpli�es tor(s) = hf(xt)f(xt+s)ix0;t � hf(xt)i2x0;thf(xt)2ix0;t � hf(xt)i2x0;t (1)Note that, again, r(s) can be viewed as a functional of f : V ! IR. While thede�nition of r(s) via the random walk is convenient for sampling data from a givenlandscape, say a particular instance of a combinatorial optimization problem ora model of the complicated energy function for protein folding, it is inconvenientfrom a mathematical point of view. Furthermore, it seems to be rather contrivedto invoke a stochastic process in order to characterize a given function de�ned ona �nite set. We will therefore derive a representation of r that does not require theexplicit assumption of a random walk, at least for the case of regular graphs. Itis not at all clear that r(s) is a useful measure of a landscape on a more irregulargraph. To this end we need two preparatory lemmata1.Lemma 1. Let � be a regular graph and let F : V ! IR be an arbitrary function.Let fxtg be a simple random walk on �. Then hF (xt)ix0;t = F .1In this contribution we will label a technical result as a \proposition" if it isknown in the literature, and as a \lemma" otherwise. The proofs of all lemmataand their corollaries can be found in the appendix.{ 5 {

P.F. Stadler: Landscapes and Their Correlation FunctionsLemma 2. Let � be a regular graph, and let F : V � V ! IR. ThenhF (xt+s; xt)ix0;t = 1jV j Xx;y2V F (x; y)[T]sxy :The desired representation of r(s) is now obtained as a corollary of the above twolemmata.Corollary 1. Let f : V ! IR be a non- at landscape on a D-regular graph �with adjacency matrix A. Thenr(s) = 1jV j hf;Tfi � f 2f2 � f2 (2)where T = (1=D)A.Remark. Both for the proof of lemma 1 and the proof of lemma 2 it is crucialthat T1 = 1. Since each transition necessarily ends in some vertex x 2 V we havealways 1T = 1. In other words, instead of insisting the � be a regular graph andthat the random walk is simple we could as well require that T is a bi-stochastic,but not necessarily symmetric, transition matrix. This might be a starting pointfor an investigation of landscapes on non-regular con�guration spaces.3.Graph Laplacians and Their Eigenfunctions3.1. Incidence Matrix and Graph LaplacianThe incidence matrix and the Laplacianmatrix of a graph can be viewed as discreteanalogues of the gradient and the Laplace operator in Euclidean spaces.De�nition. Let � be an arbitrary graph with vertex set V and edge set E. Foreach edge h = fv;wg we choose one of the two vertices as the \positive end" and{ 6 {

P.F. Stadler: Landscapes and Their Correlation Functionsthe other one as the \negative end" of the edge. The choice of this orientation iscompletely arbitrary. The matrix r+ with entriesr+ij =8<:+1 vertex vi is the positive end of edge ej�1 vertex vi is the negative end of edge ej0 otherwiseis called the incidence matrix of �.The choice of the symbol r is intentional. In fact, let f : V ! IR be an arbitrarylandscape. Then rf : E ! IR is given by(rf)(h) = f(v) � f(w) where h = fv;wgand v is the positive end of the edge h. This is as close to a di�erential operatoras one can get on a graph.De�nition. Let D be the diagonal matrix of vertex degrees, i.e., Dxx is thenumber of edges incident into x, and let A be the adjacency matrix of �. Thenthe matrix �� =D�A (3)is called the Laplacian of �. ForD-regular graphs, i.e., graphs for which all verticeshave degree D, this becomes � = A �DI.The graph Laplacian shares its most important properties with the familiar di�er-ential operator � =Pni=1 @2@x2i in IRn, as explained in some more detail in �gure 1.Proposition 1.(i) � is symmetric.(ii) �� is non-negative de�nite.(iii) � is singular; the eigenvector 1 = (1; : : : ; 1) belongs to the eigenvalue �0 = 0.If � is connected (as we will always assume), then �0 has multiplicity 1.(iv) �� = r+r, that is, it corresponds to \second derivatives" on the graphs.(v) For any two landscapes f and g Green's formula holds in the following formXx2V f(x)(�g)(x) = Xx2V g(x)(�f)(x) = �Xh2E(rf)(h)(rg)(h):{ 7 {

P.F. Stadler: Landscapes and Their Correlation Functions0

y

y

x x

4

1 2

1

2

e

e

e e

1

3 2Figure 1: The graph Laplacian � is a generalization of the discrete approximation of thefamiliar Laplacian di�erential operator in IRn.This approximation is commonly performed by replacing the continuous space by asquare lattice. Rescaling the space coordinates we can assume that the lattices pointshave integer coordinates. The �rst derivatives, evaluated at the mid-points of the edges,ei, are computed from@f@x (e2) = f(x2)� f(0) @f@x (e3) = f(0) � f(x1);and analogous expressions for @f=@y. Consequently, the second derivative evaluatedin 0 become@2f@x2 (0) = (f(x2) � f(0)) � (f(0) � f(x1)) = f(x1) + f(x2)� 2f(0);and an analogous expression for @2f=@y2 . The discrete approximation of the usualLaplacian hence coincides with the graph Laplacian of the square lattice.Proof. (i) is obvious, (ii) and (iii) are well known, see, e.g., [43, 44]. Claim(iv) is Proposition 4.8 of [43]. Green's formula (v) is easily checked by explicitcalculation:Xh2E(rf)(h) (rg)(h) = Xx2V Xy2V Xh2E f(x)rhxrhyg(y)= Xx2V Xy2V f(x) Xh2Er+xhrhy! g(y)= �Xx2V Xy2V f(x)�xyg(y) = �Xx2V f(x)(�g)(x);A similar calculation showsPh(rf)(h)(rg)(h) =Px g(x)(�f)(x).{ 8 {

P.F. Stadler: Landscapes and Their Correlation FunctionsThe graph Laplacian is central to the theory of electrical networks. As a referencewe give Kirchho�'s classical paper [45]. Let �(x) be the current owing into thenetwork at vertex x. Then there is a potential � : V ! IR satisfying �� = �, andthe vector � of currents along the edges is given by � = r�. A recent book onpotential theory on discrete lattices is [46].Finally we note the following connection between the spectrum of a graph andthe graph Laplacian: Suppose � is D-regular with adjacency matrix A. Then Aand �� have the same eigenvectors, and thus the eigenvectors of �� are givenby ��k = �k �D.3.2.Fourier Expansion of LandscapesA series expansion in terms of a complete and orthonormal system of eigenfunctionsof the Laplace operator is commonly termed Fourier expansion. We will adopt thesame terminology here following [30]. Thus, let f be a landscape on � and letf'ig denote a complete orthonormal set of eigenvectors of the graph Laplacian��. Then we call the expansionf(x) = jV jXi=1 ai'i(x) (4)a Fourier expansion of the landscape. It will often be convenient to label theeigenvectors 'i by the vertices of the underlying graph �. This is possible becausethe eigensystem of the �nite symmetric matrix � is complete. In general, thislabeling is of course arbitrary.Since we deal with a �nite vector space with a scalar product, for which we willuse the notation h : ; : i, spanned by eigenvectors f'ig of the graph Laplacian, thefamiliar properties of Fourier series, such as Parseval's equationkfk2 = hf; fi = Xy2V � hf; 'yih'y; 'yi�2 :{ 9 {

P.F. Stadler: Landscapes and Their Correlation FunctionsAnother important result is the mean square approximation theorem:Proposition 2. Consider a landscape f on � with Fourier expansion f =Xy2V ay'y. Let X be a subset of V , and consider approximations of f of the formg = Xy2X by'y. Then the squared approximation error kf � gk2 = h(f � g); (f � g)iis minimized by choosing by = ay � hf; 'yi for all y 2 X.It is clear that landscapes which are eigenfunctions of the graph Laplacian willplay a special role. It will turn out to be more useful, however, to consider aslightly larger class of landscapes.De�nition. A landscape f : V ! IR is elementary if there are constants f� and� such that �f + �(f � f�1) = 0 (5)This de�nition is motivated by Lov Grover's observation [47] that the cost functionsof a number of well-studied combinatorial optimization problems in fact ful�llequ.(5). This will be discussed in detail in section 4.Lemma 3. A non- at landscape on a connected graph � is elementary if and onlyif f(x) = f + '(x) 8x 2 V ; (6)where ' is an eigenfunction of �� with eigenvalue � > 0.3.3.\Random Walk" Correlation Functions of Elementary LandscapesThe \random walk" correlation function r(s) provides an elegant way of charac-terizing elementary landscapes.Theorem 1. Let f be a non- at landscape on a D-regular graph � and let r(s)be the \random walk" correlation function of f . Then f is elementary if and onlyif r(s) is an exponential function, i.e., i� r(s) = %s.{ 10 {

P.F. Stadler: Landscapes and Their Correlation FunctionsProof. Let f'g be an orthonormal set of real eigenvectors of the Laplacian on �,i.e., (��)'i = �i'i. Since T = I+D�1� = I+ 1D� (7)we have T'i = 1��i=D. Now substitute the decomposition f =Pi ai'i into thede�nition of r(s). One �ndsr(s) = 1jV jPi;j aiaj h'i; 'ji(1 � �i=D)s � �Pi ai 1jV jPx2V 'i(x)�21jV jPi;j aiajPx2V 'i(x)'j (x) � �Pi ai 1jV jPx2V 'i(x)�2Recall that 1 is always an eigenvector of �, belonging to �0 = 0. By orthogonalitywe have therefore Px2V 'i(x) = 0 for all i 6= 0, and consequently f = a0'0.Noting that Px2V 'i(x)'j (x) = h'i; 'ji = �ij we obtainr(s) = 1jV jPi jaij2(1 � �i=D)s � ja0j2'021jV jPi jaij2 � ja0j2'02It remains to compute '0. We know '0 = c1 with some constant c 6= 0. Normal-ization implies 1 = c2h1;1i = c2jV j, i.e, '0(x) = 1pjV j for all x 2 V . Substitutingthis into r(s) yieldsr(s) = 1jV j ja0j2 + 1jV jPi6=0 jaij2(1� �i=D)s � ja0j2 1jV j1jV j ja0j2 + 1jV jPi6=0 jaij2 � ja0j2 1jV jIt is convenient to introduce the normalized amplitudesAi def=== jaij2Pj 6=0 jaj j2 ; : (8)Note that a landscape is at if and only if Ai = 0 for all i 6= 0. Thus the normalizedamplitudes are in fact well de�ned for all non- at landscapes. FurthermoreAi = 0is true for all i 6= 0 if and only if ai = 0. The expression for r(s) simpli�esconsiderably: r(s) =Xi6=0 Ai(1� �i=D)s : (9){ 11 {

