Linear operators on correlated landscapes

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Linear operators on correlated landscapes

Peter Stadler

To cite this version:

Peter Stadler. Linear operators on correlated landscapes. Journal de Physique I, EDP Sciences, 1994,

4 (5), pp.681-696. �10.1051/jp1:1994170�. �jpa-00246941�

Classification Physics Abstracts

0250 0550

Linear operators

_on

^correlated landscapes

Peter F.

Stadler(*)

Santa Fe Institute, 1660 Old Pecos Trail, Santa

Fe,

NM 87501, U-S-A-

(Received

26 October 1993, received in final form 12

January

1994,

accepted

28

January 1994)

Abstract. In this contribution _we consider the elfect of _a class of

"averaging operators"

on

isotropic

fitness

landscapes. Expbcit expressions

for the correlation function of the

averaged landscapes

are derived. A _new class of

tunably rugged landscapes,

obtained

by

iterated snlooth-

ing

of the random _energy

model,

_is established. The correlation structure of certain

landscapes,

among them the

Shernngton-Kirkpatrick

model, remains

unchanged

under the action of ail

averaging operators considered here.

1 Introduction.

Evolutionary adaptation

_as well _as combinatorial

optimization

take

place

_on

"landscapes"

that result from

mapping (micro)configurations

to scalar

quantities

like fitness values _or

energies,

or, more

generally,

to nonscalar

objects

hke structures. The

assumption

of _a

particular

mech- anism for

mutation,

or the choice of _a

specific

move-set for

interconverting configurations,

introduces trie notions of

neighborhood

and distance between

configurations. Viewing

trie set of ail

configurations, 1e.,

trie

configuration

_space, as a non-directed

graph

r bas tumed out to be

particularly

useful. Each

configuration

is _a vertex of

r,

and

neighbonng configurations

are

connected

by edges.

^When

configurations

_{are sequences} of

equal length

and trie

neighborhood

relations _are defined

by

the number of

differing positions,

_one obtains trie _{sequence space} with the

Hamming

metric.

Among

the most

important properties

of _a

landscape

_is its

"ruggedness",

which influences the

speed

of

evolutionary adaptation

_as well _as trie

quality

of solutions of heunstic

optimization strategies.

^While

ruggedness

bas _never been

precisely defined,

_a number of

empincal

_measures bave been

proposed,

such _as trie number of local

optima

or trie average

length

of _{up- or} downhill walks

il, _2],

and the

pair-correlation

_{as a} function of distance _m

configuration

_space

_{[3, 4].}

The very notion of

"uphill"

_or

"optimum" implies

_{a mapping} to _a scalar

quantity.

Trie definition of _a

pair correlation, however,

_con be extended to mappmgs ^to

arbitrary objects,

_as

(*)

Address for

Correspondence:

Institute for Theoretical

Chemistry, University

of

Vienna,

Wàhringerstra17,

A-1090

Vienna,

Austria

long

as there is _a metric distance _measure defined for them. Extensive studies _ou

folding

of RNA _{sequences into}

secondary

structures have been based _on this fact

[5,

GI.

Trie

landscapes

of _a number of well known combinatorial

optimization problems

bave been

studied,

such _as trie

Traveling

Salesman Problem

[7],

^the

Graph Bipartitioning

Problem

[8],

and the

Graph Matching

Problem

[9].

^Detailed information _on the distribution of local

optima

and the statistical characteristics of downhill walks have been obtained for the uncorrelated

landscape

of the random energy model

[10-12]. Furthermore,

two

one-parameter

^families of

tuuably rugged landscapes

have been studied in detail: the NÉ model aud its variauts

[2, 5, 13, 14]

^aud the

p-spiu

mortels

[15, 16].

Here _we will be concerned with the eifect of

averaging procedures

related to runuing averages

ou

laudscapes.

After _some formol definitions

(Sect. 2)

_we discuss

briefly

the Fourier

decompo-

sitiou of

laudscapes (Sect. 3)

which is used in section 4 to derive the basic

properties

^of liuear operators

applied

to

landscapes.

In section 5 _we

briefly

address the relation of _{averages over} ensembles of

laudscapes

aud _{averages over}

configurations

in

single

instances of

landscapes.

In section 6

particular

operators,

namely weighted

_{averages over}

one-step neighborhoods,

_are used

to coustruct a uovel tuuable

family

of

laudscapes.

In section 7 the behavior of

p-spin

^aud NÉ

landscapes

under the iterated action of these

operators

^is

iuvestigated.

2. Some definitions.

Let r be _a

graph

with N _{< cc} vertices. A

graph autoniorphism

_{a is a} one-to-one mappmg of r onto itself such that the vertices

a(~)

aud

a(y)

_are connected

by

_an

edge

if and

only

if

~ and y are conuected

by

_an

edge.

Two vertices of r _are said _to be

equivaleut

if there is _an

automorphism

_o such that _y

=

a(~).

A

graph

is uerte~ transitive if ail vertices _are

equivalent.

A vertex transitive

graph

is

always regular,

1-e-, each vertex has the _same number of

neighbors.

