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Isotropy and Metastable States:The Landscape of the XYHamiltonian RevisitedRicardo Garcia-PelayoPeter F. Stadler

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SANTA FE INSTITUTE

Isotropy and Metastable States

The Landscape of the XY Hamiltonian Revisited

Ricardo Garc��a�Pelayoa and Peter F� Stadlerb�c

aInstituto de F��sica� Universidad Nacional Aut�onoma de M�exico� M�exico D�F�

bInstitut f� Theoretische Chemie� Univ� Wien� Austria

cThe Santa Fe Institute� Santa Fe� New Mexico� U�S�A�

�Mailing Address� Peter F� StadlerInst� f� Theoretische Chemie� Universit�at Wien

W�ahringerstr� �� A�� Wien� AustriaPhone� �� Fax� ��

E�Mail� studla�tbi�univie�ac�at or stadler�santafe�edu

Garc��a�Pelayo � Stadler� The XY Hamiltonian

Abstract

The number of local optima is an important characteristic of a landscape� It appears to dependon the pair�correlation of the landscape as well as on its deviations from statistical isotropy� Weuse Tanaka and Edward�s XY Hamiltonian as an example for an investigation of the relationshipsbetween ruggedness� metastable states� and isotropy�

�� Introduction

Spin glass Hamiltonians� cost functions of a combinatorial optimization problems

and ��tness function� in models of biological evolution can be regarded as map

pings f from the vertex set V of a usually huge� but �nite� graph � into the

real numbers� The vertex set V is the set of all possible con�gurations e�g�� spin

orientations� tours of a traveling salesman problem TSP�� or DNA sequences��

The set E of edges of the graph � V�E� is introduced by de�ning a �move

set� that allows to interconvert �neighboring� con�gurations� see e�g� ��

These �elementary moves� connect neighboring con�gurations� A move sets may

consist� for instance� in �ipping single spins� in exchanging two cities along the

salesman�s tour� or in mutating single nucleotides of a DNA sequence� A map

ping f � V � IR has been termed landscape following a picture of evolutionary

optimization originally proposed by Sewall Wright ��

Most landscape models contain a stochastic element in their de�nition� a par

ticular instance is generated by assigning a usually large� number of parameters

at random� Such models are called random �elds �� A typical example is the

SherringtonKirkpatrick Hamiltonian ��

fx� def

��

Xi�j

Jijxixj

where x x�� xn� denotes a con�gurations of spins xi �� and the �coupling constants� are identically and independently distributed i�i�d�� Gaussian

random variables� Instead of specifying the distribution of the random parameters

one may as well de�ne the joint distribution

P �u� def

��Prob�fx� � ux jx � V �

� � �


of the ��tness values� of all con�gurations x � V � Mathematically speaking� a

random �eld is the probability space whose point set is ff � V � IRg and measuredP � see e�g� �� We shall use the notation

E � � � def

��

Z� dP

for the average of a quantity w�r�t� to this measure� that is� the average over

the disorder� In the SK model this amounts to integrating over the Gaussian

distributions of the interaction coe�cients Jij � We shall restrict the use of the

term landscape to individual mappings f � V � IR� Therefore� an element a

realization� an instance� of a random �eld on � is thus a landscape� corresponding

to quenched disorder�

Because the number N jV j of vertices is very large in general� we are interested in a simple� statistical description of a landscape or a random �eld model�

The most important characteristic is probably the ruggedness of the landscape�

which is closely related to the hardness of the optimization problem for heuristic

algorithms �� Three distinct approaches haven been proposed to measure and

quantify ruggedness� Sorkin �� Eigen et al� �� and Weinberger �� used pair cor

relation functions� Kau�man and Levin �� proposed adaptive walks� and Palmer

�� based his discussion on the number of metastable states local optima�� Of

course one expects a close relationship between these di�erent characterizations of

ruggedness� In this contribution we shall concentrate on the relation between the

number of local optima and paircorrelation functions using a particular class of

spin glass Hamiltonians� the XY model� as an example�

�� The Discrete XY Hamiltonian

The discrete version of the XY model was introduced by Tanaka and Edwards in

�� There are n spins in the plane� each of which can point into one of the �

directions

��i

�cos �isin �i

�with �i

��

�xi� � � xi � � ��

� � �


The interaction energy between two spins i and j is given by

Jijh��i� ��ji Jij cos�i � �j� � ��

with coupling constants Jij � The Hamiltonian takes the form

Hx� Xi�j

Jij cos�i � �j� Xi�j

Jij cos

��

�xi � xj�

��

The coupling constants Jij Jji� i � j� are i�i�d� Gaussian random variables� The

XY Hamiltonian is thus a random �eld in the sense of �� The set V of all possible

spin con�gurations contains N �n points�

Two de�nitions of neighborhood between spin con�gurations are natural for the

XY model�

i� Two con�gurations are neighbors if they di�er in the orientation of a single

spin� Given a con�guration � let us denote by ��l�m� the con�guration that has

the same spin orientations for all spins except for spin l which has orientation

��l�m�l �l ��m

��

�� xl m�mod��

The set of all neighbors of � is thus generated when l runs over all spin

and m runs from � to � � �� This neighborhood relation arranges the �nspin con�gurations as a Hamming graph Qn

� which is the nfold Cartesian

or direct� product� of the complete graph Q� with � vertices� A graph is

complete if there is an edge between any two vertices�� Each con�guration

x � V has D n�� neighbors in Qn��

ii� Two con�gurations are neighbors of each other if a single spin orientation

di�ers by �� while all the other spins are the same� With the abovenotation� we obtain all neighbors of a con�guration when l runs over all spins

and m �� for � � �! there is only one value of m for � �� The

corresponding graphs are the nfold direct products of the cycle graphs C��which we shall denote by Cn� � This notion of neighborhood was used in ��

�The �Cartesian or direct� product �� of two graphs �� V�� E�� and �� V�� E�� hasthe vertex set V � V� � V�� Two vertices �x�� x�� and �y�� y�� of the product are connected byan edge if either �i� x� � y� and x�� y� are adjacent in �� or �ii� x� � y� and x�� y� are adjacentin �� This graph product is called the sum in ��

� � �


Each spin con�guration has D �n neighbors in Cn� if � � �� and only D n

neighbors if � ��

Note that Cn� � Qn� and Cn� � Qn

� � while Cn� �� Qn� for all � � �� Later on we

shall see that Cn� � Q�n� � Direct products of larger cycles are not isomorphic to