P.F. Stadler: Landscapes and Their Correlation FunctionsConsequently r(s) is an exponential function if and only if all nonzero Ai belongto a single eigenvalue �k of �. This is the case if and only if f is of the formf = (a0=pjV j)1+' where ' is an eigenvector of ��. Applying lemma 3 completesthe proof.The \random walk" correlation function of an elementary landscape is determinedby the single parameter % def=== r(1), which one might call the nearest-neighborcorrelation of the landscape. We have % = (1 � �k=D), where �k is a non-zeroeigenvalue of the graph Laplacian ��. Since r(s) is exponential we can de�ne acorrelation length ` by ` def=== ( 0 if % = 0� 1ln j%j if % 6= 0 (10)Thus the \random walk" correlation function is of the form r(s) = exp(�s=`) for% > 0, r(s) = (�1)s exp(�s=`) for % < 0, and r(s) = �s;0 for % = 0. Table 1 atthe end of the following section compiles numerical values of % and ` for a fewlandscapes of practical interest.4. Elementary LandscapesIn this section we will brie y discuss a number of landscapes, all of which areelementary (or at least almost elementary). We will conveniently subdivide ourdiscussion according to the type of the con�guration space underlying the land-scapes. Three classes of con�guration spaces are of particular importance: (a)landscapes de�ned on sequences, (b) landscapes de�ned on permutations, and (c)landscapes de�ned on a set of subsets of given �nite set. In this contribution wetypeset matrices only boldface if they are related to the con�guration space �in some way. Many of the landscape models discussed in the following containmatrices of parameters which will be typeset in italics.{ 12 {

P.F. Stadler: Landscapes and Their Correlation Functions4.1.Cayley Graphs and Cartesian Products of GraphsAn important class of graphs with high symmetry are obtained from �nite groups.De�nition. Let (G; �) be a �nite group, and let � be a set of generators2 of Gsuch that (i) the group identity { is not contained in �, and (ii) for each x 2 �the inverse group element x�1 is also contained in �. The Cayley graph �(G;�)is the graph with vertex set G and fx; yg 2 E if and only if there is a g 2 � suchthat y = gx, i.e., if and only if xy�1 2 �.The set of generators � can be interpreted as the set of all possible elementarymutations, or | in the context of an optimization heuristic | as the move set ofthe algorithm.Many, but by no means all, of the con�guration spaces encountered in this con-tribution are Cayley graphs, in particular the Hamming graphs (generalized hy-percubes) and the Cayley graphs of the symmetric group Sn. We will return totheir graph-theoretic properties later in this contribution. In this section we willbe content with showing that a variety of interesting landscapes are elementaryon appropriate graphs.For Cayley graphs of commutative groups the eigenvectors and eigenvalues have aparticularly simple form. We exploit the fact that a �nite commutative group hasa unique decomposition into cyclic groups, see e.g. [48, x13]: Let Nk be the ordersof the cyclic groups. The cyclic group of order Nk in turn are isomorphic to theadditive group modulo Nk, and thus G is isomorphic to the group of \vectors"x = (x1; x2; : : : ; xm), 0 � xk < Nk, under component-wise additions modulo Nk:x � y = (x1 + y1 mod N1; x2 + y2 mod N2; : : : ; xm + ym mod Nm):The characters [49] of the commutative group G are given by�g(x) = exp 2�iXk xkgkNk ! : (11)2� � G is a set of generators if each group element z 2 G can be represented as a�nite product of elements of �. { 13 {

P.F. Stadler: Landscapes and Their Correlation FunctionsLet �(G;�) a Cayley graph of G with graph Laplacian ��. It is convenient toallow also for complex eigenvectors of the symmetric matrix ��, since it can beshown [50] that(��)�g = �g�g holds with �g =Xx2�[1� �g(x)] (12)In other words, the characters �g of G are the eigenvectors of any Cayley graphderived from the commutative group G.Probably the simplest examples of Cayley graphs with commutative groups are thecomplete graphs Kn. Let G be a commutative group with n elements, for instancea cyclic group, and de�ne � = G n f{g, where { is as usual the group identity.Then �(G;�) has an edge between any two vertices, i.e., it is the complete graphKn. In terms of optimization procedures and their move-sets, the complete graphscorrespond to random search: all con�gurations are accessible in a single step.Lemma 4. Let f be a non-constant landscape on the complete graph Kn with nvertices. Then f is elementary.Elementary landscapes are thus only interesting when they \live" on non-trivialgraphs with an interestingly rich spectrum of ��.De�nition. The (Cartesian) product �1��2 of two graphs has vertex set V (�1��2) = V (�1) � V (�2). Two nodes (x1; x2) and (y1; y2) are connected if either (i)x1 = y1 and x2; y2 are adjacent in �2, or (ii) x2 = y2 and x1; y1 are adjacent in �1.Proposition 3. Let � = �1 � �2. Then(i) Let �(1)k and �(2)j be eigenvalues of the Laplacians of the two graphs �1 and�2, respectively. Then � is an eigenvalue of the Laplacian of �1 � �2 if andonly if it is of the form �(1)k + �(2)j .(ii) Let u(1)k and u(2)j be eigenvectors of the Laplacian of �1 and �2. Then u(1)k �u(2)j is an eigenvector of �1 � �2.(iii) If �1 = �(G1;�1) and �2 = �(G2;�2) are Cayley graphs then�(G1;�1)� �(G2;�) = ��G1 �G2; (f{1g ��2) [ (�2 � f{2g)�{ 14 {

P.F. Stadler: Landscapes and Their Correlation Functionsis again a Cayley graph.Proof. For (i) and (ii) see [51], claim (iii) is easily veri�ed from the de�nition ofthe Cartesian product.Our interest in the Cartesian product of graphs comes from the fact that importantclasses of graphs can be constructed as repeated Cartesian products of very simpleunits. As an example consider the sequence spaces or Hamming graphs Qn�. Thevertices of these graphs are sequences of constant length n constructed from a�xed alphabet with � letters. Two sequences are adjacent if they di�er in a singleposition along the sequence. Obviously Q1� �= K� is the complete graph with �vertices. It is easy to check that Qn� �= Qn�1� �K� for all n � 2.4.2.Landscapes on the Boolean HypercubeAn orthonormal basis of eigenvectors of the Laplacian is easily constructed ex-plicitly for Boolean hypercubes Qn2 . Without loosing generality we may use thealphabet f+1;�1g. A con�guration is then a string � of \spins" �k 2 f+1;�1g.An alternative encoding uses a binary string x, with xi 2 f0; 1g. The followingresult is well known:Proposition 4. Any landscape f on the Boolean Hypercube can be written asf(�) = J0 + nXp=1 Xi1<i2<:::<ip Ji1i2:::ip�i1�i2 : : : �ip ; (13)where the Ji1i2:::ip are constants.It is not hard to check that the products"q(�) = �i1�i2 : : : �ip where qk = 1 if and only if k 2 fi1; i2; : : : ; ipgare in fact eigenvector of the Laplacian of the Boolean hypercube because of thecorrespondence "q(�) = �q(x) with �k = 2xk � 1. Furthermore, one �nds that the{ 15 {

P.F. Stadler: Landscapes and Their Correlation Functionseigenvalue corresponding to "q depends only on the number p of non-zero entriesin the multi-index q, see e.g. [34]. One �nds�p = 2np with multiplicity m(�p) = �np� : (14)Hamiltonians of the formHp(�) = Xi1<i2<:::<ip Ji1i2:::ip�i1�i2 : : : �ip (15)play a prominent role in the theory of spin glasses; they are known as p-spin models.It was introduced by Derrida [31] in order to bridge the gap between the SK model[52], which is the special case p = 2, and the random energy model [31, 53, 32, 54].In physics the coe�cients are usually chosen i.i.d. from a Gaussian distribution.As a consequence of proposition 4 above we can represent any landscape on theBoolean hypercube as a superposition of p-spin models with suitable choices of theinteraction coe�cient Ji1i2;:::;ip . Let us now consider a few examples:Weight Partition (WP). Given a string of n \spins" xi 2 f�1;+1gn and correspond-ing weights wi, the cost function is given byf(x) = nXi=1 wixi!2 : (16)The move set is given by ipping a single spin, hence the con�guration space isagain a hypercube. Grover [47] showed that WP is elementary with � = 4 for anychoice of the weights wi.Not-All-Equal-Satis�ability (NAES). Consider a vector of n binary variables. Aliteral is a variable or its complement. A clause is a set of three literals that doesnot contain both a variable and its complement. A clause is said to be satis�ed ifat least one literal is 0 and at least one literal is 1. An instance of NAES is given bya set of c clauses, and the cost function is the number of non-satis�ed clauses. Themove set is de�ned by ipping the value of a single variable, thus the con�gurationspace is the Boolean Hypercube. Grover [47] showed that WP is elementary with� = 4 for any choice of clauses. { 16 {