Trie distance

d(~,y)

of _two vertices ~,y ^E r is defined _as the minimum number of

edges

that

separates

^{them. This distance} is _a metric. A

graph

is distance transitive if for _{any two pairs} of vertices

(z, y)

and _(~1,

u)

with

d(~, y)

_{= d(~1,}

u)

there is _an

automorphism

a such that

a(~)

= u

and

o(y)

= u

(see,

_e-g.,

_[17]).

A

graph

r is defined

by

its

adjacency

matriz

A,

with

Axy

=1 if the vertices ~ and _{y are} connected

by

an

edge,

and

Axy

= 0 otherwise.

In the

following

_we will

only

consider

graphs

that _are at least vertex transitive.

Following

Bollobàs

[18]

we may then

partition

the vertex set

by

the

following procedure:

Choose _an

arbitrary

vertex as reference

point,

^{which in the}

following

will be labeled 0. Then construct

pairwise disjoint

sets of vertices

V(0)

=

(0), V(1), V(2),

..,

V(M)

such that

â~~

_:=

£ _Axy

_is

independent

of _{y E}

V(v). (l)

x~V(p)

In other

words,

this

partitioning

^is chosen such that each _{vertex y m}

V(v)

is

adjacent

to

â~~

vertices in

V(p).

For _some

exaniples

_see

figure

1 in section 3. It is mduced

by

the

symmetry

of r: the vertices in each of the sets

V(~1)

are

permutated by automorphisms

that leave 0

invariant. In the worst _case each set contains _a

single

vertex. In

general

_we will _use

greek

letters to index trie sets of such _a

partition,

while _roman indices _are reserved for trie vertices off.

For

highly symmetric graphs

trie

collapsed adjacency

matriz

=

(â~~)

cari be much smaller than trie

adjaceucy

niatrix A. lu _case of distance transitive

graphs,

^for instance, trie distance classes

(1.e.,

the sets of ail vertices with _commou distance from the reference _vertex

0)

form _a

partition fulfilling equ.(l),

and the

collapsed adjacency

matrix comcides with the intersection

matrix _as defined

by Biggs _[19] (see

also [20]

).

We will _use d instead of

greek

letters for

mdexing

the distance classes.

The

following property

of

collapsed adjacency

matrices will be used in the

subsequent

_sec-

tions.

Lemma 1.

£ _(A~)xy

=

(Â~)~~

^for ail _{~ E}

V(p),

and ail

non-negative integers

_m, and y~v(~)

hence for each

polynomial

_p holds

£~~~~~~lp(A)]xy

=

[p(Â)]~~, i-e-, jJ(A)

₌

p(Â).

Proof. We

proceed iuductively:

The assertion is true

by

^{definitiou for} m = 1. For _m

= 0 it

holds

trivially.

Now _suppose it is true for _m 1; theu

£ £ (A~~~)xzAzy

"

&<)~ [ ^(~[[

~~~ii~i~lll z(~mi)«~Àv<

-

(Â~~«~

~ ~~~~~, ~~v~~~ v

The

geueralizatiou

to

polyuomials

_is

straight

^forward.

^~

Definition 2. A

landscape f

r _- lit is _a raudom field F _ou the set of vertices of r defined

by

the distribution fuuction

P(yi,

_y2>

_UN)

" Prob

( f(~~) §

_j,

§1 § N) (2)

where N is the number of vertices of r.

Let

El-j

denote the mathematical

expectation.

For

example,

the

expected

value of the

product

of the values

f(~k)

and

f(~i)

at the vertices _~k and _{~i is}

Eifl~k)fl~i)i

=

/vkvidPlvi>v2>. ^{>vN). (3)}

Definition 3. A

landscape

_is

isotropic

if

Ii) El f(~)]

₌

f

for ail

configurations

_~ E

r;

(ii)

for any ^two

pairs

of

configurations (~, y)

and _(~1,

u)

such that there is a

graph

automor-

phism

_a with

o(~)

= ~land

a(y)

= v halas

£[f(~) f(y)]

=

El f(~1) f(v)].

The second condition _nleans that _any two

pairs

of

configurations

that _are

equivalent

in

config-

uration space have the _same

pair-correlation.

The _covanauce matrix of F is defined

by

Cxy

=

Eif(~)f(v)i Elf(~)iElf(v)1 ₁₄₎

By

vertex

transitivity

there _{is an}

automorphism

_a such that

a(~)

= 0.

Isotropy

hence

implies C~y

=

Co,~(y)

=

c(p),

where

V(~t)

is the

partition

to which

o(y) belongs.

The variance is given

by

Varjfj

=

Cxx

₌

Coo

_#

c(0) (5)

The a~ltocorrelation

f~lnction

_is defiued _as

p(p)

=

c(~1)/c(0).

3. Fourier series _on

landscapes.

Let C be the _covanauce matrix of _a raudom field F _{ou some}

graph

r. Deuote

by (#~)

_a

complete

set of orthonormal

eigenvectors (which

exists

by symmetry

of

C). Although (#~)

can be chosen

real,

_we will admit

complex

vectors as well. Then the

landscape f

_on r _cari be

represented

in _mean square sense as

f(~)

=

£ _{a(v)4y(~) (6)}

(The labeling

of the

eigenvectors #y

with vertices is

arbitrary.)

This series

representation

_ta known _as the Karh~lnen-Loèue

expansion.

Remark. The coefficients

a(y)

_are uncorrelated. For limite _{sets the} Karhunen-Loève

expansion

coiucides with the well known

principal compouent aualysis

mtroduced

by Hotelliug [21] (see,

e-g.,

[22]).

For _au

arbitrary graph

r with

adjaceucy

matrix A the

graph Laplacian

is defined

by

/h

=

D~~A E,

where D is the

diagonal

matrix of _vertex

degrees (see,

_e-g.,

_[20])

For vertex transitive

graphs

this becomes /h

=

(1/D)A E,

where D is _now the constant vertex

degree

of r. Denote

by (Ù~)

a

complete

orthonormal set of

eigenvectors

of /h. A _serres

representation

of the form

f(~)

=

£ _{b(v)Ùy(~) (7)}

y~r

is called Fourier _expansion of the

landscape

_[2].

Definition 4. Let

(G,o)

be _a limite _group, aud let 4l be _a set of

geuerators

of G such that the group

identity

is flot contained in

4l,

and for each _{~ E} 4l the _inverse group element