Hamming graphs�

�� Elementary Landscapes

A landscape is a function de�ned on the vertex set V of a graph �� It is convenient

to describe this graph by its adjacency matrix A which has the entries Axy �

if the vertices x and y are connected by an edge� and Axy � if x and y are not

neighbors of each other� The number of neighbors of a vertex x is called the degree

of x� It is customary to arrange these numbers in a diagonal matrix D such that

Dxx is the degree of vertex x� A graph � is regular if all vertices have the same

number of neighbors� i�e�� if D DI� where I denotes the identity matrix�

For our purposes it will be more useful to represent � by its Laplacian �" def��D�

A� The matrix " is a discretization of the familiar Laplacian di�erential operator!

see �� for reviews on graph Laplacians� We shall use that�" is symmetric�nonnegative de�nite� and its smallest eigenvalue is #� �� which corresponds to

a constant �at� eigenfunction� Furthermore� the multiplicity of #� equals the

number of connected components of �! hence � is connected if and only if #�

is a simple eigenvalue of �"� Throughout this paper we shall assume that � isconnected and regular�

De�nition� The landscape f is elementary if there are constants # � � and $f

such that

�"f�x� #fx�� $f� x � V � ��

The notation "f�x� denotes the xcomponent of the function "f � V � IR

which is de�ned pointwise asP

y�V "xyfy��

Equivalently� f is elementary if and only if

�x� def

�� fx��

N

Xz�V

fz� ��

� � �


is an eigenfunction of the graph Laplacian with a nonzero eigenvalue # � �

�� Elementary landscapes play an important role because of their algebraic

properties �� and because a large number of wellstudied examples

from spin glass physics such as Derrida�s pspin models� and from combinatorial

optimization among them the TSP� are elementary� see ��

A landscape f can be decomposed into a superposition of elementary landscapes�

Since �" is symmetric there is an orthonormal basis f�kg of eigenvectors� Theeigenvector �� belonging to the eigenvalue #� � is constant� Such a basis has

been termed Fourier basis� and an expansion of the form

fx� N��Xk��

ak�kx� ��

may be called a Fourier series of the landscape �� see section � for more details�

Two types of correlation functions have been investigated as a means of quantifying

the ruggedness of a landscape� Eigen and coworkers �� introduced d� which

measures the pair correlation as a function of the distance between the vertices of

�� Weinberger �� used the �time series� ffx�� fx�� g generated by a simplerandom walk �� on � for measuring properties of f � The autocorrelation of this

�time series� is

rs� def

��hfxt�fxts�ix��t � hfxt�ix��t hfxts�ix��tq� hfxt��ix��t � hfxt�i�x��t

� �hfxts��ix��t � hfxts�i�x��t� � ��

The notation h � ix��t emphasizes that the expectation is taken over all �times� tand all initial conditions x�� The relation between rs� and d� is discussed in ��

��

The correlation function rs� is intimately related to the Fourier series expansion

of the landscape �� Elementary landscapes belonging to the eigenvalue #p have

exponential autocorrelation functions of the form rs� � � #pD�s if � is aregular graph� In general� the correlation function can be written as

rs� Xp��

Bp�� #pD�s � ��

� � �


The amplitudes Bp can be obtained from the Fourier expansion of the landscape�

Bp Xk�Ip

jakj��X

k ��

jakj� � ��

where Ip denotes the set of the indices j for which �"�j #p�j � Consequently�Bp is nonnegative� It is easy to check that a� f � and hence the �component of

f is canceled by the subtraction of f in the de�nition of the correlation function�

Consequently� the sum in equ�� runs only over the non�at components� k � ��The amplitudes Bp determine the relative importance of the di�erent modes� A

landscape is elementary� therefore� if it contains only a single non�at mode p�

The crucial information about a landscape is therefore contained in the eigenvalues

#p and the amplitudes Bp which describe the relative importance of the di�erent

elementary components� In particular� the correlation length

� def

��

�Xs��

rs� DXp��

Bp

#p��

is a very useful measure for the ruggedness of the landscape f �� It

will play a prominent role in this contribution� Alternatively� one might de�ne a

correlation length %� such that r%�� e� as in most of the work on Nk and RNA

landscapes� see e�g� �� In practice� the value of %� depends on the details of

the interpolation procedure that is used to extend rs� to noninteger values of s�

For an elementary landscape we have � D#p while

%� � �

ln�� #pD� D#p � �� O#pD� �� O#pD� � ��

In our applications we shall always �nd #p O�� while D On�� Thus thediscrepancy between � and %� becomes negligible� We shall use � instead of %� because

i� it is de�ned unambiguously for all autocorrelation functions rs�� and because

ii� it leads to much simpler algebraic expressions�

In the reminder of this section we show that each instance of the XYHamiltonian

is elementary for arbitrary � and with respect to both de�nitions of neighborhood

between spin con�gurations�

� � �


Theorem �� The discrete XYHamiltonian is elementary on Qn� with eigenvalue

# �� for all n � � and all � � ��

Proof� The Laplacian of the Hamiltonian is given by

"H�� nXl��

��Xm��

hH��l�m�

��H��

i

Xi�j

Jij

nXl��

��Xm��

hcos�

�l�m�i � �

�l�m�j �� cos�i � �j�

i

�Xi

nXl��

Jil

��Xm��

cos��i � �l � ��m

�� cos�i � �l�

�

The addition formula for the cosine yields

cos��i � �l � ��m

� cos

��

�m

�cos�i � �l�� sin

��

�m

�sin�i � �l�

The sinterms vanish when summed over all m� For the cosineterms we observe

��Xm��

cos

��

�m

�

��Xm��

cos

��

�m

��

Thus we have �nally

"H�� Xi�l

Jil�� cos�i � �l� ��H��

i�e�� the XY Hamiltonian is an eigenfunction of the graph Laplacian of the Ham

ming graph Qn� with eigenvalue # ��

Corollary� The correlation length of the XY Hamiltonian on Qn� is

� D

#

n��

� ��

Theorem �� The discrete XYHamiltonian is elementary on Cn� with eigenvalue

#

�� if � �� sin�� if � � �

� � �


for all n � � and all � � ��

Proof� The proof is similar to the above theorem� The only di�erence is� that

the sum over m now runs only over m �� when � � �� Since Cn� Qn� � this

cases is already taken care of by theorem �� After using the addition formula for

the cosine we observe that the sineterms cancel since sin�x� � sinx�� Hencewe are left with