P.F. Stadler: Landscapes and Their Correlation FunctionsLow Autocorrelated Binary Strings (LABSP). The LABSP [55, 56] consists of �ndingbinary strings � over the alphabet f�1;+1g with low aperiodic o�-peak auto-correlation Rk(�) = PN�ki=1 �i�i+k for all lags k. These strings have technicalapplications such as the synchronization in digital communication systems andthe modulation of radar pulses. The quality of a string � is measured by the�tness function f(�) = n�1Xk=1Rk(�)2: (17)In most of the literature on the LABSP the merit factor F (�) = n2=(2f(�)) is used,see [56] for details.Lemma 5. The landscape f of the LABSP can be written asf(�) = a0 + dn2 e�1Xk=1 n�2kXi=1 2"i;i+k(�) + n�1Xk=1 n�1Xi=1 Xj 6=i�k;i;i+k "i;i+k;j;j+k(�): (18)Corollary 2. The \random walk" correlation function of the LABS is of the formr(s) = [1�O(1=n)]�1� 8n�s +O(1=n)�1� 4n�s : (19)The landscape of the LABPS is thus not elementary, it consists of a superpositionof two modes, namely p = 2 and p = 4. The smoother p = 2 contribution becomesnegligible for large n, so that f behaves for long strings almost like an elementaryp = 4 landscape. This fact explains why the LABPS has been found to be muchharder for simulated annealing than, say, the SK spin glass [56]. This author [13]has computed the \random walk" autocorrelation function r(s) numerically basedon the merit factor F . The numerical estimate for the correlation length` � 0:123 � n� 0:983is in excellent agreement with the asymptotic value ` = n=8+O(1) implied by thecorollary. { 17 {

P.F. Stadler: Landscapes and Their Correlation Functions4.3.Landscapes on Hamming GraphsBoolean hypercubes are of course a special case of Hamming graphs. We havediscussed them in a separate subsection because of their particular importance.Hamming graphs with larger alphabets (� > 2) are on particular importance inbiology: the sequences of nucleic acids, RNA or DNA, contain four di�erent bases,and proteins use 20 di�erent amino-acids. Just as for the hypercube, the Laplacianspectrum and an ONB of eigenvectors can be constructed explicitly [57, 58].Graph Coloring Problem (GCP). An instance of a graph coloring problem consistsof a graph G(V;E) with n vertices and a collection of � di�erent colors. A con�g-uration x is an �-coloring of the vertex set V , i.e., an assignment of one color xpto each vertex p of the graph. The cost function is the number edges fp; qg 2 Esuch that xp = xq : f(x) = Xfp;qg2E �x(p)x(q): (20)A move is the replacement of one color by another one at single vertex. Thecon�guration spaces are thus the general Hamming graphs, i.e., sequence spacesover the alphabet of the � colors. Grover [47] has shown that each instance of GCPis elementary on Qn� with � = 2�.4.4.Permutations: Travelling Salesman and Graph MatchingThe con�gurations of a family of optimization problems can be represented aspermutations of �nite number n of objects. Hence the symmetric group Sn takesthe role of V . Natural choices of move sets are sets of generators of the Sn, andthus the con�guration spaces are Cayley graphs of the symmetric group. The mostconvenient sets of generators are: the set T of all transpositions (i; j), the set Kof all canonical transpositions (i; i + 1), and the set J of all reversals [i; j], whichare also called inversions or 2opt-moves, [59].{ 18 {

P.F. Stadler: Landscapes and Their Correlation FunctionsTravelling Salesman Problem (TSP). An instance of a TSP [9] is de�ned by a set ofn cities and matrix W of costs for connecting them. A tour is a permutation � ofcities, and its cost is f(�) = nXi=1W�(i);�(i�1) (21)where the indices are taken modulo n. Di�erent versions of the TSP are de�ned bythe properties of W (arbitrary, or symmetric, or with entries additionally obeyingthe triangle inequality, etc.; see [60]).In the following it will be convenient to usef�(�) = f(��) =Xj w�(j)�(j+1):The permutation �� is the \reverse order" permutation of �, i.e., f�(�) = f(��) isthe cost of the tour � when traveled in the opposite direction. Thus f(�) = f�(�)for all � 2 Sn is true if and only if the cost matrix W is symmetric. Recallingthat any matrix W can be uniquely decomposed into its symmetric componentW � = (W +W+)=2 and its antisymmetric component W� = (W �W+)=2 weintroducef�(�) =Xj w��(j)�(j�1) = f(�) + f�(�)2 ; f�(�) =Xj w��(j)�(j�1) = f(�) � f�(�)2 :Note that f� and f� can be viewed as cost functions of TSPs with \distancematrices" W � and W�, respectively.Lemma 6. Both f� and f� are elementary landscapes on the Cayley graphsof the symmetric group with the transpositions and the inversions as generators,respectively. In particular we have for transpositions�f� + 2(n� 1)(f� � �f ) = 0 �f� + 2nf� = 0 ; (22)and for inversions (2opt moves [59]) we �nd�f� + n(f� � �f) = 0 �f� + n(n+ 1)2 f� = 0 : (23){ 19 {

P.F. Stadler: Landscapes and Their Correlation FunctionsCorollary 3. The landscape of a TSP with transpositions or inversions is ele-mentary if and only if W is either symmetric or antisymmetric.Asymmetric TSPs hence provide an example of fairly simple composite landscapes.They consist of two modes corresponding to the symmetric and the antisymmet-ric part of the distance matrix W , respectively. It is also interesting to note inthis context that canonical transpositions (i; i + 1) do not lead to an elementarylandscape. Numerical data [61, 11] have indicated that the \random walk" cor-relation functions r(s) of both the symmetric and the antisymmetric componentsare exponential. Theorem 1 now provides a mathematical explanation for thisobservation.The nearest neighbor correlations of the symmetric and antisymmetric componentsof a TSP with transpositions are % = 1� 4=n and % = 1� 4=(n� 1), respectively,i.e., very similar. In fact, numerical estimates [11] are consistent with % � 1� 4=nfor large n in both cases. In the case of inversions we have a symmetric mode withnearest neighbor correlation % = 1� 2=(n� 1) and an antisymmetric contributionwith a vanishingly small contribution % = �2=(n� 1) � 0.It is interesting to correlate these values of % with known facts about the per-formance of heuristic optimization algorithm, in particular with the simulatedannealing. It has been observed by several authors that simulated annealing onsymmetric TSP is much more e�ective when reversals instead of transpositions areused as move set, see, e.g., the books [62, 63]. Furthermore, Miller and Pekny [64]have observed that reversals are a remarkably ine�cient move set for asymmetricTSPs. These observations are in accordance with the conjecture that landscapewith smoother correlation functions have fewer local optima and are thus easierto optimize on [11]. In particular the di�erence between symmetric and asymmet-ric TSPs when reversals are used is easily explained in these terms: while for thesymmetric TSP the landscape is as smooth as possible, it is extremely rugged forthe antisymmetric case.Graph Matching Problem (GMP). Given a graph G with n vertices and a symmetricmatrix W of edge weights, the task is to partition the graph into n=2 pairs of{ 20 {

P.F. Stadler: Landscapes and Their Correlation Functionsvertices such that the sum of the edge weights corresponding to these pairs isoptimal. A convenient encoding of the problem is the following. Let � 2 Sn be apermutation of the vertices. We assume that the vertices are arranged such that[�(2k � 1); �(2k)] form a pair. The cost function is thenf(�) = n=2Xk=1W�(2k�1);�(2k) : (24)Again, the con�guration space is the symmetric group, and hence the set of alltranspositions is a reasonable move set.Lemma 7. The landscape of the graph matching problem GMP with transpositionmetric is elementary with � = 2(n� 1).Note that this result is false if W is not symmetric.4.5.Landscapes on Johnson GraphsTable 1. Summary of Elementary Landscapes.Problem � D � % `NAES Qn2 n 4 1� 4n 14n� 12� 13 1n+O( 1n2 )WPP Qn2 n 4 1� 4n 14n� 12� 13 1n+O( 1n2 )p-spin Qn2 n 2p 1� 2pn 12pn�12� p6 1n+O( 1n2 )GCP Qn� (��1)n 2� 1� 2�(��1)n ��12� n�12+ �6(��1) 1n+O( 1n2 )symmetric �(Sn;T ) n(n�1)=2 2(n�1) 1� 4n 14n� 12� 13 1n+O( 1n2 )TSP �(Sn;J ) n(n�1)=2 n 1� 2n�1 12n�1� 16 1n+O( 1n2 )anti-symmetric �(Sn;T ) n(n�1)=2 2n 1� 4n�1 14n� 34� 13 1n+O( 1n2 )TSP �(Sn;J ) n(n�1)=2 n(n+1)=2 � 2n�1 1lnn�ln 2 1ln2 n+O(ln�3 n)GMP �(Sn;T ) n(n�1)=2 2(n�1) 1� 4n 14n� 12� 13 1n+O( 1n2 )GBP J(n;n=2) n2=4 2(n�1) 1� 8n+ 8n2 18n� 38� 1324 1n+O( 1n2 ){ 21 {