~~~

is also coutained in 4l.

(4l

^is a set of

generators

meaus that each _group element _{z E} G _can be

represented

_{as a} truite

product

of elemeuts of

4l,

with

multiplication

defiued

by

the _group

operatiou o,)

Let

r(G, 4l)

be the

graph

with vertex set G aud _au

edge couuectiug

two vertices

~ aud _y if and

ouly

if

~y~~

E

4l,

1-e-, two vertices _are couuected if and

ouly

there _{is a't E} 4l such that _y

= ~/~.

r(G, _4l)

is called _a

Cayley graph (see,

_{e-g., [23]}

).

Remark. A

Cayley graph

^is vertex transitive.

A commutative group

(G, _o)

_can be writteu _as trie direct

product

of _a finite uumber L of

cyclic

_groups

Cj

of order

Nj.

A group element cau be

represeuted by

_{~ =}

(~i,

_~2,

~L)

with 0 < ~j ^<

Nj;

theu the group

operatiou

becomes

z=~oy -

zj=~j+yjmodNj j=1,. ,L. (8)

It is easy ^to check that the functions ep, p E

G,

^defiued

by ep(z) IL

= exP

2«1£ ^(~~ ₍₉₎

~=i ^j

fulfill

ep(~)eq(~)

=

epoq(x).

Furthermore

they

_are

orthogonal

with

respect

to the scalar

product

£~~~ eq(~)eq(~)

=

ôpq.

^A

simple

cousequence of these facts _is trie well knowu

Lemma 5. Let

r(G,4l)

be _a

Cayley graph

of the commutative group

G,

aud let

Hi

₌

h(j

_o

1-1) depeud ouly

_ou the "diifereuce" of the group elemeuts1 aud

j.

Theu ep as defiued

in

equatiou (9)

is _au

eigeuvector

of H. In

particular,

the set

(ep)p

E

G)

forms _au orthouormal basis of

eigeuvectors

of the

adjaceucy

matrix of

r(G, 4l).

Remark. Lovàsz

[24]

^{used this result to show that that the}

eigenvalues

of the

adjacency

matrix of _a

Cayley graph r(G, 4l)

with commutative group G _are

giveu by

_Àp

=

£~~~ ep(k).

Weiuberger _[14]

showed

usiug

lemma 4 that Karhuueu-Loève _expansion aud Fourier _expan- sion coiucide for

isotropic landscapes

_ou

Cayley graphs arisiug

from comm~ltatiue groups.

(He

used _a

slightly

_more restrictive defiuitiou of

isotropy, requinug

the _{same covanauce} for ail

pairs

of

configurations

with the _same

distance.)

We will show that _au

aualogous

result holds for distance transitive

configuration

_space.

01o 0110

Â=

_{3 01}

Â-

²¹⁰¹

0 2 2

~0121

10 01

Fig.

1. Petersen's

graph

is distance transitive, but it is not _a

Cayley graph (see,

_e-g.,

[23]).

It has three classes of vertices: the refereuce vertex

(top),

its

ueighbors,

aud ail other vertices. Its collapsed adjacency ^{matrix is} given below the

graph.

The

trigonal

_prism

graph

T is the

Cayley graph

of C2 x C3 with set of generators 4l

=

((~,e), (e,~),(e,~~~)),

_{with C2}

#

le, ~)

and

C3

₌

(e,~,~~~)

and _e

denoting

the _group

identity.

An

edge connecting

the two

triangles

_is

clearly

not

equivalent

to _an

edge

within _a

triangle,

hence _{T is not} distance transitive. There

are 4 symmetry classes of vertices: the reference vertex

(top),

its two

neighbors

_m the upper

triangle,

its

neighbor

in the lower

triangle,

and the remammg ^two vertices, which have distance 2 from the reference.