"H�� Xi�l

Jil

� cos

��

�

�� cos�i � �l� ��

�� cos

��

�

� H��

The identity � sin� x �� cos�x� completes the proof�

Corollary� The correlation length of an XY Hamiltonian on Cn� is

� n

� sin��

In order to compare and interpret these �ndings we shall need more information

about the spectra of the graph Laplacians of Qn� and Cn� � To this end it will be

convenient to consider a broader class of highly symmetric graphs�

�� Cayley Graphs and their Spectra

Both the complete graph Q� and the cycle C� can be derived from the cyclic

group ZZ� by means of the the socalled Cayleygraph construction� A subset &

of a group G� �� is called a set of generators if each group element u � G can berepresented in the form u x� � x� � � � � � xk as a �nite �product� of generatorsxi � &� Now suppose that i� the group identity � is not contained in & and thatii� x � & implies that the inverse element x�� &� The Cayley graph �G�&�has then the vertex set G and there is an edge connecting two x and y if and onlyif xy�� &� It is trivial to check that the graphs Q� and C� are Cayley graphsof the commutative group ZZ� f�� g� The corresponding sets ofgenerators are

&Q ZZ� n f�g &C f�� g �

� � �


It is not hard to show �� that the product graph of two Cayley graphs �G��&��and �G��&�� is the Cayley graph �G� � G��&� � f��g f��g � &�� where ��

and �� are the group identities of G� and G�� respectively� Consequently� both theHamming graphs Qn

� and the graphs Cn� are Cayley graphs of the nfold Cartesianproduct of the group ZZ� with itself� which is of course again a commutative group�

Eigenvalues and eigenfunctions of the Laplacian of a Cayley graph of a commuta

tive group are not hard to construct explicitly� see e�g� �� One can decompose

any �nite commutative group G into a direct product of m cyclic groups ZZNk of

order Nk �� We can choose a componentwise representation of G such thatcomponentwise representation of the group elements such that the group action

becomes componentwise addition

x � y x� y� mod N�� x� y� mod N�� xm ym mod Nm� �

A set of basis functions is then given by the characters �� of G�

�gx� exp

��iXk

xkgkNk

� ��

The corresponding eigenvalues of �" are #g P

x�

�� gx�

�� Alternatively

one can use the properties of the graph product in order to construct the spectrum

and an eigenbasis for the product graph �� from its components� see e�g� ��

Lemma �� #i�i� #i� #i� is an eigenvalue of �� with eigenfunction�i�i�x�� x�� i�x��i�x��

The latter method was used to compute the eigenvectors and eigenvalues of the

Hamming graphs Qn� in �� starting with the spectrum of the complete graph

Q�� see also �� sect� �� There are n � distinct eigenvalues #p p�� with p

�� n� The XYHamiltonian belongs therefore to the third smallest eigenvalue

i�e�� the second excited state� #� of the graph Laplacian of the Hamming graph

Qn��

The eigenvalues and eigenvectors of the cycle graphs C� are also well known� Anorthogonal system of eigenfunctions is�

�� cos

��k

�x

�� sin

��k

�x

��

�� k � �

�

��

� � �


where the associated eigenvalues �k of the adjacency matrix and #k of the Lapla

cian are

�k � cos

��k

�

�and �k � sin

�

��k

�

��

The multiplicities mk � except for k � and k �� where mk ��

Lemma �� The XY Hamiltonian belongs to the thirdlargest eigenvalue #�� of

the Laplacian of the graph Cn� for all � � ��

Proof� We have of course #� � as a simple eigenvalue� The smallest nonzero

eigenvalue #� is composed of �� from a single component and �� from the

remaining n� � cycles� The next largest eigenvalues are #�� sin��which has the XY model as eigenvector� and #� �� sin�� which

can be constructed for � � � only� It is easy to check that the �k are strictly

increasing with k� Thus the lemma follows if #� � #�� for � � �� In fact� we

have cos�� p� for all � � � equality holds i� � �� Multiplying by

� sin�� yields

� sin�� cos�� sin�� p#��

p� sin��

p#��

Since the expressions on both sides are positive we can take the squares and the

lemma follows�

The multiplicities of the eigenvalues of Cn� will be discussed in some detail at theend of the following section�

�� Isotropy

A quantity that is closely related to the autocorrelation function discussed in

section � is the covariance matrix C with the entries

Cxy E �fx�fy�� E �fx��E �fy��

Clearly� C does not depend on the neighborhood structure among the con�gu

rations� hence we obtain the same covariance matrix for the XYHamiltonian on

� ��


both Qn� and Cn�� We have E �fx�� for all x because E �Jij� �� Thus the

entries of the covariance matrix are

Cxy Var�Jij �Xi�j

cos

��

�xi � xj�

�cos

��

�yi � yj�

�

Var�Jij �

�

Xi�j

cos

��

��xi yi�� xj yj��

�cos

��

��xi � yi�� xj � yj��

�

where Var�Jij � denotes the common variance of the coupling constants� The co

variance matrix therefore depends not only on the angles ��xi� yi� between

two orientations of the same spin but also on their sum except for � � where

sum and di�erence are the same thing�� There is large amount of symmetry in this

matrix� an arbitrary permutation of the indices does not change the covariance

matrix�

The variance of the Hamiltonian at a given con�guration x � V is given by

Var�Hx�� Cxx Var�Jij �Xi�j

cos��

�xi � xj�

��

If � � then jxi � xj j is either � or �� Thus the argument of the cosine is either� or �� and we have

Var�Hx�� Var�Jij � �n

�

��

The same is true for arbitrary � if all spins are aligned in the con�guration x� If

there are di�erent spin orientations� however� then some of the cosines are smaller

than �� provided � � �� and the variance in no longer independent of x�

Another interesting property of the covariance matrix of the XYHamiltonian is

wx� def

��

Xy�V

Cxy Var�Jij �Xi�j

cos

��

�xi � xj�

�Xy

cos

��

�yi � yj�

� � �

Equivalently� � �� is an eigenvector of C� In �� a random �eld was

termed pseudo�isotropic if E �fx�� Var�fx�� and wx� are independent of x� TheXYHamiltonian is hence not pseudoisotropic for � � � because Var�fx�� is notconstant�

