Physics on Random Graphs - Univerzita...

Physics on Random Graphs

Lenka Zdeborová (CNLS + T-4, LANL)

in collaboration with Florent Krzakala (see MRSEC seminar), Marc Mezard, Marco Tarzia ....

Saturday, January 30, 2010

Take Home Message

Solving problems on random graphs =

I. Gaining insight about the origin of algorithmic complexity of NP-hard optimization problems.

II. More realistic mean field models for glass formers and other disordered materials.


(Periodic) Lattice modelsIsing model, lattice gas, percolation, Potts model, xy model, Heisenberg model, models from Baxter’s book, etc ....

Disordered or frustrated models on (periodic) lattices are mostly not solvable, e.g. random field Ising model, Edwards-Anderson model

Let’s take a lattice on which even (many) disordered systems are solvable!


Fully connected lattice

Disadvantage: No notion of distance and locality.

Curie-Weiss model (for Ising model)

m = tanh (!Jm)


Bethe lattice(Cayley tree)


=!h’!

h1 h2 h3

Fixed point:!h = (c! 1)atanh(tanh!h tanh !J)

!c(2d) = 0.346!c(3d) = 0.203!c(4d) = 0.144!c(5d) = 0.112

!c(2d) = 0.440!c(3d) = 0.221!c(4d) = 0.149!c(5d) = 0.114

<- Bethe & True ->


Bethe lattice

Trouble: When results depend in a complex way on boundaries,

the boundaries need to be specified. Complicated! Moreover unreasonable for simulations ...


Random graphs4-regularErdos-Renyi

p =c

N ! 1

Q(k) = !(k ! 4)limN!"

Q(k) =e#cck

k!Saturday, January 30, 2010

Why Random Graphs?

Still solvable and we like solvable models.

Complex systems (web, internet, social contacts, food chains, gene regulation) live on networks not periodic lattices

Interesting physics (ideal glass transitions, no crystal, enhanced frustration).


Solvable How?

Shortest cycle going trough a typical node has length log(N).

Bethe lattice = Random graphLocally:


Less trivial example

Graph ColoringH =

!

(ij)

!Si,SjSi ! {1, . . . , q}

Coloring = antiferromagnetic

Potts model at zero temperature


Less trivial exampleLess trivial example

Graph ColoringH =

!

(ij)

!Si,SjSi ! {1, . . . , q}

Coloring = antiferromagnetic

Potts model at zero temperature


Bethe-Peierls = Belief Propagation = Replica Symmetric = Liquid Solution

q!

si=1

!i!jsi

= 1

ij

k !i!jsi probability that node i

takes color conditioned on absence of link ij.

si

!i!jsi

=1

Zi!j

!

k"!i\j

[1! (1! e#")!k!isi

]


Point-to-set correlations

BP solution asymptotically exact on random graphs if and only if point-to-set

correlation decay to zero.

Divergence of point-to-set correlation length = divergence of equilibration time (Montanari,Semerjian’06)= dynamical glass transition (Kirkpartick, Thirumalai’87)


Point-to-set correlations

BP solution asymptotically exact on random graphs if and only if point-to-set

correlation decay to zero.

Divergence of point-to-set correlation length = divergence of equilibration time (Montanari,Semerjian’06)= dynamical glass transition (Kirkpartick, Thirumalai’87)

?


Glassy (1RSB) solution(Parisi’80, Mezard,Parisi’01)

Point-to-set correlation finite => decompose Boltzmann measure into many Gibbs states.

Structural entropy, complexity = logarithm of the number of dominating Gibbs states that can be induced in the bulk of the tree.

T

!(T )

TdTK


Glassy cavity equationP i!j(!i!j) =

1Zi!j

! "

k"!i\j

dP k!i(!k!i)(Zi!j)m"[!i!j ! F({!k!i})]

Closed equations for probability distributions -

population dynamics (Mezard,Parisi’01) technique for

numerical solution

!i!jsi

=1

Zi!j

!

k"!i\j

[1! (1! e#")!k!isi

]


The phase diagram

0

0.1

0.2

0.3

0.4

12 13 14 15 16 17 18

Kauzmann transition

dynamicalglass transition

Tem

pera

ture

Average degree

5-coloring of E-R random graphs


Relation to structural glass transition


Structural Glass transition

(from Debenedetti, Stillinger’01)Saturday, January 30, 2010

Angell’s plotlo

g(visc

osity

)

inverse temperature

! ! e!T

! ! e!(T )

T!TK


Dynamical transition (diverging equilibration time and point-to-set correlation length, system trapped at higher free energy, mode-coupling-like) In finite dimension smeared out, barrier always finite (due to nucleation), but grow with .