Remark. There _are

graphs

that _are distance transitive but flot

Cayley graphs,

aud vice _versa, there are

Cayley graphs

of commutative groups that _are not distance transitive. The two most

simple examples,

the Petersen

graph

aud the

trigoual

_pnsm are shown _m

figure

1.

Lemma 6. Let

a(~

^{= if}

d(1, j)

₌

t,

and 0

otherwise,

for

ail1, j

E r. Denote the

corresponding

matrix

by ^A(~)

Let r be distance transitive. Then

A(~)

_{is a} linear combinatiou of the first t powers of A.

Proof. The assertion _is

trivially

true for t

= 0 aud t

=

1,

_as

A(°)

= E aud

A(~)

= A.

Now calculate A

A(~~~)

_The

triangle mequahty

_assures that

only

_pairs of indices

(1, j)

with

distances

t,

t

-1,

^{aud t 2} can have _non-zero entries. Distance

transitivity

^of the

graph guarantees

that ail entries

belonging

to the _same distance Mass _are

equal,

and heuce

A

A(£-1)

= c ~

A(£)

_{+ c~}

A(£-1)

_~

~~

~(t-2) _(~~)

where c+ > 0 _as

long

as t ares _flot exceed the diameter of r.

~

Theorem 7. Let

f

be _au

isotropic laudscape

ou a

Cayley graph r(G, _4l)

with a commutative group

G,

_{or on a} distance transitive

graph

r. Then trie

adjacency

matrix A of r and the

covariance matrix C of the

landscape f

_commute, AC

= CA.

Proof.

(a)

For

Cayley graphs

of commutative groups the

proposition

_{is a} direct cousequeuce of leninia

5,

_see

[14].

(b)

If

f

is

isotropic

aud r is distance

transitive,

theu

Cxy depeuds ouly

ou the distance of the vertices

d(~, y).

Heuce C _cau be written _{as a} hnear conibiuatiou of the matrices

A(~),

and

by

lemnia

6,

it _cari be writteu _{as a}

polyuoniial

in terms of

A,

C

=

p(A).

In both _cases C aud A commute because

they

bave trie _same

eigeuvectors. ~

4. Linear

operators

_on

landscapes.

Consider

landscape

_g obtained

by

_some

weighted averaging procedure

from _a

landscape f,1

_e.,

g(~)

=

£~~r flxy f(y).

In the

following

_we will _use the obvious matrix notations g =

Qf.

Lemma 8. Let C be the covariance matrix of the

landscape f,

and let fl be _a hnear

symmetric operator.

^{Then the covariance matrix} of g = fl

f

is given

by

C~

= flcfl

(11)

Proof. We _assume without

loosing generality

that

El _fp]

= 0 for ail _{p E} r.

c)~

₌

Ej(n1)~(n1)~j

= E

£ _{n~~ i~} £ _n~~ il

p q

=

L "xpElfpfqlflqy

₌

(flcfl)xy

p,q

Corollary

9. Let

f

be

isotropic

and _assume that the

configuration

_space r is distance transi- tive _or the

Cayley graph

of _a commutative group.

Suppose

^fl is _a

polynoniial

of the

adjacency

matrix. Then _we have

C~

=

(fl~)C.

Proof. From theorem 7 we know that C and the

adjacency

niatrix A of r comniute. Thus

C comniutes with ail powers of A and

consequeutly

with ail

polynomials

of A. The

corollary

follows _now

imniediately

from lemma 8.

~

Theorem la. Let

f

be

isotropic

^with autocorrelatiou functiou _p aud let r be distance tran- sitive

(or

_a

Cayley graph

of _a comniutative

group).

Furthermore let fl be _some

polynomial

in

A,

Q

=

P(A).

Then _g

= Q

f

is agam

isotropic

_on r and

p~

= c

ù~p ₍₁₂₎

where the constant c =

1/(ù~p)(0)

is chosen such that

p~(0)

=1.

Proof. From

corollary

9 _we kuow

Cg

=

(fl~C)ox.

With

Q

"

fl~

we may ^wnte this _as

~~ ~Î0 ~ ~ _QxyCy0 ~ _ÔÀpCp0

P y~V(p) P

for ail _{~ E}

V(À),

_{as a} cousequence of lemma 1.

Dividing by

the variauce

completes

the

proof.~

There exists _a set of

nght eigenvectors

^{of that} are

orthogonal

with

respect

to the

weight

function

w(~t)

=

)V(~t)) (for

details _see

[25]).

^We _may ^therefore

expand

the autocorrelation function with

respect

to this

basis, p(~t)

₌

£~ a~p~(~t).

Corollary

ii. Uuder the

assumptions

of theorem 10 _we have

p~(~t)

= c

£ a~P~(À~)p~(~t) (13)

where À~ ^denotes the

eigenvalue

of

belonging

to p~. In

particular,

_{if p}

= pj for _some

j,

then

we have

p~

= p.

5.

Empirical landscapes

and ensembles of

Iandscapes.

The

empirical

_vanance

^a~

of _an

isotropic landscape f,

1-e-, of _a

particular

instance of the random field F is measured

by picking

a

sample

of values at

configurations randomly

distributed _on r.