� ��


For a detailed discussion of isotropy we shall need some information on the sym

metries of the con�guration spaces� An automorphism of a graph � is a onetoone

map a � V � V that preserves adjacency� i�e�� fax�� ay�g is an edge of � if andonly fx� yg is an edge� The set of all automorphisms forms the automorphismgroup Aut�� A graph is vertex transitive if Aut�� acts transitively on V � i�e�� if

for any two vertices x and y there is an automorphism a such that y ax�� All

Cayley graphs have this property� see e�g��

The action of Aut�� on the set of all ordered pairs of vertices induces a partition

R of V � V the classes of which are the orbits Aut�� The class X containing

a particular pair x�� y�� of vertices is X � �

ax�� ay�� j a � Aut��

�� In

particular� the diagonal I fx� x�jx � V g is class of R if and only if � is vertex

transitive�

All vertex pairs that belong to the same class are equivalent w�r�t� the symmetry

of �� It makes sense then to ask whether a given random �eld model has the same

symmetries as its underlying con�gurations space�

De�nition� A random �eld is isotropic if and only if there is a constant a� and

a function c � R � IR such that i� E �fx�� a� and ii� Cxy cX � holdsfor all x� y� � X � i�e�� the covariance matrix the random �eld is constant on the

symmetry classes of underlying graph ��

We de�ne isotropy here as a second order quantity� A stronger version of isotropy

is obtained by requiring that the distribution function P �u� is invariant under all

automorphisms of �� The two de�nitions become equivalent of P �u� is Gaussian�

A fairly general algebraic theory of isotropy is laid out in �� We shall not use

the most general form of these results but rather restrict our discussion to a class

of graphs that contain both the Hamming graphs Qn� and the direct products of

cycles Cn��

Proposition� A random �eld is isotropic on a Cayley graph �G�&� of a commutative group G if and only if its Fourier coe�cients ful�lli� E �ak� � for all k � �!ii� E �aka�l � �klE �jakj��!

� ��


iii� E �jajj�� E �jakj�� p whenever the corresponding eigenfunctions �j and �k

belong to the same eigenvalue #p of the graph Laplacian�

Proof� The proposition is a specialization of �� thm��

In the case of Ising models this condition means that there is no structure in the

interaction coe�cients Jij � i�e�� all the Jij are assigned as i�i�d� random numbers�

Thus the SherringtonKirkpatrick Hamiltonian is isotropic� while short range spin

glasses do not have this property�

In general� the number of uncorrelated nonzero Fourier coe�cients in an isotropic

elementary random �eld equals the dimension m#p� jIpj of the correspondingeigenspace of the graph Laplacian �"� This observation suggests to interpretisotropy as a maximum entropy type condition� Given the parameters �p� the

�most random� choice of coupling constants are Gaussian random variables ful�ll

ing i� through iii�� On the other hand� the �p are closely related to the expected

autocorrelation function E �rs�� in the isotropic case because

E�� Xk�Ip

jakj�� m#p��p � ��

where m#p� denotes the multiplicity of the eigenvalue #p� Equ�� then implies

that the autocorrelation functions is determined by the �p�s� The converse follows

from the discussion in �� In particular� a Gaussian isotropic elementary

random �eld is the maximum entropy model subject to prescribed parameters �p

in equ�� Derrida�s pspin models �� for instance� are the maximum entropy

models with the single constraint that only one order of interaction contributes to

the Hamiltonian� The random energy model �� can be regarded as the maximum

entropy model subject to the constraint that the constants �p are all equal� see ��

for a proof�

Theorem � in ref�� implies that an isotropic random �eld on any Cayley graph of a

commutative group is pseudoisotropic� Thus the XYHamiltonian is not isotropic

on either con�guration space for � � � because it is not even pseudoisotropic

as we have seen above� For � �� however� the XY model coincides with the

SherringtonKirkpatrick Hamiltonian which is known to be isotropic� It will be

� ��


interesting to see how and to what extent the XY model deviates from isotropy�

To this end we need to examine the multiplicities of the Laplacian eigenvalues of

the graphs Qn� and Cn�� respectively�

The eigenvalues of Qn� are #p p� for � � p � n with the multiplicities mp

�� p�np�� see e�g� �� The �n�� independent coupling constants imply that H is

contained in a�n�

�dimensional subspace of the ��n�� dimensional eigenspace

of #�� Restricting the spinorientations to a plane hence severely restricts the

structure of the Hamiltonian for larger � in comparison to an isotropic model

with the same correlation structure�

For the graphs Cn� we obtain similar results� see Lemma � in section �� The eigenvalue #�� is �

�n�

�fold degenerate if � � �� An orthogonal set of basis functions

for its eigenspace is

fsin �i sin �j � cos �i sin �j � sin �i cos �j � cos �i cos �j��i j g ��

A simple transformation yields the basis function from which the XY Hamiltonians

are constructed�

fcos�i � �j� � cos�i �j� � sin�i � �j� � sin�i �j�� i j g ��

As in the case of the Hamming graphs� the XY models do not span a complete

eigenspace� However� their contribution does not decrease with increasing � on the

graphs Cn� � Intuitively one would expect that the XY models behave �almost� likeisotropic random �elds� and that deviations become more prominent on Hamming

graphs with large ��

Remark� The multiplicities of the eigenvalues of �su�ciently� symmetric graphs

are large� As a consequence the cost function of an isotropic random �eld is

composed of a large number uncorrelated random variables at each point x in

con�guration space�� Thus the central limit theorem tells us that the distribution

of the cost function itself should be approximately Gaussian in such systems�

� ��


�� The Correlation Length Conjecture

Metastable states or local optima are an intrinsic property of a rugged landscape�

In fact� Palmer �� used the existence of a large number of local optima to de

�ne ruggedness� We say that x � V is a local minimum of the landscape f if

fx� � fy� for all neighbors y of x� The use of � instead of is conventional

�� ! it does not make a signi�cant di�erence for spin glass models� Local

maxima are de�ned analogously� The number N of local optima of a landscape�

however� is much harder to determine than its autocorrelation function rs� or its

correlation length �� We shall sometimes use the subscript n in order to emphasize

the dependence on the number of spins�

As it appears that Nn and �n are two sides of the same coin we search for a con

nection between the two quantities� For random �eld models de�ned on Hamming

graphs Qn� it will be convenient to use the constant

� def�� lim

n��

n

rE �Nn�

�n��

as a description of the scaling of the number of local optima� Some papers use

� def

�� limn��

�

nlog�E �Nn�

��

or � log� log� instead of �� The constants are related by

� �� e�

��

The expected number E �N � of metastable states has been determined by variousstatistical mechanics methods for a selection of Ising models di�ering in the as

signment of the interaction coe�cients� These can be drawn independently from

a Gaussian distribution as in the SherringtonKirkpatrick model �� or one may

assume that the spins are arranged on a two or threedimensional lattice with

nonzero interactions only between neighbors on the lattice� The Sherrington

Kirkpatrick model received considerable attention around ��! at least three

groups have computed the number of local minima of the SK model by means

of what are now considered standard methods in Statistical Mechanics� Tanaka

� ��


and Edwards �� computed the expected number of local optima E �N �� while Brayand Moore �� and De Dominicis et al� �� used a replica approach to evaluate