Kauzmann transition (vanishing structural entropy) In finite dimension (Kauzmann’48), barriers grow with and diverge at (Adam, Gibbs’85 relation, Bouchaud, Biroli’04)

1/!(T )

1/!(T ) TK


Applications

Algorithmic hardness

Lattice model of colloidal glass


Algorithmic hardness


Where the REALLY hard problems are? (Cheeseman, Kanefsky, Taylor, 1991)Random K-SAT

What makes problems hard to solve ?

Experiment :

random 3-SAT

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 100

500

1000

1500

2000

2500

3000

3500

4000

pS

AT

med

ian

com

p. ti

me

!

pSAT N = 100pSAT N = 71pSAT N = 50

comp. time N = 100comp. time N = 71comp. time N = 50

average degree of the graph

time

to d

ecid

e

prob

ability

of

colo

rabi

lity


Answer 2: Glassiness makes problems hard(Mezard, Parisi, Zecchina’02)


Answer 2: Glassiness makes problems hard

BUT!(Mezard, Parisi, Zecchina’02)




Many simple algorithms work in the glass phase.



Zero energy states

Positive energy states

Zero energy states

Positive energy states

Canyon dominated Valley dominated vs.


Many simple algorithms work in the glass phase.


Valleys

Canyons4-coloring of 9-regular random graphs

solvable by reinforced belief propagation

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

-0.1 -0.05 0 0.05 0.1

E(S)

S

3-XOR-SAT with L=3solvable only by Gauss

0

0.005

0.01

0.015

0.02

0.025

0.03

-0.05 0 0.05

E(S)

S


Model for colloidal glass


Model for colloidal glassHard spheres with attractive potential (Lennard-Jones, square well)

Colloids in polymer suspension - depletion induced attraction

Valency limited colloids, patchy colloids - given number of attractive (sticky) sites

CONTENTS 38

Figure 18. Top: Adapted with permission from [223]. Copyright 2005: AmericanChemical Society. Experimental particles realized from bidisperse colloids in waterdroplets. Courtesy of G.-R. Yi. Bottom: Reprinted with permission from [27].Copyright 2006 by the American Physical Society. Primitive models of patchy particlesused in the theoretical study of Ref.[27].

possibilities o!ered by the realization of new colloidal molecules[5]. Numerical studies

are being used to design specific self-assembly [228, 229, 230, 231, 232, 233], as well

as to determine optimized circularly (spherically-)symmetric interactions in 2d (3d) for

producing targeted self-assembly with low-coordination numbers: by inverse methods,

square and honeycomb lattices[238, 239] have been assembled in 2d, and a cubic lattice

in 3d[240]. In this paragraph, we will only focus on the knowledge about phase diagram

and gelation of patchy colloids that is being recently established.

Models have started to appear in the literature, taking into account not only

a spherical attraction, but also angular constraints for bond formation[241]. Similar

ideas have been exploited in the study of protein phase diagrams[242, 243]. However,

these earlier works have not addressed the important question of how to systematically

a!ect the phase diagram of attractive colloids in order to prevent phase separation

and allow ideal gel formation. To this end, we recently revisited [28, 244] a family of

limited-valency models introduced by Speedy and Debenedetti[245, 246], where particles

interact via a simple square well potential, but only with a pre-defined maximum number

of attractive nearest neighbours, Nmax, while hard-core interactions are present for

additional neighbours. This model can be viewed as a toy model for particles with

randomly-located sticky spots, due to the absence of any angular constraint. Moreover,

the sticky spots are not fixed, but can roll onto the particle surface, also relatively to each

other. The disadvantage of such model is that the Hamiltonian contains a many-body

term, taking into account how many bonded neighbours are present for each particle

at any given time. Notwithstanding this, the model is the simplest generalization of

attractive spherical models, and its study can be built on the vast knowledge of phase

(Cho, Yi, et all, 2005)Saturday, January 30, 2010

The lattice modelEach site has c neighbors

Occupied site can have up to l occupied neighbors (Biroli, Mezard’01)

Attraction between nearest occupied neighbors

c = 4, l = 2

1Z(µ,!)

e17µ+16!