Denote

by p(~t)

₌

)V(~t)) IN

the

probability

that two

randomly

chosen vertices _~ and _y

belong

to

partition

_~t, 1-e-, that there _{is an}

automorphism

_a with

a(~)

₌ 0 and

a(y)

_E

V(~t).

^In case

of _a distance transitive

configuration

_{space we can}

mterpret p(d)

_as the

probability

to

pick

at random two

configurations

with distance d. The

empirical

_{vanance cari} be written _as [5]

a~

=

jr _{(f(~) f(Y))~ (14)}

~,y

and its

expected

value _cari be calculated:

Eia21 =j _{~l (2Eii(~)21} 2Eii(~)i(y)1)

=varifi

LP(JL)jv)~~j L Eif(°)f(Y)i

=

varifi Ii P(~1)P(~1)j

~~~~

~ y~v~~> ~

=varlfl(i à)

Hence the

expected enipirical

variance of _a

landscape f

is related to the ensemble

variance,

i e., the _vanance of the randoni field F at _a

particular vertex, by

E[a~]

₌

Var[f](1 fl) (16)

The

parameter

p nieasures the average correlation of the

landscape.

As shown _m

[25]

fl is the contribution to p

corresponding

to the

largest eigenvalue

of

A,

Ào " D.

The ensenible-autocorrelation function _p and the

expectation

^{of the}

enipincal

correlation function fi are hence related

by

P(11) =

~l~ j~ (17)

Equation (17) iniplies

that the

oninipresent 'back-ground'

correlation p has to be subtracted.

We reniark that for

practical

_purposes it is the

empincally

observable correlation function fi(~1) that _niatters. The autocorrelation function _is said _to be

self-averaging

_{if fi}

= p

(see,

_e-g.,

[26]).

Corollary

12. Under the

assumptions

of theorem 10 it follows from _p

= fi ^that

p~

=

fi~,

_1-e-,

if the autocorrelation function of

f

is

self-averaging,

then the autocorrelation function of the

landscape

Q

f

is also

self-averaging.

6. Iterated

smoothing

_maps A tunable

family.

By N(z)

_we will denote the _set of

neighbors

of _~

plus

_~

itself,

_i-e-, ail vertices y E r with

d(~,y)

< 1. We derme the

smoothing

_operator it

by

it ₌

~

(A

+

E) (18)

The snioothed

landscape

is then defined

by

_{g =}

itf.

Let r be distance transitive. Then _we obtain _{as an} immediate consequence of theoreni 10

f a~(à)p(à+k) p~(à)

=

k=-j (ig)

La~(o)p(k)

~mo

where _ah

(d)

=

(Î~)d,d+k.

The coefficients _{have a}

siniple

combiuatorial

iuterpretatiou:

_ah

(d)

is the uumber of _{pairs (~1,}

v)

with _u E

N(~),

_v E

N(y)

with d(~1,

v)

= d + k where

d(~,y)

= d is

fixed.

For

particular graphs

r it is

fairly

_easy to calculate the coefficients _ah

(d) explicitly.

Cousider

a

geueral hypercube

of diameter _{n over au}

alphabet

of _{size ~.} The

configuration

_space has N

= ~" vertices.

By

distance

trausitivity

_{we may} choose

~

=(000...0000000....0000)

y

=(111.. 1111000....0000)

^~~~~

with

d(~, y)

₌ d. We

partition

the set of _pairs

lu, v)

with _~1 E

N(~),

_v E

N(y)

iuto _urne subsets

depending

on whether _u

= ~, u is

neighbor

of _~ with the diiference

occurring

m one of the first d

positions,

_or u is a

neighbor

with the diiference _occurnng in the latter _n d

positions,

^and

the _same for _v. If _u

= ~ aud _v

= y then

d(u,v)

= d. If _u diifers form _~ in _oue of the latter

n d

positions

and _v

= y then d(~1,

v)

= d

+1;

there _are

(n d)(~c -1)

such

pairs.

If _~t differs from _~ in the first

section, however,

there _are d(~c

2)

_pairs with distance d aud d

pairs

^with distance d

1,

the latter

ansiug

from

mutatiug

a 0 to _a 1. The combinatious of mutations m

both _z and y cau be treated

similarly. Collecting

ail terms

fiually yields

a~~(à)

=

à(à i)

a-i(d)

= 2d + (~c

2)(2d 1)d

Où

(d)

₌ 1 _{+ (lG}

1)(n

₊

2d(n d))

+ _(lG

2)(lGd

+

d(d

_1)(lG

2)) (21)

a+i

(d)

=

2(~ l)(n d)

+

(~ 2)(~ l)(n d)(2d

₊

₁₎ a+2(d)

= (lG

1)~(n d)(n

d

1)

Let

fo

be the random _euergy mortel

[27, 28].

^The autocorrelation function is

p(d)

=

ôo,à

^and

fi(0)

₌

1, fi(d)

₌

-1/(N -1)

for d >

0, respectively,

where N denotes the number of

points

m the

configuration

_space. Define

fr

₌ it

fr-i

Let fi(~) deuote the autocorrelation fuuctiou of

fr.

We will refer to this

family

of

laudscapes

_as iterated

smoothing landscapes,

ISLS.

Numerical evaluatiou of

equation (19)

with the initial condition fi ou Boolean

Hypercubes

and their ~c-letter

generalizatious

shows that

(fr)

iudeed forms _a

family

of

tuuably rugged landscapes (Fig. 2).

In

particular,

_we have