E �lnN �� These papers provide also a detailed analysis of the distribution of localminima as a function of their energies� The common result of the three groups is

limn��

�

nlog E �Nn� lim

n��

�

nE �logNn� ��

The numerical value is obtained as the solution of a set of coupled algebraic equa

tions� For the case of short range spin glasses� in which only a small number z of

coupling constants Jij are nonzero for any given spin i� a slightly larger number

of local optima has been found

limn��

�

nlog E �Nn� lim

n��

�

nE �logNn� ��

��

z Oz��

where �� The only known case in which the logarithmic averagedeviates from the direct average is the linear spin chain� Derrida and Gardner ��

found log E �Nn�n � ln�� and E �logNn�n � ln �� for

this example�

Since all Ising models have the same correlation length �n n� �� but

somewhat di�erent values of N � we cannot hope for a simple formula relating� and E �� for general random �eld models� Isotropy de�nes what a �typical�

landscape looks like when the amplitudes E �Bp�� i�e�� the autocorrelation function

E �rs�� are prescribed� Using this maximum entropy condition we should be ablein principle to estimate the density of metastable states in isotropic Gaussian

random �eld models from the correlation function� simply because the model does

not contain more information� The correlation length � determines the single non

zero amplitude in elementary random �elds� We would therefore expect a close

relationship between � andN for Gaussian elementary isotropic random �elds� The

functional form of this relation will of course depend on the geometric properties

of the underlying graph�

A heuristic argument linking local optima and correlation measures runs as follows�

For a typical elementary landscape we expect that the correlation length � gives

a good description of its structure because the landscape does not have any other

distinctive features� Since � de�nes the size of the mountains and valleys of the

� ��


landscape it should also allow to estimate the number of local optima� As there are

many directions available at each con�guration we expect there are only very few

metastable states besides the summit of each of these �sized mountains � almost

all of the con�gurations will be saddle points with at least a few superior neighbors�

This picture was �rst developed in �� with the symmetric Travelling Salesman

Problem as an example� where it gave a surprisingly accurate estimate for the

number of local optima� Since we measure � along a random walk but attempt

to de�ne the radius of a mountain in terms of the intrinsic distance measure on

� is seems reasonable to de�ne R�� as the average distance that is reached by

the random walk in � steps� With the notation BR� for the number of vertices

contained in a ball of radius R in � we have therefore the following

Conjecture� In an isotropic elementary random �eld we expect

E �N � � N

BR��

Table ��Metastable States in Elementary Isotropic Random Fields�

Model Graph � lim npProbfloc�opt�g Relative

best estimate� conjecture ErrorRef� Ref� in '

SK Qn� �� S ��

�spin Qn� �� N ��




GBP� Jn� n�� N �� symmetric TSP �Sn� T � nonexp� N �� n

�n�� y� The best estimates for � are either obtained by statistical mechanics methods �S� such asreplica calculations or by means of numerical simulations �N��

y Relative error of log�Nn�n�� compared with log�� n��n�� averaged over the availablenumerical data�

Derrida�s p�spin Hamiltonian is H�� X

i��i��ip

Ji�i��ip�i��i� � � � �ip with i�i�d� Gaussian

coupling constants Ji�i��ip �� The graph bipartitioning problems is discussed in detail in ��

� ��


There is a fair amount of computational evidence for this conjecture� see table ��

For the TSP example from �� using R�� instead of � makes no di�erence since

R�� for large TSP problems� In �� it is shown that the conjecture

works extremely well for Derrida�s pspin Hamiltonians including the Sherrington

Kirkpatrick model� A similar result is obtained for the landscape of the Graph

Bipartitioning Problem �� In addition� one obtains reasonable estimates for

Kau�man�s Nk landscapes �� A few counterexamples are known as well! all of

them strongly violate the maximum entropy assumption�

In the following two sections we evaluate the correlation length conjecture for

the XYHamiltonian� This model is interesting for a number of reasons� It is

elementary on two fairly di�erent types of graphs� in both cases it belongs to

the thirdsmallest eigenvalue of the graphs Laplacian� It shares this property

among others with the symmetric TSP� the graph bipartitioning problem� the

graph matching problem �� The XY Hamiltonian violates isotropy� but not very

strongly� The extent of the violation is constant for the graphs Cn� while it getsworse with increasing � on Hamming graphs� This fact makes the XY Hamiltonian

an ideal model for a systematic investigation of the e�ects of anisotropy� And

�nally estimates for � obtained by a cumulant expansion technique have been of

obtained by Tanaka and Edwards ��

The strategy for the evaluation of the correlation length conjecture is the following�

By virtue of equ�� the conjecture can be recast in the form

� � limn��

np�B�n� � ��

De�ning the scaled correlation length

� limn��

�nn ��

and the scaled correlation radius

(� def

�� limn��

�

nRn ��

we �nally arrive at

� � limn��

n

q�Bn (��

We have hence to evaluate the relaxation behavior of simple random walks on the

con�guration space and the number of vertices contained in ball of given radius in