P.F. Stadler: Landscapes and Their Correlation FunctionsAnother class of con�guration spaces arises if the con�gurations can be regardedas subsets of some �nite set.De�nition. Let X be a �nite set, n = jXj, and let Sk be the collection of k-element subsets of X. The graph J(n; k) has vertex set Sk and two vertices areadjacent if the corresponding subsets of X have k� 1 vertices in common. J(n; k)is called Johnson graph.Only one example of this class has received extensive attention so far.Graph Bipartitioning Problem (GBP). G is a graph with an even numbern of verticesand H is a symmetric matrix of edge weights. A con�guration is a partition of thevertex set into two subsets A and X nA of equal size. Two bi-partitions [A;X nA]and [B;X nB] are neighbors if B is obtained from A by exchanging a vertex fromA by a vertex from X n A. Thus the con�guration space is the Johnson graphJ(n; n=2). The cost function isf([A;X nA]) =Xi2A Xj2XnAHij ; (25)i.e., f is the total weight of all edges connecting A and X nA. As a close relativeof the Sherrington-Kirkpatrick model the GBP has received considerable attention[10]. Grover [47] found that each instance of the GBP is an elementary landscapewith � = 2(n�1), see also table 2. Stadler and Happel [23] have shown by explicitcalculations that the \random walk" correlation function of the GBP on J(n; n=2)is r(s) = �1� 8n + 8n2�s ; (26)in accordance with Theorem 1.Table 1 summarizes the known elementary landscapes.{ 22 {

P.F. Stadler: Landscapes and Their Correlation Functions4.6.Nodal Domains of Schr�odinger OperatorsIn this contribution we are only interested in eigenfunctions ' of the Laplacianoperator ��. The following interesting result can, however, be proved for a muchlarger class of operators acting on landscapes. Let a be a weight function on theedges of �, conveniently de�ned as a : V �V ! IR such that a(x; y) = a(y; x) > 0 iffx; yg 2 E and a(x; y) = 0 otherwise. Furthermore let v : V ! IR be an arbitrary\potential". A linear operator H, de�ned by the actionHf(x) def=== Xx�y a(x; y) [f(x) � f(y)] + v(x)f(x) ; (27)is called a Schr�odinger operator. Setting a(x; y) = 1 if fx; yg 2 E, i.e., a(x; y) =Axy, and choosing v(x) = 0 shows that the graph Laplacian �� is in fact aSchr�odinger operator. Discrete Schr�odinger operators without potential can beinterpreted as Laplacians of edge-weighted graphs [65].The Perron-Frobenius theorem implies that the smallest eigenvalue �1 of H is non-degenerate and the corresponding eigenfunction f1 is positive everywhere if � isconnected. Let�1 < �2 � �3 � : : : � �k�1 � �k � �k+1 � : : : � �jV jbe the list of eigenvalues of H arranged in non-decreasing order and repeated ac-cording to multiplicity3. Let fk be any eigenfunction associated with the eigenvalue�k. Without loosing generality we may assume that ffig is a complete orthonor-mal set of eigenfunctions satisfying Hfi = �ifi. Since H is a real operator, wetake all eigenfunctions to be real.A con�guration x 2 V is a local minimum if f(x) � f(y) for all neighbors y 2@fxg of x. Correspondingly x is a local maximum if f(x) � f(y) for all y 2@fxg. Local optima are the very feature of a landscape that makes it rugged.3Note that in this subsection we count the eigenvalues starting at 1 instead of 0as in the rest of this contribution { 23 {

P.F. Stadler: Landscapes and Their Correlation FunctionsLocal optima are traps of optimization heuristics and evolutionary adaptation.An understanding of the distribution of local optima is thus of utmost importancefor the understanding of a landscape. Little can be said about the geometricarrangement of local optima for an arbitrary landscape f . The situation is slightlybetter for elementary landscapes.Proposition 5. Let f be an eigenfunction of H with eigenvalue � > 0. If x0 isa local maximum of f then v(x0) � � implies f(x0) � 0 and v(x0) � � impliesf(x0) � 0. If x0 is a local minimum of f then v(x0) � � implies f(x0) � 0 andv(x0) � � implies f(x0) � 0. In particular, if v � 0 then all local maxima arenon-negative and all local minimal are non-positive.Proof. If x0 is local maximum then f(x0) � f(y) � 0, and hence Hf(x0) �v(x0)f(x0). Using that f is an eigenfunction yields �f(x0) � v(x0)f(x0). Usingthat � � 0 yields the assertion for a local maximum. The argument for localminima is analogous. If v � 0, i.e., if H is a Laplacian of an edge-weighted graph,the case � � v(x0) contradicts � > 0. The special case H = �� is theorem 6 in[47].Some more global information can be obtained on the distribution of positive andnegative valued of an eigenfunction f of a Schr�odinger operator. This leads usto the notion of nodal domains. In a continuous setting one de�nes the nodalset of a continuous function f as the preimage f�1(0). The nodal domains arethe connected components of the complement of f�1(0). In the discrete case thisde�nition does not make sense since a function f can change sign without havingzeroes. Instead we use the followingDe�nition. X is a nodal domain of a function f : V ! IR if it is a maximalsubset of V subject to the two conditions(i) X is connected as an induced subgraph of �;(ii) if x; y 2 X then f(x)f(y) � 0.The following properties of nodal domains of an eigenfunction f of a Schr�odingeroperator can be easily veri�ed. { 24 {

P.F. Stadler: Landscapes and Their Correlation Functions(a) Every point x 2 V lies in some nodal domain X � V .(b) If V is a nodal domain then it contains at least one point x 2 V with f(x) 6= 0and f has the same sign on all non-zero points inX. Thus each nodal-domaincan be called either \positive" or \negative".(c) If two nodal domains X and X 0 have non-empty intersection the f jX\X0 = 0and X;X 0 have opposite sign or X = X 0.The following result generalizes Courant's nodal domain theorem for Riemannianmanifolds, see [66], to arbitrary connected graphs.Proposition 6. The eigenfunction fk has at most k nodal domains.Proof. See [67].Remark. The second-largest eigenvalue of Graph, often called the algebraic con-nectivity, and its eigenvectors, which are sometimes referred to as characteristicvaluations, have received some attention in graph theory [68, 69, 70]. The casek = 2 of proposition 5 was proved already in 1975 by Fiedler [71]. Stuart Kau�-man [72] calls this type of landscape \Fujijama", because they have only a singlemountain massive (positive nodal domain).5.Autocorrelation Functions and Coherent Con�gurations5.1.Autocorrelation Functions on PartitionsThe use of \random walk" correlation functions for characterizing landscapes hastwo disadvantages: (i) the information on the landscape is \blurred" by the tran-sition matrix T of the walk, and (ii) samples along a random walk converge slowlycompared to sampling independent pairs of con�gurations. For these reasons mostof the results on the correlation structure of RNA landscapes, where the evalua-tion of f is extremely costly in terms of computer resources, have been obtained in{ 25 {

P.F. Stadler: Landscapes and Their Correlation Functionsterms of the correlation functions �(d), which is de�ned as the average correlationof all pairs (x; y) of vertices with distance d(x; y) = d [15, 16, 24]. In this sectionwe link this approach to the theory of the \random walk" correlation functionsdeveloped above.It is convenient to formulate our discussion in terms of partitions of the set of(ordered) pairs of vertices V � V . Recall that there is a canonical metric on thevertex set V of any connected graph �. The distance d : V � V ! IN+0 on V isde�ned as the minimumnumber of edges separating two vertices. If � is �nite, thenthere is a maximum distance, which is called the diameter d� of �. The propertiesof the distance on V are discussed in detail in the book [73]. The metric d( : ; : )induces in a natural way the distance partition D of V � V which has the classesDd def=== �(x; y) 2 V ����d(x; y) = d� : (28)The distance partition seems to be the most natural and useful partition of V �V .The subsequent discussion will show, however, that this is true only for su�ciently\symmetric" graphs �. For the moment we consider arbitrary partitions of V �V ,postponing the problem of choosing the partitions until the following subsections.De�nition. Let R be a partition of V � V and suppose f : V ! IR is non-constant. Then the correlation function � : R ! [�1; 1] of f with respect to R isde�ned by �(X ) def=== 1jX j � �2 X(x;y)2X � (f(x) � f )(f(y) � f) � (29)where X is a class of the partition R .Let us note a few general properties of correlation functions on partitions. Amatrix X with entries Xxy def=== � 1 if (x; y) 2 X0 otherwise (30)is associated with each class X 2 R . In order to simplify the formalism in thefollowing we will always assume that f = 0. Since both r(s) and �(X ) are in-variant under the transformation f(x) ! f(x) � f we can do this without loosinggenerality. Thus we have �(X ) = jV jjX j hf;Xfihf; fi : (31){ 26 {

P.F. Stadler: Landscapes and Their Correlation FunctionsLemma 8. Let R be a partition of V � V and let f be a non-constant landscapeon V . Then XX2R �(X ) � jX j = 0.Lemmarl-0 means that there is no \average correlation" in a sample of randompoints, because jX j=jV j2 is the probability that two randomly picked points x andy form a pair (x; y) 2 X .The diagonal of V � V is I def=== �(x; x) �� x 2 V ; the associated matrix is theidentity matrix I. The transpose of a class X 2 R isX+ def=== �(y; x) �� (x; y) 2 X :If X is the matrix associated with the class X , then its transpose X+ is associatedwith the class X+. The de�nition of � immediately implies�(X ) = �(X+) and �(I) = 1 : (32)The following two conditions seem to be quite natural for our purposes:De�nition. Let R be a partition of V � V . Following Higman [74] call R apre-coherent con�guration if the implication X \ I 6= ; =) X � I holds and iffor any X 2 R we have also X+ 2 R .The symmetry of � under transposition might suggest to require X = X+ andI 2 R . For technical reasons (which will become clear later on) it is wise toabstain from these more stringent conditions. Since we are given not only the setV but also the neighborhood structure of � it seems natural to require that it isin some way respected by the partition R . The edge set E of � translates into tothe set A def=== �(x; y) �� fx; yg 2 E = D1: (33)of ordered pairs of vertices. The matrix associated withA is of course the adjacencymatrix A of �.De�nition. A partition R of V � V compatible with � if X \ A 6= ; impliesX � A for all X 2 R . { 27 {