Theorem 13. Let r be _a

generalized hypercube

_{over an}

alphabet

with _~c > 2 letters. Then

lim

fi(~'(d)

= 1-

Îd ₍₂₂₎

Ou _a Boolean

Hypercube

_(~c

=

2)

_we have

lim

fi(~)(d)

=

~(l _~~)

₊

(-1)~ ₍₂₃₎

Proof. The

eigenvalues

of the

adjaceucy

matrix of the

geueral hypercubes

_are

[29,30]

ÀJ "

n(~t ₁₎

_~cj

(24)

aud heuce the

eigeuvalues

of the

smoothiug _operator

it _are 1

j

^~ for

j

=

0,

,

n(~c -1)

+ n.

The

largest eigeuvalue

of

it~, apart

^from

1,

which is obtaiued from

j

=

0,

is fouud be

settiug j

₌ 1 for _{~c >} 2. In the _{case ~c}

= 2 the second

largest eigenvalue

of

it~

is

degenerate,

the

corresponding eigenspace

_is

spanned by

_pi and _p~. As _an immediate cousequence of

equation (28)

in

[25],

the autocorrelation function of the random energy mortel _con be written _as

>(°) (à)

= c

f(~ _{i)£ (])}

p~(à) (25)

tmi

By corollary ii,

^the iterates of fi ^will converge ^to a vector _tu the eigenspace of the

largest eigenvalue

of

Î~

_{that is not}

_orthogonal

_{to the initial condition. The}

_eigeuspace correspoudiug

to Ào has beeu omitted

by

construction of

fi. By equatiou (25) p(°),

_is flot

orthogonal

to auy of the

eigenvectors

_pi

through

_p~. For _{~ >} 2

hence,

_{p converges} to pi while for _~c

= 2 _we obtaiu

a mixture of _pi aud _p~. The ratio of their contributions does _flot

change

^{uuder trie action of}

flÎ _{as a} consequence of

coroiiary

II.

.

o

0.8

îl.

~ ~,

>i _~

O.G _~ ',

i1,

"~

'i

~

'~

0,4 ~ ',

', i '~

i

1,

_"~,

°.~

",, ',,~ ",_~

~

oo

Î""-=.-m-ÎÎÎ-=~Î~

~ --_

0.2 '~~~"--

0.4

0.6

0.8

0

0 10 20 30 40 50 60 70 80 90 100

Hamming

Distance d

a)

0

;,

°'8 _' ',

' ',

' ',

0.6 1 ~,

[

i '~

', ' '~~

°'4

', _"~

' , ',

' , ',

~ '

0'2

' '~ "~

" ~~ '~,

~~ ~,

~o ',_ ~-__ ~

6 O.O '~~'"'~~~~~~

~~ '- "~-,

-0.2

-0.4

0.6

o.8

1.o

0 10 20 30 40 50 60 70 80 90 100

Hamming

Distance d

b)

Fig.

2. Autocorrelation function of ISLS _on

generahzed hypercubes

with

a)

_<

= 2, and

b)

_<

= 4

for with _n

= 100. The number of

smoothings

_are

a)

_{r =} 25

(dotted),

_r

= 50

(short dashed),

_{r =} 75

(long dashed),

_{r =} 100

(dot,dashed),

_r

= 200

(sohd),

and

b)

_{r =} 50

(dotted),

r = 100

(short dashed),

r = 150

(long dashed),

_r

= 200

(dot,dashed)~

and _r

= 400

(sohd), respectively.

We remark that trie contribution of _p~ decreases with the size of the

configuration

_space.

The

hmiting

autocorrelation function

(22) corresponds

to both the trivial

spin glass

mortel with Hamiltonian

7i(a)

=

£~ Ajaj,

and to the Nk,mortel with k

= 0. We also note that for

r

In

< 1

the

landscapes

_are

essentially uncorrelated,

_1-e-, the correlation

length

is

o(n).

Numerical estimates show that the nearest

neighbor

correlation fi(~)

(1) depends only

_on the ration _r

In

for

large

_n

(See Fig. 3).

As _a consequence of theorem

13,

we find that

limr-m pli

= 1

~ _~

n ~c 1

for _{~c >} 2. For _~c

= 2 _we find

lim

p(~)(d)

= lim

p(~)(d ₁₎

= 1-

~~

for _even d

(26)

r-ce r-ce 7Î +1

The data _in

figure

3 indicate that this limit is

already

obtained for

r/n

> 2 to _a

good approximation.

7.

Averaging operators

on

spin glass

and Nk

landscapes.

We first consider the p-spm

mortels,

1-e-, ^the

long

_range

spin glasses

with Hamiltonian

7ip(a)

₌

£

Aj~j~__,j~ aj~aj~ aj~

(27)

31<32<...<3P

with _a~ E

(-1, +1)

and

coupling

constants A~j___k drawn from _{some common} distribution. The

ruggedness

of the

landscapes

increases _as the order _p of the

spin-interactions

^increases. We have then

Theorem 14. Let fl be _any

polynomial

of the

adjacency

matrix of the Boolean

Hypercube.

Then the

landscapes 7ip

^and

Q7ip

^{have the} same autocorrelation functions.

Proof. As shown in

[16]

^the autocorrelation functions of the

p-spin

^mortels are

PV

id)

=

(il

^~

(ii)~ Ill Ii _Il (28)

These _are

smtably

normalized Krawtchouk

polynomials,

which _are known to be the

eigenvectors

of the

collapsed adjacency

matrix of the Boolean

Hypercube (see,

_e-g.,

_[25, 29]). Application

of

Corollary

ii

completes

the

proof. ~

In Kauifman's Nk-mortel the fitness of _a

binary string

_{~ is} defined _as the average over

"Site fitnesses"

f~,

^which

depend

_{on ~~} and k additional

positions ii through ik.

The site fitnesses f~ =

f~(~~,

_~~~,

..,