�� The technical details of these computations can be found in appendix A�

� ��


�Metastable States on Hamming Graphs

The average radius Rs� that is reached by an unbiased walker after s steps on a

Hamming graph Qn� is

Rs� n��

��

n

�

�� s�

� ��

as shown in appendix A� The scaled correlation radius is thus

(� ��

h�� e� �

��i� ��

From equ�� we obtain �nally

Lemma �� (� ��

�� p

e

�

Having estimated the radius of the mountains in our landscape we now turn to

computing their volume�

De�nition� �� Let V be the vertex set of a graph� For each vertex x � V let

�xd� we denote the number of vertices in distance d from x� i�e��

�xd� �� f y � V j dx� y� d g ��

The sequence �xd�� is called the distance degree sequence of vertex x� A graph

is distance degree regular if �xd�� is independent of x� which is true for both Qn�

and Cn�� We shall therefore drop the index x�

The distance degree sequences are known explicitly for many highly symmetric

families of graphs� In particular we have

Lemma �� The Hamming graph Qn� is distance degree regular� Its distance degree

sequence is

�d� �� d�n

d

��

The volume of a ball is of course

BR� RXd��

�d� � ��

� ��


We have (� � � � �� on Hamming graphs� thus Bn(�� n(�� up to a mul

tiplicative correction of at most a factor of n� Using Stirling�s approximation for

the factorials in the binomial coe�cients yields

�(�� (��

(�

�� (��

��

The numerical values for �(�� in table � have been obtained with the help of

Mathematica�

Table ��Local Minima of the XY Hamiltonian on Hamming graphs�

� �(�� Stat�Mech Numerical� �� y��

� Exact result�y From a cumulant expansion� not exact�

Numerical estimates for the fraction Nn�n of metastable spin con�gurations are

straight forward but quite time consuming� at least for larger values of � because

local optima become an exceedingly rare phenomenon� In fact� the total number

of metastable states increases exponentially with n� but with a much smaller rate

than the total number of con�gurations� Computer experiments hence are looking

for the proverbial needle in the haystack� We have been able to obtain reasonably

accurate estimates only for � �� see table �� Figure �a shows our results�

Data for � �� the SherringtonKirkpatrick spin glass� are reported in �� We

note an increasing discrepancy between the numerical estimates and the predic

tions from the correlation length conjecture� which we attribute to the increasing

deviations from isotropy for large values of �� We shall return to this point in the

discussion�

� ��


5 10 15 20Number of Spins n

-20

-15

-10

-5

0lo

g(P

rob{

loc.

opt.}

)/lo

g(al

pha)

3

4

5

5 10 15 20Number of Spins n

-20

-15

-10

-5

0

log(

Pro

b{lo

c.op

t})/

log(

alph

a)

4

5

a b

Figure �� Numerical estimates for the number of metastable states of the XY Hamiltoniana� Hamming graphs� Data for � � � and � � � are o�set by �� and �� respectively�b� Direct products of cycles� Data for � � � are o�set by ��

�Metastable States on Direct Products of Cycles

The relaxation of simple random walks on Cn� is discussed in appendix A� One�nds

(� �� Xk��

w��k exp

�� cos ��k

�

��

��

for � � �� The coe�cients �� and w��k are tabulated in the appendix for

� � � � �� In particular we have

(� ��

�e�� for � �

(� �� e� for � �(� �

� � ��e��

�e�� for � �

��

The scaled correlation length of the XYHamiltonian on Cn� is � �� sin�� In

table � we show the scaled correlation radii (� for the XY Hamiltonian on Cn��

� ��


Table �� Scaled correlation radii for the XY Hamiltonian on Cn� �

� � �n (��

The distance degree sequence of the graphs Cn� is much more complicated to evaluate� This is due to the fact that Cn� is not vertex transitive� i�e�� not all pairs ofsequences with a given distance are equivalent in this graph� see ��

Lemma �� Cn� is isomorphic with the Hamming graph Q�n� �

Proof� In the graph Cn� we relabel the vertices using the translation table

��

Each spinorientation then corresponds to a string of length �� and neighboring

con�gurations)strings di�er in exactly one of the two positions� i�e�� the distance

of two singlespin con�gurations is the Hamming distance of their twoletter codes�

Consequently the graph Cn� is isomorphic to the Boolean hypercube of strings oflength �n� i�e�� Cn� � Q�n

� �

As an immediate consequence we have ��d�

��n

d

�and

��(�� limn��

n

qBn(�� lim

n��

n

q�n(��

�

�� (��

�� (�(�

�

��

for d n(� since (� � �� We have not succeeded in �nding explicit expressions for�d� for � � �� It is possible� however� to compute

�(�� limn��

�

nlogBn(�� lim

n��

�

nlog �n(��

� ��


Table ��Local Minima of the XY Hamiltonian on Direct Products of Cycles�

� �(�� Stat�Mech Numerical� �� y�� y�� y�� y�� y�� y�� y�� y��

� Exact result�y From a cumulant expansion� not exact�

see appendix B�

Numerical values of �(�� can be found in table �� These can be obtained with

arbitrary precision� The cases � � and � � are Hamming graphs and have

already been dealt with in the previous section� In the case � � we have(� �� pe � �� Thus we have Bn(�� n(�� n� and

�� log�� log�� This is in excellent agreement with the data reported in �� Explicit expres

sions for �(�� can be obtained very easily using Mathematica or another symbolic

mathematics package� As an illustration we display here the result for � ��

�(�� (� � �� (��

with � ��

(� � �

q�� (� � ��

� For larger values of � we obtain much more

complicated expressions which we shall not reproduce here�

Numerical simulations for � � and � � are summarized in �gure �b� The data

agree very well with the statistical mechanics calculations in �� The quality of

the estimate from the correlation length conjecture is not perfect� but much better

than for the Hamming graphs see the previous section�� We �nd a relative error

of less than �' for �� which translates to about ��' relative error for ��

� ��


��Discussion

In this contribution we have investigated the discrete XY Hamiltonian on two

di�erent types of con�guration spaces corresponding to the two natural de�nitions

of neighborhood between spin con�gurations� The reorientation of a single spin

into an arbitrary direction gives rise to the Hamming graphs Qn�� while changing

the current orientation of a single spin by the smallest possible angle� ��yields a direct product of cycles Cn� � We have focused our attention at the thegeometric properties of an individual XY landscape with �xed coupling constants

