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.
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(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!
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Fully connected lattice
Disadvantage: No notion of distance and locality.
Curie-Weiss model (for Ising model)
m = tanh (!Jm)
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Bethe lattice(Cayley tree)
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=!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 ->
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Bethe lattice
Trouble: When results depend in a complex way on boundaries,
the boundaries need to be specified. Complicated! Moreover unreasonable for simulations ...
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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).
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Solvable How?
Shortest cycle going trough a typical node has length log(N).
Bethe lattice = Random graphLocally:
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Less trivial example
Graph ColoringH =
!
(ij)
!Si,SjSi ! {1, . . . , q}
Coloring = antiferromagnetic
Potts model at zero temperature
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Less trivial exampleLess trivial example
Graph ColoringH =
!
(ij)
!Si,SjSi ! {1, . . . , q}
Coloring = antiferromagnetic
Potts model at zero temperature
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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
]
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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)
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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)
?
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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
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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
]
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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
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Relation to structural glass transition
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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
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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
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Applications
Algorithmic hardness
Lattice model of colloidal glass
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Algorithmic hardness
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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
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Answer 2: Glassiness makes problems hard(Mezard, Parisi, Zecchina’02)
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Answer 2: Glassiness makes problems hard
BUT!(Mezard, Parisi, Zecchina’02)
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Answer 2: Glassiness makes problems hard
BUT!(Mezard, Parisi, Zecchina’02)
Many simple algorithms work in the glass phase.
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Answer 2: Glassiness makes problems hard
Zero energy states
Positive energy states
Zero energy states
Positive energy states
Canyon dominated Valley dominated vs.
BUT!(Mezard, Parisi, Zecchina’02)
Many simple algorithms work in the glass phase.
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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
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Model for colloidal glass
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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
Saturday, January 30, 2010
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.
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Exactly Solvable on Random Graphs
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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
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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?
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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
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Conclusions
Random graphs - solvable (tree-like!)
Ideal glass phase transition
Easy / Hard transition in optimization
Models for structural glasses
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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).
Saturday, January 30, 2010