P.F. Stadler: Landscapes and Their Correlation FunctionsUnfortunately, pre-coherent con�gurations compatible with � are still too gen-eral to allow for an interesting theory. In the following section we will thereforeintroduce one more condition on R .5.2.Coherent Con�gurationsDe�nition. A pre-coherent con�gurationR is coherent if for all classes X ;Y;Z 2R the following statement is true:The numbers ���z 2 V �� (x; z) 2 X and (z; y) 2 Y �� def=== pZXY are the same for allpairs (x; y) 2 Z.The numbers pZXY are called the intersection numbers of R .Coherent con�gurations have been studied in detail by Higman [74, 75, 76]. Thehighlights of his theory will be outlined later on in this subsection.Let R = fR1;R2; : : :Rrg and S = fS1;S2; : : :Ssg be two partitions of a setW . We say that R is a re�nement of S if for all k 2 [1; r] there is a j 2 [1; s](depending on k) such that Rk � Sj . The number of classes in a partition is calledits rank. We write S � R because the ranks ful�ll jS j � jR j.The partition M of V � V with classes f(x; y)g for all x; y 2 V is called themaximum con�guration, and the partition N consisting of the two classes I andK = f(x; y)jx 6= yg is called the minimum con�guration of V � V . It is easy tocheck that both M and N are coherent. The following result provides a usefulcondition for compatibility with �:Lemma 9. A coherent con�guration R on V is compatible with � if and only ifR is a re�nement of the distance partition D on V � V , i.e., D � R .Let � be an arbitrary graph with vertex set V , then the maximum con�gurationM on V is consistent with �. Of course M is not interesting in itself becauseit does not contain any information about �. It guarantees, however, that the{ 28 {

P.F. Stadler: Landscapes and Their Correlation Functionstheory developed in the remainder of this contribution is well de�ned for all graphs.Clearly, our considerations will be of interest only if � admits compatible coherentcon�gurations that are much coarse than M . It will be shown in the followingsection that the most interesting landscapes in fact \live" on con�guration spacesthat admit very coarse coherent con�gurations. The only graph with which theminimum con�gurationN is compatible, however, is the complete graphKjV j withvertex set V .The importance of coherent con�gurations comes from the algebraic properties ofthe matrices associated with its classes.Proposition 7. The partition R of V �V is a coherent con�guration if and onlyif the matrices associated with the classes of R ful�ll:(i) XX2R X = J.(ii) I is the sum of some elements of R .(iii) X 2 R implies X+ 2 R(iv) XY = XZ2R pZXYZProof. See [74, 76].The matrices associated with the classes of R form therefore (the standard ba-sis of) an algebra hR i that is closed (i) under the component-wise (i.e. Schuror Hadamard) product, (ii) closed under ordinary matrix multiplication and (iii)closed under transposition. Conversely, any matrix algebra ful�lling (i), (ii), and(iii) has a standard basis of the above form, i.e., there is a one-to-one correspon-dence between coherent con�gurations and matrix algebras of this type, for whichHigman [76] has introduced the term coherent algebra. The condition for compat-ibility with � translates to \hR i is compatible with � if and only if the adjacencymatrix A is contained in hR i".Proposition 8. The matrices de�ned by YXZ = pZXY for all X ;Y;Z 2 R form amatrix algebra hR i, the so-called intersection algebra. hR i is isomorphic to hR i.{ 29 {

P.F. Stadler: Landscapes and Their Correlation FunctionsProof. See [74, 76].It will sometimes be necessary to consider additional conditions on R . The mostimportant ones have been considered already in Higman's papers:De�nition. A coherent con�guration R is� homogeneous if I 2 R ;� commutative if pZXY = pZYX for all X ;Y;Z 2 R , i.e., if XY = YX for allX;Y 2 hR i;� symmetric if X = X+ for all X 2 R .Higman [74] showed that symmetry implies commutativitity which in turn im-plies homogeneity. Conversely, a homogeneous coherent con�guration with rankjR j � 5 is already commutative [74, (4.1)]. Commutative coherent con�gurationshave important applications in coding theory [77, 78]. They have been stud-ied extensively under the name association schemes by Delsarte [79]. Symmetriccoherent con�gurations are oftentimes termed symmetric association schemes orsimply association schemes; the matrix algebra associated with them is known asBose-Mesner algebra [80].The adjacency matrix A of a graph � generates an algebra hA i, the so-calledadjacency algebra of �, see e.g. [43]. Now let R be a coherent con�gurationcompatible with �. Since A 2 R it is clear that the adjacency algebra is a sub-algebra of the coherent algebra hR i. It has been shown [74] that hA i = hR i ifand only if R is commutative. This result can be used to explicitly construct Rfor a given graph, see [81].Not all graphs admit interestingly small coherent con�gurations. On the otherhand there are interesting classes of graphs that give rise to very coarse coherentcon�gurations. A prominent example are distance regular graphs, which haveattracted a lot of interest in discrete mathematics, as the recent monograph [82]shows. As an immediate consequence of their de�nition we have the following{ 30 {

P.F. Stadler: Landscapes and Their Correlation FunctionsRemark. � is distance regular if and only if its distance partition D is a coherentcon�guration. Examples include the Hamming graphs and the Johnson graphs butnot the Cayley graphs of the symmetric group discusses in the previous section.In the following subsection we will brie y review a very general construction forcoherent con�gurations. All this material is known in the literature, it will serve,however, as an additional motivation for the application of coherent con�gurationsto the study of landscapes.5.3.Graph AutomorphismsAn automorphism of the graph � = (V;E) is a one-to-one map � : V ! V suchthat �(x) and �(y) are adjacent if and only if x and y are adjacent. The setof all automorphisms of a graph forms a group under composition, the so-calledautomorphism group Aut[�]. It is a permutation group acting on the point setV .Now consider a permutation group G acting on a �nite set V . On V � V is actscomponent-wise: �(x; y) def=== ��(x); �(y)�. The orbits of G on V � V are calledorbitals. They form a partition G on V � V .Proposition 9. The partition G of V � V induces by the orbitals of a group Gacting on V is a coherent con�guration. The coherent algebra hG i coincides withthe centralizer algebra of the permutation representation of the group G.Proof. See [83, 84, 74].Thus any graph � with a non-trivial automorphism group Aut[�] admits a coher-ent con�guration G �M that is strictly coarser than the maximum con�gurationM . Important properties of the permutation groups translate into the propertiesof coherent con�gurations discussed at the end of the previous section:{ 31 {

P.F. Stadler: Landscapes and Their Correlation FunctionsProposition 10. Let G be a permutation group acting on V and let G be thecoherent con�guration induced on V � V . Then:G is transitive () G is homogeneousG is multiplicity-free () G is commutativeG is generously transitive () G is symmetricProof. See [84].The �rst condition is of particular importance: All Cayley graphs have transitivegroups of automorphisms [85], and thus the corresponding coherent con�gurationsG are homogeneous. The Hamming graphs and the Johnson graphs, for example,are distance transitive, i.e., for any two pairs of vertices (u; v) and (x; y) ful�llingd(x; y) = d(u; v) there is an automorphism � such that �(x; y) = (u; v), see, e.g.,[82]. This implies in particular that Aut[�] is generously transitive, i.e., that forany two vertices u; v 2 V there is an automorphism ful�lling �(u; v) = (v; u).5.4.Equitable PartitionsDe�nition. Let $ be a partition of the vertex set V of �. $ is called equitableif the number of neighbors which a vertex y 2 Y has in class X is independent ofthe choice of the vertex in Y . In other words, $ is equitable if for all X;Y 2 $holds AXY def=== j@fyg \Xj = Xx2XAxy for all y 2 Y: (34)We call A the collapsed adjacency matrix of � with respect to $. If an equitablepartition $ contains a class $0 = fug consisting of a single vertex u 2 V we willsay that it is anchored at (the reference vertex) u.With each partition $ of V intoM +1 classes there is an associated (M +1)�jV jmatrix, which we will also denote by $. Its entries are$Xx def=== � 1 if x 2 X0 otherwise (35){ 32 {

P.F. Stadler: Landscapes and Their Correlation FunctionsWe remark that this de�nition is the transpose of the convention in Godsil's book[77], while it conforms the notation in Bollob�as' book [86]. Equitable partitionshave been introduced by Schwenk [87]; more recently they have been used byPowers and coworkers under the name colorations, see, e.g., [88, 89], see also [51,Chap.4]. The following lemma collects their most useful properties.Proposition 11. Let $ an equitable partition of the vertex set V of a graph �.Then the following statements hold.(i) $A = A$.(ii) If ' is an eigenvector of A with eigenvalue �, then v = $' is a right eigen-vector of A with the same eigenvalue � provided v 6= 0. For later referencewe note the explicit formula v(X) = Xx2X '(x) : (36)(iii) If v is a right eigenvector of A, then u(X) = (1=jXj)v(X) de�nes a lefteigenvector of A.(iv) If u is a left eigenvector of A, then u$ is an eigenvector of A which isconstant on all classes of $.(v) jXj[As]XY = jY j[As]Y X for all s.(vi) The characteristic polynomial of A divides the characteristic polynomial ofA.(vii) If � is connected then A and A have the same spectral radius.(vi') If $ is anchored at some reference vertex x0 2 V then A and A have thesame minimal polynomial.(vii') If $ is anchored at some reference vertex x0 2 V then � is an eigenvalue ofA if and only if � is an eigenvalue of A.(viii) p(A) = p(A) for any polynomial p.(ix) If $ is anchored at x0 2 V then [As]yx0 = [As]y0x0 whenever y and y0 belongto the same class of $.Proof. See [77, chap.5] for the proofs of (i) through (vii). Properties (vi') and (vii')are [86, Thm.8.6], (viii) is [90, Lem.1], and (ix) follows from (viii) and [As]yx0 ={ 33 {