Jij and to the properties of the random �eld model that is obtained by assigning

the coupling constants i�i�d� from a Gaussian distribution�

Many important examples of landscapes are elementary� i�e�� up to an additive

constant� they ful�ll a discrete analogue of the Helmholtz equation "f �f ��

where " is the graph Laplacian of the con�guration space on which the landscape

is de�ned� Examples include Derrida�s pspin models� and the landscapes of the

best known combinatorial optimization problems such as the traveling salesman

problem� The XY Hamiltonian is elementary on both Qn� and Cn� � In both cases

it belongs to the third smallest eigenvalue� i�e�� the second �excited state�� just

like most of the combinatorial optimization problems studied so far �� Ele

mentary landscapes exhibit a characteristic distribution of local optima on the

con�guration space which depends crucially on the corresponding eigenvalue # of

the Laplacian� In particular� the location of # in the spectrum of the Laplace

operator determines the maximum number of nodal domains� that is the maxi

mum number of disconnected islands of values of f that are below average ��

The nearest neighbor correlation is directly linked to the eigenvalue #� which can

therefore serve as a measure of the ruggedness� In fact� the correlation length of

an elementary landscape is given by � D#� where D is the number of neigh

bors� The �correlation length conjecture� �� suggests that the number of local

optima of a �typical� landscape can be estimated from its correlation length ��

More precisely� one expects on the order one local optimum on a mountain with a

radius that is determined by the correlation length ��

� ��


The notion of statistical� isotropy was introduced from a purely geometric point of

view �� It translates� however� to a notion of a �typical landscape� with a pre

scribed correlation function� Given an orthonormal basis f��k g of the eigenspace

belonging to the eigenvalue # of the graph Laplacian of the con�guration space�

we can write f $f P

k ak��k � In fact� it can be shown �� for the types of con�g

uration spaces considered in this contribution that f is isotropic if and only if the

coe�cients ak are uncorrelated random variables� Assuming in addition that the

ak are drawn from a Gaussian distribution we observe that a Gaussian isotropic

random �eld is exactly the maximum entropy model with a prescribed autocorre

lation function� Isotropy is an important property also from a practical point of

view because it has become clear in computer simulations �� that plateaus�

ridges� and other anisotropies strongly in�uence adaptation on a landscape ��

��

Recent numerical surveys provided good evidence that the �correlation length

conjecture� allows for a fairly accurate prediction of the number of local optima

metastable states� of isotropic elementary random �eld models ��

Systems that are known to deviate from the conjecture� on the other hand� have

strongly constrained coe�cients ak� For instance� in shortrange Ising spin glasses

most of the coupling coe�cients are zero �� and the graph matching problem

can be treated as a TSP with a severely constrained distance matrix ��

The XY model is not quite isotropic on both types of con�guration spaces� The

deviations from isotropy increase with � on the Hamming graphs * and so do

the discrepancies between the prediction of the correlation length conjecture and

numerical estimates of the number of local optima� For the graphs Cn� the relativedeviation from isotropy as measured by the fraction of independent coe�cients�

does not depend on �� The quality of the predictions obtained by the correlation

length conjecture is reasonable but not as good as for the isotropic cases discussed

in the literature� The data presented in this contribution thus support the idea that

there could be quantitative relation between the deviations from the �correlation

length conjecture� and the �degree of anisotropy�� For Ising models it has been

found �� that the deviation is proportional to �z� where z is the number of

nonzero coupling constants per spin� Note that �z can be regarded as a measure

for the deviations from isotropy� More work will be necessary in order to establish

� ��


a detailed understanding of di�erent aspects of ruggedness in landscapes and their

random �eld models�

Acknowledgments

This work on this study began while P�F�S� was a guest of UNAM in Mexico City in

summer �� Thanks to the colleagues at the Instituto de F��sica and the Instituto

de Ciencias Nucleares for stimulating discussions and for their hospitality� R�GP�

wants to thank the Departamento de F��sica Fundamental� UNED� Madrid� for

their hospitality during the �nal stages of this work�

Appendix A� Relaxation of Simple Random Walks

Consider a graph X n� the graph product of n identical copies of a regular graph X �The distance in X n is given by the sum of the distances in the individual factors�

d�x� �y� nXi��

dixi� yi� � A��

In this section we shall be concerned mainly with the expected distance Rs� of

a simple random walk of s steps on the graph� A closely related quantity is the

distance from the starting point within a single copy X � To this end we consider arandom walk on X that pauses with probability �� and moves to one of the K

neighbors in X with probability �K� The transition matrix of this random walk

is

T� �� I �

KAX ��

K"X � A��

where I is the identity matrix� AX is the adjacency matrix of the Kregular graph

X and �"X def��AX �KI is its Laplacian� Let �� s� denote the average distance

from the origin at time step s�

Lemma A�� The average distance Rs� of a simple random walk on X n is given

by

Rs� n��n� s� � A��

� ��


Proof� The simple random walk on X n performs a step in a particular factor Xwith probability �n at each given timestep� Thus the expected distance after

s steps within a given factor X is exactly ��n� s�� Since there are n identical

factors the lemma follows�

The next step is to derive an explicit expressions for �� s�� Let f�ig denote acomplete set of eigenvectors ofAX � with associated eigenvalues �i� Furthermore let

� denote the starting point of the random walk� and let pxs denote the probability

that the walk is in position x at time s� Then

pxs Xy

�Ts�xy��y Xy

�Ts�xyXi

�i��

k�ik��iy� Xi

�i��

k�ik� �si �ix� � A��

where �idef

�� K�i �� iK is the eigenvalue of the transition matrix

T belonging to the eigenvector �i of the graph Laplacian �"X � As usual� �i�j isKronecker�s symbol� The norm of the eigenvectors is of course k�ik�

Px �ix�

��

The expected distance �� s� is then given by

�� s� Xx

pxsdX �� x� � A��

This method was explored in �� In general it is not easy to evaluate since it

requires detailed knowledge of the eigenvalues and eigenvectors of the graph X �

Lemma A�� Let Q� be the complete graph on � vertices� Then

�� s� ��

��

�

�� s�

� A��

Proof� For complete graphs we have dX �� x� � for all x � �� The eigenvalues ofthe complete graph Q� are �� which is simple� with eigenvector �� and �� which is �� fold degenerate and all its eigenvectors are orthogonal to�� see� e�g�� The eigenvalues of the transition matrix T are

��

�� and �i ��

��

��

� ��


for � � i � n� Thus

�� s� �� p��s ��nXi��

�i��

k�ik� �si ��

��

k��k� ��s � �s�

nXi��

�i��

k�ik�

��

� �s�

��

k��k� � �s�

nXi��

�i��

k�ik� ��

�s�

�

��

�

and the lemma follows�

Corollary� On a Hamming graph Qn� we have

Rs� n��

��

n

�

�� s�

� A��

Proof� Follows immediately from the above two lemmas� The same result was

obtained by di�erent methods in ��

In order to use this formalism for the graphs Cn� we need to compute the seriesexpansion of the Kronecker delta in terms of the eigenfunction of the cycle graphs