~~~)

are assumed to be uncorrelated random variables drawn from

a common distribution

iii.

The

ruggedness

of the

landscape

_{mcreases as} the number k of

positions contributing

to a

surgie

site fitness _mcreases. Diiferent variants of these

landscapes

have been

investigated, depending

on which

positions

contribute to a given site fitness

(see,

e-g.,

[31]).

In contrast to the p-spm

mortels,

the

landscapes

of NÉ

type

are aifected

by

the

smoothing

operator

it.

Theorem 15. Let

f

be _an

arbitrary

NÉ

landscape.

Then the autocorrelation function of the

landscapes it~f

_converges to

p(d)

=

((1- 2d/n)

₊

(1- ()(-1)~

with o _<

(

< 1 _{as r} tends towards

infinity.

o

0.8

0.6

0.4

n=25 n=50 n=100 n=200 0.2

ù-ù

ù-ù 0.5 1-Ù 5 2.0

r/n

a)

i o

o.8

0.6

0.4

n=25 n=50 n=100 n=200

o-ù

o-o 0.5 1-o 1.5 2.0

r/n

b)

Fig.

3. Nearest

neighbor

correlation for ISLS _on

generalized hypercubes

_{as a} function of the number of iterated

smoothings. a)

_~

= 2,

b)

_«

= 4.

Proof. It is necessary and sufficient to show that the autocorrelation function _{p of}

f

and _pi

are not

orthogonal,

_1-e-, that

£ _{p(d)pi (d)} (")

⁼

^{( #}

⁰

⁽²⁹⁾

"

Î~

d

Note that _pi

(d)

_{= -pi}

(n d). Equation (29)

_cari hence be rewritten _as

f

₌

£ (()

pi

(d) lp(d) p(n

_d)1

(30)

d<n/2

If

p(d)

>

p(n d)

for ail d _<

n/2,

and if the

inequality

holds

strictly

for at least _one d <

n/2,

we have

(

> 0. It follows

immediately

from the very definition of the NÉ mortel

(irrespective

of how the

neighboring

sites are

chosen)

that p is

monotonically decreasing

and non-constant-

'

Remark. The vanants of the NÉ mortel do trot behave

identically

under the action of it.

Whenever the autocorrelation function is _a

polynomial

of

degree

less than _n, then _we have

(

₌ 1 in theorem 15. For exact

expressions

^of the autocorrelation functions of varions NÉ

niodels _see

[2, 5]. Recently

Nk-mortels _{on sequences} of

arbitrary alphabets

have been considered

[2].

^{Theorem 15 holds with}

(

₌ for ail

alphabets

with three _{or more} letters.

As _a final

exaniple

consider _an

isotropic landscape

_on the circle of

length

_n > 4

(Millonas,

pers.

comniunication).

The

configuration

_space hence is the

Cayley graph

of the

cyclic

_group

C~

₌

l't~)0

< k <

n)

with _set of

generators

^4l =

16't,'t~~).

Its _vertex

degree

is D

= 2. The

eigenvectors

^of the

adjacency

niatrix _{are eh}

(~)

₌

exp(2grk~ In),

_{as an} immediate consequence of lemma 5. The

eigenvectors

^{of the}

collapsed adjacency

matrix are

pk(d)

=

cos(2grkd/n)

for 0 < k <

n/2,

associated with the

eigenvalues

Àk _#

2cos(2grk/n).

The

eigenvalue

of

Î~

corresponding

to

À~/2

is niuch smaller that the _one

corresponding

to

Ài

for _{n >}

4,

hence _p converges ^to pi under the iterated action of it whenever _{p is not}

orthogonal

to pi

Suppose f

has _{a monotone}

non-increasing

autocorrelation function _p. As _a

particular

_ex-

ample

consider the random _energy niodel _on the circle.

By

the _{same reasoning as} in the

proof

of theorem

15,

_{p is} not

orthogonal

to pi, and hence iterated

smoothing eventually

leads to a

landscape

with autocorrelation function _pi Note that the smoothed

landscapes

_on the Boolean

Hypercube (and

_on related

graphs) eventually

have autocorrelation functions of the form

p(d)

= 1-

)d

₊

O(d~)

while for the circle

graphs

_we find

p(d)

= 1

ad~

₊

O(d3)

The

landscape

_on the circle hence becomes

exceptionally

smooth [16] as a consequence of the

special geometry

of the

configuration

_space.