P.F. Stadler: Landscapes and Their Correlation Functions[As]x0y = [As]x0Y for all y 2 Y which is true because fx0g is a class of $ on itsown.De�nition. Let R be a partition of V � V , and let x0 2 V . Then the partitionR x0 of V anchored at (the reference vertex) x0 2 V has the classesXx0 = fx 2 V j (x; x0) 2 X g ; (37)where X 2 R .It is trivial to check that R x0 is in fact a partition of V . Note that Xx0 asde�ned above can be empty. The interest in this constructions originates from thefollowing result:Lemma 10. Let R be a coherent con�guration on V and let x0 2 V . Then R x0is an equitable partition of V anchored at the reference vertex x0. The entries ofthe collapsed adjacency matrix ful�llAXx0Yx0 = XZ2AZ\A6=; pZXY : (38)By equ.(38) the entries in A do not depend explicitly on the reference vertex x0.If R is homogeneous, then none of the classes Xx0 in the \projection" is empty,and we can assume that the rows and columns of A are indexed by the classes ofthe coherent con�guration R .Note that the collapsed adjacency matrix A obtained from a coherent con�gurationR as described above is contained in the intersection algebra hR i. In fact, a basisof hR i is obtained by the same procedure:Proposition 12. Let R be a coherent con�guration and x0 2 V . ThenpZXY = YXZ = Xx2Xx0Yxz for any z 2 Z : (39){ 34 {

P.F. Stadler: Landscapes and Their Correlation FunctionsProof. See [74].Proposition 13. Let R be a coherent con�guration compatible with �. ThenhR i is commutative if and only if all eigenvalues of A are simple.Proof. See [74].The following proposition is the main result of [91]. In this paper $ has beenassumed to arise from the orbits of a transitive automorphism group of �. However,only the properties of equitable partitions anchored at a reference vertex x0 2 Vare actually used for its proof.Proposition 14. Let � be a D-regular connected graph with adjacency matrixA and let $ be an equitable partition of V anchored at x0 2 V . Let g : $ ! IRbe a real valued function of the classes of $ and de�ne g� : IN! IR byg�(s) def=== XY 2$0@Xy2Y [Ts]yx01A g(Y ) : (40)Then g� is an exponential function if and only if g is a left eigenvector of thecollapsed adjacency matrix A of �. In this case g�(s) = g(fx0g) � (�=D)s.Proof. We sketch only the main ideas of the proof here. The details can be foundin [91].(i) The �rst step is to construct bases of left eigenvectors fuig and right eigenvec-tors fvig of A that ful�ll hui; vji = ci�ij with constant ci > 0. This is possiblesince A is diagonalizable. One can show furthermore that one may choose thesebases such that u(fx0g) = v(fx0g) = '(x0) = 1=pjV j.(ii) The next step is to express [Ts]yx0 in terms of the collapsed adjacency matrix.One �nds explicitly Xy2Y [Ts]yx0 = 1pjV jXk 1ck ��kD �s vk(Y ):(iii) The �nal step consists in expanding g w.r.t. the left eigenvectors of A, g(Y ) =Pk bkuk(Y ). Substituting this into the de�nition of g� yields after some calcula-tion g�(s) = (1=pjV j)Pk bk(�k=D)s.{ 35 {

P.F. Stadler: Landscapes and Their Correlation Functions5.5.Random Walks and Coherent Con�gurationsIn this section we link the simple random walks on � with the properties of coher-ent con�gurations that are compatible with �. Our goal of course is to eventuallyderive a relation between the \random walk" correlation function r(s) of a land-scapes and its autocorrelation function � with respect to a coherent con�gurationR on �. The �rst step is to consider the structure of T in some detail.Lemma 11. Let � be a D-regular (connected) graph with adjacency matrix A,and let T = (1=D)A be the transition matrix of a simple random walk on V .(a) If $ is an equitable partition of V anchored at x0 2 V , then we have for eachclass Y 2 $ and all y 2 Y : [Ts]yx0 = 1jY jXy2Y [Ts]yx0 :(b) If R is a homogeneous coherent con�guration on V and compatible with �,then for each class Y 2 R and each x0 2 V holdsXy2Y [Ts]yx0 = 1jV j X(y;x0)2Y[Ts]yx0 def=== #sY : (41)We are now in the position to derive a simple geometric relationship between r(s)and �(X ). The following lemma establishes a generalized version of a well knownresult for the distance classes of a distance transitive graph, see e.g. [15, 41].Lemma 12. Let � be a connected D-regular graph, let R be a homogeneouscoherent con�guration compatible with � and let f be a non-constant landscapeon �, with \random walk" correlation function r(s) and correlation function �with respect to R . Then r(s) = XX2R #sX �(X ) : (42)Theorem 2. Let � be a connected D-regular graph with adjacency matrix Aand let R be a homogeneous coherent con�guration on V compatible with �.{ 36 {

P.F. Stadler: Landscapes and Their Correlation FunctionsFurthermore let f be a non- at landscape on � with \random walk" correlationfunction r(s). Then:The autocorrelation function � of f with respect to R is a left eigenvector of A ifand only if r(s) is an exponential.Proof. Choose an arbitrary x0 2 V and consider the equitable partition R x0 .Furthermore de�ne g(Xx0) def=== �(X ) for all X 2 R . Theng�(s) def=== XYx02R x0 Xy2Yx0 [Ts]yx0g(Yx0) = XY2R #sY�(Y) = r(s)is the \random walk" correlation function of f as an immediate consequence oflemma 12 above. Proposition 14 guarantees that g� is exponential if and only ifg is a left eigenvector of A. Recalling that we may consider A indexed by theclasses of R instead of by the classes of R x0 completes the proof.5.6.Association Schemes and Distance Regular GraphsSuppose that R is a symmetric coherent con�guration (i.e., a (symmetric) asso-ciation scheme) compatible with the D-regular graph �. Then the algebra hR icoincides with the adjacency algebra of �, i.e., any X 2 hR i is a polynomial in A[74]. Thus the ONB f'ig of the Laplacian simultaneously diagonalizes all the ma-trices associated with the classes of R . The corresponding eigenvalues are knownas the eigenvalues of the association scheme, X'i = pi(X )'i for all X 2 R , see[77, p.225].Now consider the correlation function of a basis vector 'i with respect to R :!i(X ) def=== jV jjX j h'i;X'iih'i; 'ii = jV jjX jpi(X ) : (43)As a consequence of theorems 1 and 2 we know that !i is a left eigenvector of thecollapsed adjacency matrix, !iA = �i!i. Proposition 11.iii implies that vi(X ) ={ 37 {

P.F. Stadler: Landscapes and Their Correlation FunctionsjX j!i(X ) is the corresponding right eigenvector. Consequently we �nd that piitself is a right eigenvector of A. This is a well known result in the theory ofdistance regular graphs [82, sect. 4.1].The functions !i ful�ll the orthogonality relationXX2R jX j!i(X )!j(X ) = jV jm(�i) �i;j ; (44)where m(�i) is the multiplicity of the i-th eigenvalue of the Laplacian of �, see[82, Prop. 2.2.2].Even more is known if the graph � is distance regular. An association scheme iscalled P-polynomial if the matrices Dd corresponding to the distance classes canbe written as polynomials of degree d in terms of the adjacency matrix A. Anassociation scheme is P-polynomial if and only if it consists of the distance classesof a distance regular graph, see e.g. [77, sect. 12.3]. The Hamming graphs Qn�and the Johnson graphs J(n; k) are of this type, while the Cayley graphs of thesymmetric group are not distance regular, see [92] for the adjacency algebra of�(S4;T ).Suppose � is distance transitive; then for any two points (x; y) 2 V with distanced(x; y) = d there are c(d) neighbors of y with distance d � 1 from x and b(d)neighbors of y with distance d + 1 from x. The functions !i(d) def=== !i(Dd) ful�llthe following recursion:!i(d + 1) = 1 + c(d)� �ib(d) !i(d) � c(d)b(d)!i(d � 1) !i(0) = 1 (45)for 1 � d � d�. In this expressions �i is as usual the eigenvalue of the graphLaplacian corresponding to !i. Furthermore it can be shown that the !i for asystem of orthogonal polynomials [93]. We present here the explicit expressionsfor the Hamming and Johnson graphs, see also table 2.Hamming graphs lead to Krawtchouk polynomials:!l(d) = 1�nl�(�� 1)lKn;�l (d) = 1�nl�(�� 1)l lXj=0(�1)j(��1)l�j�dj��n� dl � j� (46){ 38 {

P.F. Stadler: Landscapes and Their Correlation FunctionsTable 2. Parameters for Hamming and Johnson graphs.Qn� J(n; k)Order jV j �n �nk�Degree D (� � 1)n k(n� k)Diameter d� n min[k; n� k]Eigenvalue �j j � j(n� 1 + j)Multiplicity m(�j) (�� 1)j�nj� n+1�2jn+1�j �nj�Polynomial !j Krawtchouk, equ.(46) Hahn, equ.(47)b(d) (�� 1)(n � d) (k � d)(n � k � d)c(d) d d2Krawtchouk polynomials play an important role in coding theory [94, 95].The Hahn polynomials are the terminating hypergeometric series of type3F2(a1; a2; a3; b1; b2; z) def=== 1Xj=0 (a1)j(a2)j (a3)j(b1)j(b2)j zjj!where Pochhammer's symbol (a)j is de�ned by (a)j = a(a+ 1) : : : (a+ j � 1) and(a)0 = 1. The polynomials associated with the Johnson graphs are!n;kl (d) =3 F2(�d;�l;�n+ (l � 1);�(n � k);�k; 1)= lXj=0 (�l)j (�n+ (l � 1))jj! (k � n)j (�k)j (�d)j : (47)6. SummaryA mathematical framework for the analysis of (�tness) landscapes on regulargraphs has been developed. By a landscape we mean a given function f de�nedon �nite set V of con�gurations together with a neighborhood relation betweencon�gurations that allows us to consider the \con�guration space" as an undi-rected graph. The basic ingredient of this theory presented here is the \Fourierseries" implied by the geometry of the graph that underlies the landscape. More{ 39 {