C�� Let us label the vertices by �� ! depending on whether � is even orodd we have either one vertex labeled �� or two vertices labeled �� atthe maximum distance from �� We have already encountered the eigenvalues and

eigenvectors of the cycle graphs at the end of section �� The eigenvalue ��

with eigenvector � and the eigenvalue �� for even �� are simple� All the other

eigenvalues have multiplicity mk �� In order to �nd the eigenvalues of the

transition matrix T we substitute the eigenvalues of the cycles from equ�� and

set K � since each vertex has � neighbors� We obtain

�k ��

�� cos

��k

�

�� A��

Using equ�� and A�� we can now compute the probability that the walker is

in position x at time step s� for the eigenvalues of the transition matrix T we have

pxs �

��s�

��Xk��

sin��

k sink k� sin��k

�x

��sk

��Xk��

cos��

k cosk k� cos��k

�x

��sk �

� ��


The sineterms vanish of course� The eigenvectors of a distance regular graph with

N vertices ful�ll �� p��k�kk��

N

mk� A��

Thus we have

pxs �

��s�

��Xk��

mk

�cos

��k

�x

��sk A��

and therefore

�� s� Xx

jxjpxs �

�

Xx

jxj ��Xk��

mk

�

�Xx

jxj cos��k

�x

��sk � A��

The brackets can be evaluated using equ�� in �� One obtains di�erent

expressions for odd and even values of �� respectively�

%w��kdef

��

Xx

jxj cos��k

�x

�

��

�

��Xj��

j cos

��k

�j

�

�

��Xj��

j cos

��k

�j

�

�

�cosk��

� �

�� cos ��k�

� ��

k cos

�k�

�

�for odd �

�� k for even �

For the constant term we �nd �� for odd � and �� for

even �� respectively� Setting w��k mk�� %w��k we obtain the desired expansion

for the distance on the cycle graph�

�� s� �� Xk��

w��k �sk � A��

The coe�cients for small values of � are tabulated in table ��

It is now easy to obtain numerical values for

(� limn��

��n� n�� Xk��

w��k limn��

��

n

�kK

n�

�� Xk��

w��k limn��

��

n�

�n��k�K ��

��Xk��

w��ke��k�

K

� ��


Table ��Coe�cients w��k for the relaxation of random walks on cycle graphs�

� �� w�� w�� w�� w�� w��

� ��

Substituting the explicit expression for �k and K � �nally yields

(� �� Xk��

w��k exp

�� cos ��k

�

��

A��

for � � ��

Appendix B� The Distance Degree Sequence of Cn�

Since Cn� is vertex transitive we may choose an arbitrary reference con�guration�say all spins +up�� The con�gurations can be classi�ed by the numbers nk of spins

that di�er by ��k�� from the reference con�guration� The common distance

of all these con�gurations from the reference con�guration is of course

d X

k��

knk � B��

The number of con�gurations with given values of nk can be estimated by the

largest term with this property up to an error of at most a factor On�� Theseare

Sn

�n

n�n��

n�� n

�and Sn

�n

n�n��

n�� n�

�n��

�B��

for even and odd values of �� respectively� The factors �� arise from the fact that

there are two con�gurations� �k� which give rise to the same distance contribution�

� ��


Summing over all values of nk that yield a total distance d provides us with the

desired number ��d�� Since there are � sums running from � up to at most n we

�nd that the largest individual term of the above form approximates ��d� with

an error of at most On ��

It is convenient to introduce the fractions qkdef

��nkn and the function

Gq�� q�� q � limn��

�

nlogSn

��q� log q�

��Xd��

qd logqd�� q logq m��

B��

with m� � or � depending on whether � is even or odd� We have used Stir

ling�s approximation logn,� � n logn� n here� which again introduces only non

exponential errors� Maximizing the multinomial coe�cient is equivalent to �nding

the maximum Gmax(�� of Gq�� q�� q � subject to the constraints

Xj��

qj � and X

j��

j qj d

n (� � B��

Hence we may estimate ��d� � expnGmaxdn�� with a nonexponential error

term� This approach has been termed maximum entropy approximation because

G has the form of Boltzmann�s entropy� It should not be confused with the �max

imum entropy� interpretation of isotropy in section ��

Using the method of Lagrange multipliers we can �nd the maximum of this function

of � � variables subject to the constraints that the distribution is normalized

and that its mean is (�� i�e�� we have to solve for the unconstrained maximum of

F �q!u� v� def

��Gq�� q�� q � u

� X

j��

qj � �!A v

� X

j��

jqj � (�!A � B��

where u and v are Lagrange multipliers� From �F�qk � we obtain the condi

tions

logqk %mk� u� � kv B��

� ��


for the stationary point� where %mk is � or � according to the corresponding entries

in the multinomial coe�cient� We set s def

�� expu� �� and t def

�� expv� � ��

The equations for the maximum become

qk s %mktk � s

Xk��

%mktk � � s

Xk��

k %mktk (� � B��

Eliminating s we are left with the algebraic equation

(� X

k��

%mktk

Xk��

k %mktk � B��

which has a unique positive solution (t(�� for � (� �� It is straight forward to

check that (t is a monotonously increasing function of the parameter (�� We obtain

a simple expression for the maximum value of G in terms of the solutions (t(�� and

(s(��

Gmax(�� X

j��

(s(�� %mj(t(��j log(s(��(t(��j�

� log (s(�� (s(�� X

j��

%mj(t(��j � log (t(�� (s(��

Xj��

%mj(t(��jj

� log (s(�� (� log (t(��

B��

It is now easy to show that Gmax(�� is unimodal on �� In fact� taking the

derivatives in equ�B�� yields

dGmax

d(� � log (t(�� B��

which is positive if (t(�� and negative for (t(�� larger than �� Since (t is

monotonously increasing there is a unique value $� such that (t$�� The de�ni

tion of (t implies

$� X

k��

%mk $�� X

k��

k %mk

�� if � is even

�� if � is oddB��

i�e�� $� is the average distance of two randomly chosen points on C�� We �ndGmax$�� log (s$�� log�� Adding the distance classes up to n(� introduces an

� ��


error of at most a factor On� in comparison with the largest term in the sum�

Since Gmax(�� is monotonously increasing for (� � $�� we have

Bn(�� expnGmax(�� for (� � $�

�n for (� � $�B��

up to nonexponential corrections� Our �nal result is thus

�(�� Gmax(��

log�B��

for the graphs Cn� � because (� $� for all �� Note that this is an exact result because

all errors in the above derivation are nonexponential� Thus we can compute

Gmax(�� with arbitrary precision�

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