Z(µ,!) =!

allowed {n}

eµP

i ni+!P

(ij) ninj


Motivated by patchy colloidsCONTENTS 38

Figure 18. Top: Adapted with permission from [223]. Copyright 2005: AmericanChemical Society. Experimental particles realized from bidisperse colloids in waterdroplets. Courtesy of G.-R. Yi. Bottom: Reprinted with permission from [27].Copyright 2006 by the American Physical Society. Primitive models of patchy particlesused in the theoretical study of Ref.[27].

possibilities o!ered by the realization of new colloidal molecules[5]. Numerical studies

are being used to design specific self-assembly [228, 229, 230, 231, 232, 233], as well

as to determine optimized circularly (spherically-)symmetric interactions in 2d (3d) for

producing targeted self-assembly with low-coordination numbers: by inverse methods,

square and honeycomb lattices[238, 239] have been assembled in 2d, and a cubic lattice

in 3d[240]. In this paragraph, we will only focus on the knowledge about phase diagram

and gelation of patchy colloids that is being recently established.

Models have started to appear in the literature, taking into account not only

a spherical attraction, but also angular constraints for bond formation[241]. Similar

ideas have been exploited in the study of protein phase diagrams[242, 243]. However,

these earlier works have not addressed the important question of how to systematically

a!ect the phase diagram of attractive colloids in order to prevent phase separation

and allow ideal gel formation. To this end, we recently revisited [28, 244] a family of

limited-valency models introduced by Speedy and Debenedetti[245, 246], where particles

interact via a simple square well potential, but only with a pre-defined maximum number

of attractive nearest neighbours, Nmax, while hard-core interactions are present for

additional neighbours. This model can be viewed as a toy model for particles with

randomly-located sticky spots, due to the absence of any angular constraint. Moreover,

the sticky spots are not fixed, but can roll onto the particle surface, also relatively to each

other. The disadvantage of such model is that the Hamiltonian contains a many-body

term, taking into account how many bonded neighbours are present for each particle

at any given time. Notwithstanding this, the model is the simplest generalization of

attractive spherical models, and its study can be built on the vast knowledge of phase

l = 2 l = 3 l = 4 l = 6

Parameter c, graph degree, e.g. the kissing number, i.e. c=12 in 3d.


Exactly Solvable on Random Graphs


Attractive colloids, c=6, l=4

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

T

Packing fraction !

spinodals

transient percolation

Td

TK

TLL

c=6,l=4

Verduin, Dhont’95; Sastry PRL’00;Foffi, McCullagh et al PRE’02; Dawson’02; Shell, Debenedetti PRE’04; Sciortino, Tartaglia, Zaccarelli J. Phys. Chem’05; Manley, Wyss et al, PRL’05; Ashwin, Menon, EPL’06; Lu, Zaccarelli et al, Nature’08; and many others

T• Liquid-gas coexistence

• Glass-gas coexistence at low T


Patchy colloids, c=10, l=3

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4

T

Packing fraction !

transient percolation

spinodals

TdTK

TLL

c=10,l=3

Bianchi, Largo et al PRL’06; Zaccarelli, Buldyrev et al, PRL’05; Zaccarelli, Saika-Voivod et al, J.Phys‘06; Sastry, Nave, Sciortino, J.Stat.Mech’06; and many others

T• Coexistence region shrinks,

• Kauzmann transition on at large density,

• Bellow dynamical transition - ideal gel?


Re-entrance

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3 0.4 0.5 0.6

T

Packing fraction !

Td

TK

c=4,l=2, sp=2

0 0.2 0.4 0.6 0.8

1

0.6 0.62 0.64

Fabbian, Gotze er all PRE’99; Dawson, Foffi et all, PRE’01; Sciortino, Nature Materials’02; Frenkel, Science’02; Pham, Puertas, et al Science’02; Eckert, Bartsch PRL’02; Dawson’02; and many others

Z(µ,!) =!

allowed {n}

eµP

i ni+!P

(ij) ninj!spP

i "(l!P

j!!i nj)

T• Entropic penalty when max # of neighbors

• High T - particle run out of space - repulsive glass• Low T - particle stick together - attractive glass


Conclusions

Random graphs - solvable (tree-like!)

Ideal glass phase transition

Easy / Hard transition in optimization

Models for structural glasses


ReferencesPotts glass / Coloring / Hardness

Krzakala, Montanari, Ricci-Tersenghi, Semerjian, LZ, PNAS 104, 10318 (2007).

LZ, Krzakala, PRE 76, 031131 (2007).

F. Krkazala, LZ, EPL 81 (2008) 57005.

LZ, PhD thesis, Acta Phys. Slov. 59, No.3, (2009).

Colloidal Glass

F. Krzakala, M. Tarzia, LZ; PRL 101, 165702 (2008).


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