P.F. Stadler: Landscapes and Their Correlation Functionsprecisely, we use an expansion of the landscape in terms of an orthonormal baseof eigenfunctions of the Laplacian operator of the con�guration space upon whichthe landscape f is built, see Sect 3.We call a landscape elementary if it consists of an eigenfunction of the graphLaplacian plus an arbitrary constant function. The landscapes of a large numberof important examples from spin glass physics to combinatorial optimization are ofthis type, among them the Sherrington-Kirkpatrick spin glass and the TravellingSalesman Problem, see Sect. 4.Two types of correlation functions are commonly used for characterizing and com-paring landscapes: \Time-series" sampled along random walks give rise to the\random walk" correlation function r(s) of a landscape, see Sect. 2. Partition-ing the set of all pairs of con�gurations into suitable classes, such as the classesinduced by the natural distance measure between con�gurations. The autocorre-lation function �(X ) of a given landscape with respect to a given partition of thecon�guration space can be de�ned in a canonical way. It has interesting propertiesprovided the partition is consistent with symmetry properties of the con�gurationspace. For example, the distance partition turns out to be the natural choice onsequence spaces. In more technical terms, the partition must form a coherentcon�guration compatible with the graph-structure of the con�guration space, seeSect 5.This contribution has been concerned with elucidating the relationships betweenthe Fourier-expansion of a given landscapes and its two correlation measures r(s)and �(X ). The main result of this paper characterizes elementary landscapes interms of their autocorrelation functions. Summarizing theorems 1 and 2 we have:Main Result. Let � be connected D-regular graph, and let R be a homogeneouscoherent con�guration which is compatible with �. Furthermore let f be a non- atlandscape on � with \randomwalk" autocorrelation function r and autocorrelationfunction � with respect to R . Then the following statements are equivalent:(i) f is elementary. { 40 {

P.F. Stadler: Landscapes and Their Correlation Functions(ii) f � f1 is an eigenvector the Laplacian �� of � with a positive eigenvalue.(iii) r is of the form r(s) = a�s (with �1 � a < 1).(iv) � is a left eigenvector of the collapsed adjacency matrix A of � (with aneigenvalue smaller than D).Since any landscape on � is necessarily a superposition of elementary landscapes, soare their correlation functions. Thus suppose f =Pj aj'j, where ��'k = �k'k.Let �p, p = 0; : : : ;M designate only the distinct eigenvalues, and let Ip be anindex set such that �k = �p if and only if k 2 Ip. As usual we use �0 = 0 for thesmallest eigenvalue of ��. De�neBp def=== Xk2IpAp = Pk2Ip jakj2Pk 6=0 jakj2for all p 6= 0. The we have r(s) =Xp6=0Bp(1� �p=D)s:If furthermore all eigenvalues of the collapsed adjacency matrix A for which thereare non-zero coe�cients in the Fourier expansion of f are simple then we have�(X ) =Xp6=0Bpup(X )up(I) ;where up is a left eigenvector of A belonging to the eigenvalue �p. This is true inparticular for all landscapes on Hamming graphs and Johnson graphs.7.DiscussionMany important examples of landscapes are elementary, i.e., (up to an additiveconstant) they ful�ll a discrete analogue of the Helmholtz equation �f + �f = 0.Among them are Derrida's p-spin models, and the landscapes of the best knowncombinatorial optimization problems. Elementary landscapes exhibit a character-istic distribution of local optima on the con�guration space which depends crucially{ 41 {

P.F. Stadler: Landscapes and Their Correlation Functionson the corresponding eigenvalue � of the Laplacian. In particular, the \location"of � in the spectrum of the Laplace operator determines the maximum number ofnodal domains, that is the maximum number of disconnected islands of values off that are above average. The analogy with the properties of the eigenfunctionsof the Laplacian on a Riemannian manifolds strongly suggests the conjecture thata landscapes has to be more rugged if it corresponds to a higher \excited state"in the spectrum of the Laplacian. In fact, the nearest neighbor correlation onthe landscape is directly linked to the eigenvalue �. More precisely, the set ofall autocorrelation functions forms a simplex spanned by suitably normalized lefteigenvectors of the collapsed adjacency matrix of the con�guration space. Thecorresponding eigenvalues determine the decay of the correlation function along asimple random walk on the landscape.The relation between the Fourier expansion of a landscape and the computationalcomplexity of the optimization problem on the landscape is of great importancefor devising practical optimization heuristics. Disappointingly, there seems to beno simple relation. Most of the elementary landscapes discussed in this paper,such as the TSP, are NP-complete, but have only a fairly small number of non-zero Fourier coe�cients, namely those corresponding to a single eigenvalue of thecon�guration space.The formalism derived in this contribution suggests to approximate a given land-scape by a superposition of elementary landscapes, in particular if we do not havea closed form but only a computationally costly algorithm for evaluating it at par-ticular con�gurations such as, for example, in the case of RNA secondary structuremodels. The elementary landscapes depend only on the con�guration space andcan be obtained explicitly in many cases, in particular for all sequences spaceswith constant chain length (i.e., Hamming graphs). In practice it is even easierto directly compute the correlation functions of elementary landscapes. By theMain Theorem one needs only the collapsed adjacency matrix of the con�gurationspace, which is small enough in many cases that numerical solutions can be obtainseven if an analytical expression is unknown, as in the case of TSPs. A comparison{ 42 {

P.F. Stadler: Landscapes and Their Correlation Functionsof the autocorrelation functions obtained from computational studies of the land-scape of interest on the correlation functions of elementary landscapes on the samecon�guration space allow to estimate the amplitudes Bi, that is the coe�cients ofthe decomposition of the \experimental" correlation function into the elementaryones. These amplitudes can be used as an easy means of describing the landscape.On a Boolean hypercube, for instance, the amplitudes Bp have a very intuitiveinterpretation: they measure the relative importance of p-ary (spin) interactions.As an example we have seen in sect. 4.2 that the landscape of the \Low Auto-correlated Binary String Problem" consists of two \modes": An asymptoticallyvanishing contribution of a 2-spin model and a dominating mode correspondingto a 4-spin model. Asymmetric TSPs, as another example, consists of two modes,corresponding to the symmetric and the anti-symmetric part of the cost matrix.A study on the decomposition of RNA landscapes into \modes" corresponding toelementary landscapes will be reported elsewhere [96]. Of course, even a completeunderstanding of elementary landscapes is only the �rst chapter in the story oflandscapes: What can be said about the superposition of elementary landscapes,even if the structure of the elementary parts is known in detail?Many open questions remain. Are there better bounds than Courant's theoremon the number of nodal domain of an elementary landscape? What is the relationbetween eigenvalue and number of local optima in an elementary landscape? Howmuch of the theory outlined in this contribution carries over to con�guration spacewith less symmetry, such as spaces of �nite trees? Can we devise a comparableformalism for the combinatory maps of sequence-structure relations [16], where theimage of a con�guration is not a real-valued �tness but an element of an abstractmetric space?AcknowledgmentsStimulating discussions with P. Anderson, B. Davies, A.W.M. Dress, W. Fontana,R. Palmer, P. Phillipson, Chr. Reidys, and G.P. Wagner are gratefully acknowl-edged. The author is grateful to the GermanMax Planck Gesellschaft for support-ing his stay at the Santa Fe Institute in 1993 where the work on this manuscriptbegan. { 43 {

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P.F. Stadler: Landscapes and Their Correlation Functionsof [Ts�x0]. ThushF (xt)ix0;t = limT!1 1T + 1 TXt=0 1jV j Xx02V Xz2V F (z)[Tt�x0 ]z= limT!1 1T + 1 TXt=0 1jV jXz2V F (z)"Tt Xx02V �x0!#z= limT!1 1T + 1 TXt=0 1jV jXz2V F (z)[Tt1]zSince � is regular we have T1 = 1, and thereforehF (xt)ix0;t = limT!1 1T + 1 TXt=0 1jV jXz2V F (z) = limT!1 1T + 1 TXt=0 F :Since F is constant in \time" we have hF (xt)ix0;t = F .Proof of Lemma 2.As in the proof of lemma 1 we start withhF (xt+s; xt)ix0;t = limT!1 1T + 1 TXt=0 1jV j Xx02V Xz;z02V F (z0; z)[Ts�z ]z0 [Tt�x0 ]z= limT!1 1T + 1 TXt=0 1jV j Xz;z02V F (z0; z)[Ts�z]z0 "Tt Xx02V �x0!#z= limT!1 1T + 1 TXt=0 1jV j Xz;z02V F (z0; z)[Ts�z]z0 � 1= 1jV j Xz;z02V F (z0; z)[Ts�z]z0By de�nition we have [Ts�z]z0 = [Ts]z;z0 ; this completes the proof.Proof of Corollary 1.Use lemma 1 with F = f and F = f2, respectively, and lemma 2 with F (x; y) =f(x)f(y). Substituting the results into the de�nition of r(s) completes the proof.{ 52 {