This

example

also shows that the

degeneracy

of it _on the Boolean

Hypercube

is trot a

necessary consequence of the fact that the

configuration

_{space is}

bipartite,

1-e-, that there _is

an

eigenvalue

-D of the

adjacency matrix,

^{since trie} circle

graphs

_are

bipartite

for _even n, and it is not

degenerate

there.

8 Discussion.

A

general theory

of the action of linear

(averagmg)

operators on

isotropic landscapes

is devel-

oped

which allows to

predict

their eifect _on the correlation structure of the

landscape.

We have shown that there _are

particular Iandscapes (characterized by

autocorrelation functions that _are

eigenvectors

^{of the}

collapsed adjacency

matrix of the

underlying configuration space)

which _are

not aifected

by "smoothing" operators.

The

p-spin models,

and in

particular

the

Sherrington- Kirkpatrick [32],

^model are members of this dass _on the Boolean

hypercube.

Another

example

is the

landscape

of the

graph-bipartitioning problem.

A _new

family

^of

tunably rugged landscapes

is

established,

which arises

by

iterated

smoothing

of the random

landscape.

These "iterated

smoothing landscapes" (ISLS)

have _an

advantage

over the other two known

tunably rugged

families: The

p-spin

mortels _are defined

only

for Boolean

hypercubes,

and I<auifman's Nk

family

lives _on

general

_{sequence spaces,} while ISLS _can be constructed _on

arbitrary configuration

_spaces. Their

simple

structure

suggests

furthermore

that

analytical

solutions of these mortels

might

be feasible. Iterated

smoothing

of the random energy mortel

eventually

leads to _a

landscape

with

linearly decreasing

autocorrelation function

on

general

_{sequence spaces} with

alphabets

of size _~c >

2,

while _on Boolean

hypercubes,

~c =

2,

one obtains _a

superposition

^of a linear function and _an oszillation of the form

(-1)d.

On _a circle

graph

trie _same

procedure

leads to an autocorrelation function of trie form 1-

ad~

_+.

This shows

again

that trie structure of _a

landscape

cannot be

separated

from the

geometry

of the

underlying configuration

_space.

Evolutionary adaptation

_cari be descnbed in the

simplest

_{case as} the motion of _a

population

in

configuration

_space. Almost ail

analytical work, however,

has been doue with

adaptive

walks

(or

related

processes)

which

neglect

the

population aspect.

For _a recent review _see, e-g-,

[31].

^TO a first

approximation

_a

population

_cari be

replaced by

its "center of mass"

(consensus configuration),

which _moves in the

landscape "experienced" by

the entire

population: again

to _a first

approximation,

this

landscape

is _a

weighted

_average of the fitnesses of

configurations

centered around the _consensus.

Consequently,

the width of the

population

should

strongly

influence the

dynamics

of evolution: broader

populations

"feel" in

general

smoother

landscapes,

and hence

optimization

should be easier.

Usually

the broadness of the

population

is controlled

via the mutation _rate

[3,33]

^{which at the} same time controls the

exploration

rate. Further

investigations

_{are necessary} in order _to separate the eifects of enhanced

exploration

and

implicit smoothing

^{of the}

underlying landscape

caused

by

_an increased mutation rate.

It should be

kept

in mina that the results

presented

here

depend

_on the

assumption

of

isotropic landscapes.

While this requirement ^is often fulfilled for

simple

statistical

mortels,

it is

unlikely

to holà for real

biophysical

or

physical systems.

^It bas been

shown,

for

instance,

that trie

landscapes

of RNA free

energies

_are not

isotropic

for natural _sequences

[34].

Much work _on random

landscapes

such _as Derrida's REM bas been motivated with the

notion that a

sufficiently

"coarse

grained"

version of _any

rugged landscape

would look random.

More

precisely,

this is to say that it should be

possible

to construct renormalization _groups

having

random

landscapes

_as attractive fixed

points.

^{While the}

averaging ^procedures

discussed

m trie this contribution _are trot

exactly

trie kind of _coarse

graining

_one would have in mmd for such _a

construction,

the methods

developed

here _seem to be useful for such _a task.

The

theory

outlined in this contribution _is trot

applicable

for the

traveling

salesman

problem

since the raturai

configuration

_spaces of the TSP _are

Cayley graphs

of the

symmetric

_group with the _set of ail

transposition,

_or the set of ail inversions

(20pt-moves),

_as

_generators

_[7].

It is easy ^to check that these

graphs

are trot distance

transitive,

and of _course the

symmetric

group is non-commutative-

Acknowledgements.

thank the Max Planck

Society, Germany,

for

supporting

_my stay ^{at the Santa Fe Institute.}

Stimulating

discussions with P-W-

Anderson,

W.

Fontana,

S-A-

Kauifman,

M.

Millonas,

R.

Palmer,

and M. Virasoro _are

gratefully acknowledged.

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