P.F. Stadler: Landscapes and Their Correlation FunctionsProof of Lemma 3.�1 = 0 implies that ~f = f � f�1 is an eigenvector of �� with eigenvalue �. Thus~f either constant and � = 0, or � > 0 and h ~f ;1i = 0. In the �rst case f itself is a at landscape with f(x) = f� = f for all x 2 V . In the second case we havejV jf = Xx2V f(x) = h ~f ;1i + f�h1;1i = 0 + f�jV j :Consequently, if f is elementary, then the constant f� coincides with the meanvalue f of the landscape f . Connectedness of � implies that 0 is a single eigenvalueand thus ~f must be an eigenvector belonging to an eigenvalue � > 0.The converse is trivial: a landscape of the form (6) is always elementary.Proof of Lemma 4.The Laplacian��ofKn has only two distinct eigenvalues, �0 = 0 with multiplicity1 and eigenvector 1, and the n � 1-fold degenerate eigenvalue �1 = n [51]. Thusany non-constant f is of the form c1+' with some constant c 2 R and h';1i = 0,i.e., ' is an eigenvector belonging to �1.Proof of Lemma 5.It is convenient to allow arbitrary indices, setting �q = 0 for all q < 1 or q > n.With this convention we havef(�) = n�1Xk=1 n�kXi=1 n�kXj=1 �i�i+k�j�j+k= n�1Xk=1 n�kXi=1 24 Xj 6=i�k;i;i+k �i�i+k�j�j+k + �i�k�2i �i+k + �2i �2i+k + �i�2i+k�i+2k35= n(n� 1)2 + n�1Xk=1 n�kXi=1 [�i�k�i+k + �i�i+2k] + n�1Xk=1 n�kXi=1 Xj 6=i�k;i;i+k �i�i+k�j�j+kUsing the de�nition of ":::(�) and eliminating all terms containing a �q with q < 1or q > n completes the proof. { 53 {

P.F. Stadler: Landscapes and Their Correlation FunctionsProof of Corollary 2.The functions "i1;i2;:::;ip , where all indices ik are di�erent, are eigenfunctions of theLaplacian �� of the Boolean hypercube belonging to the eigenvalues �p. Equ.(9)in the proof of theorem 1 implies immediately that r(s) = aa+b(1�4=n)s+ ba+b(1�8=n)s where a and b are the sums of the squares of the coe�cients of "2 and "4,respectively. Since all coe�cients are 0 or 1 we simply need to count all non-zeroterms in lemma 5. We �nd that a = O(n2) and b = O(n3), and the corollaryfollows.Proof of Lemma 6.Without loosing generality we can consider the neighborhood of the identity per-mutation {, since the numbering of the cities is arbitrary.(i) In the case of transpositions we start with Grover's [47] formulaf((i; j)) � f({) = wj;i�1 + wi+1;j + wi;j�1 + wj+1;i� wi;i�1 � wi+1;i � wj;j�1 � wj+1;j+ (�j;i+1 + �j;i�1)(wij + wji):Summing over all i 6= j yields 2�f({) = 4X � 4nf({) + 2f({) + 2f�({) and analo-gously we obtain 2�f�({) = 4X � 4nf�({) + 2f�({) + 2f({), where X is the sumover all non-diagonal entries of W . The proposition follows immediately.(ii) In case of inversions we �ndf([k; l]) � f({) = wl;k�1 + wl+1;k �wk;k�1 �wl+1;l + l�1Xj=k(wj;j+1 � wj+1;j) :Summing over all k 6= l yields 2�f({) = 2X � 2nf({) + n(n � 1)2 [f�({) � f({)]. Acompletely analogous result is obtained for �f�({), and a short calculation thencompletes the proof. { 54 {

P.F. Stadler: Landscapes and Their Correlation FunctionsProof of Lemma 7.Without loosing generality we may evaluate the cost function of the identity per-mutation { since the labeling of the vertices is arbitrary. Using the labels k ofthe matched pairs as indices we have to following �ve types of neighbors, and thecorresponding changes of the cost function for each of them[11'] [W2j�1;2i +W2i�1;2j ]� [W2i�1;2i +W2j�1;2j ],[12'] [W2j;2i +W2j�1;2i�1]� [W2i�1;2i +W2j�1;2j ],[21'] [W2i�1;2j�1 +W2i;2j ]� [W2i�1;2i +W2j�1;2j ],[22'] [W2i�1;2j +W2j�1;2i]� [W2i�1;2i +W2j�1;2j ], and[11] W2i;2i�1 �W2i�1;2i.Summing the �rst four terms over all i 6= j (i.e., counting each neighbor twice)yields[oe] + [oe]� 2f({) � 2(n=2� 1)f({) + [ee] + [oo]� d� 2(n=2� 1)f({) +[oo] + [ee]� d� 2(n=2� 1)f({) + [oe] + [oe]� 2f({) � 2(n=2� 1)f({);where [oe], etc., denotes the sum over all Wij with odd i and even j, and d is thesum over all diagonal entriesWkk. Using thatW is symmetric, i.e., [oe] = [eo], thissum my be written as 2�f({) = 2Xi6=j Wij � 4(n � 1)f({), since the transposition(2i � 1; 2i) gives a contribution of 0. Thus we have �f({) = 2(n � 1) [f � f({)],i.e., � = 2(n� 1), and f is elementary on the Cayley graph �(Sn;T ).Proof of Lemma 8.From the de�nition of � we �ndXX2R �(X ) � jX j = jV jhf; fi *f;0@ XX2R X1A f+ = jV jhf; fi hf;Jfi = jV jhf; fi :where J is the matrix with all entries 1. It is straightforward to check Jf = jV jf1,and thus hf;Jfi = jV j2f2 = 0. { 55 {

P.F. Stadler: Landscapes and Their Correlation FunctionsProof of Lemma 9.This statement is equivalent to claiming that X \ Dd 6= ; implies X � Dd if andonly if R is compatible with �. The \only if" part is trivial.Assume thus that R is compatible with �. We proceed by induction in d. Theclaim is true for d = 0 since R is pre-coherent, and for d = 1 by compatibilitywith �. Suppose now, that the claim is true for all distances up to d and considera pair of vertices (x; y) with distance d(x; y) = d + 1. Let Z be the class of Rto which (x; y) belongs. The triangle inequality implies that there is z 2 V suchthat d(x; z) = d and d(z; y) = 1 and let (x; z) 2 X and (z; y) 2 Y � A. ThereforepZXY > 0. Now consider an arbitrary pair (x0; y0) 2 Z. Since pZXY > 0 there isat least one z0 2 V which ful�lls d(x0; z0) = d and d(z0; y0) = 1. The triangleinequality implies d(x0; y0) � d + 1. If d(x0; y0) < d + 1 we have Z \ Dh 6= ; forsome h � d and thus, by the induction hypothesis, Z � Dh. This contradictsd(x; y) = d+ 1. We conclude d(x0; y0) = d+ 1 and therefore Z � Dd+1.Proof of Lemma 10.Consider three classes X ;Y;Z 2 R and an arbitrary vertex x0 2 V . We havepZXY = ���z 2 V ��(x; z) 2 X and (z; y) 2 Y�� 8 (x; x0) 2 Z= ���z 2 Yx0��(x; z) 2 X�� 8x 2 Zx0On the other hand we obtainAXx0Yx0 = ���z 2 Yx0��(x; z) 2 A�� = XX�A ���z 2 Yx0��(x; z) 2 X��for all x 2 Zx0. Comparing this with the above representation of the intersectionnumbers yields AXx0Yx0 = XZ2AZ\A6=; pZXY ;which is independent of the representatives of the classes by de�nition. Thus R x0is equitable. It remains to show that R x0 is in fact anchored in x0. X � I impliesthat Xx0 is either empty Xx0 = fx0g. By de�nition there is some X � I thatcontains (x0; x0), and thus fx0g 2 R x0.{ 56 {

P.F. Stadler: Landscapes and Their Correlation FunctionsProof of Lemma 11.(a) By proposition 11.ix we know that [As]yx0 = [As]y0x0 whenever y and y0 arein the same class of $, and thus the same holds true for the powers of T. Thuswe have for all y 2 Y Xy2Y [Ts]yx0 = jY j [Ts]yx0 :(b) Since R is a coherent con�guration and compatible with � we may writeAs =PX2R bXX, and thus As is constant on the classes of R . Furthermore, wehave shown in lemma 10 that the projection Yx0 is an equitable partition anchoredat x0. Thus we have Xy2Yx0[Ts]yx0 = jYx0j [Ts]yx0 , where the second factor on ther.h.s. depends only on the class Y to which the pair (y; x0) belongs. The �rstfactor isjYx0 j = �y 2 V �� (y; x0) 2 Y = �y 2 V �� (y; x0) 2 Y and (x0; y) 2 Y+ = pXY+Y ;where X is the class of R to which (x0; x0) belongs. Since R is homogeneous byassumption we have X = I, and thus jYx0j = pIY+Y for all x0 2 V . Consequentlywe have jYj = jV j � jYx0 j and the �rst factor is independent of x0 as well. We haveX(y;x0)2Y[Ts]yx0 = Xx02V Xy2Yx0[Ts]yx0and part (b) of the lemma follows immediately.Proof of Lemma 12.We begin with the de�nition of r(s). Without loosing generality we assume f = 0and �2f = 1. Thusr(s) = 1jV j Xx02V Xy2V [Ts]yx0f(y)f(x0) = 1jV j Xx02V XYx02R 0 Xy2Yx0[Ts]yx0f(y)f(x0 ) :As a consequence of lemma 11.a we haver(s) = 1jV j Xx02V XYx02R 00@ Xz2Yx0 [Ts]yx01A� 1jYx0j Xy2Yx0 f(y)f(x0) ;{ 57 {

P.F. Stadler: Landscapes and Their Correlation Functionsand using lemma 11.b we may rearrange this asr(s) = 1jV j Xx02V XYx02R 0 #sY 1jYx0 j Xy2Yx0 f(y)f(x0)= XY2R #sY 1jV j Xx02V 1jYx0j Xy2Yx0 f(y)f(x0) = XY2R #sY 1jYj X(x;y)2Y f(x)f(y) :Substituting the de�nition of �(Y) completes the proof.

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