Physics on Random Graphs

Transcription

1 Physics on Random Graphs Lenka Zdeborová (CNLS + T-4, LANL) in collaboration with Florent Krzakala (see MRSEC seminar), Marc Mezard, Marco Tarzia...

2 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.

3 (Periodic) Lattice models Ising 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!

4 Fully connected lattice Curie-Weiss model (for Ising model) m = tanh (βjm) Disadvantage: No notion of distance and locality.

5 Bethe lattice (Cayley tree)

6 h 1 h 2 h 3 =! h! Fixed point: βh = (c 1)atanh(tanh βh tanh βj) β c (2d) = β c (3d) = β c (4d) = β c (5d) = <- Bethe & True -> β c (2d) = β c (3d) = β c (4d) = β c (5d) = 0.114

7 Bethe lattice Trouble: When results depend in a complex way on boundaries, the boundaries need to be specified. Complicated! Moreover unreasonable for simulations...

8 Random graphs Erdos-Renyi p = c N 1 4-regular lim Q(k) = e c c k N k! Q(k) = δ(k 4)

9 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).

10 Solvable How? Locally: Bethe lattice = Random graph Shortest cycle going trough a typical node has length log(n).

11 Graph Coloring H = (ij) δ Si,S j S i {1,..., q} Coloring = antiferromagnetic Potts model at zero temperature

12 Graph Coloring H = (ij) δ Si,S j S i {1,..., q} Coloring = antiferromagnetic Potts model at zero temperature

13 Bethe-Peierls = Belief Propagation = Replica Symmetric = Liquid Solution i j k ψs i j i probability that node i takes color s i conditioned on absence of link ij. ψ i j s i = 1 Z i j k i\j [1 (1 e β )ψ k i s i ] q s i =1 ψ i j s i = 1

14 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)

15 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)

16 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. Σ(T ) T K T d T

17 Glassy cavity equation P i j (ψ i j ) = 1 Z i j k i\j dp k i (ψ k i )(Z i j ) m δ[ψ i j F({ψ k i })] ψ i j s i = 1 Z i j k i\j [1 (1 e β )ψ k i s i ] Closed equations for probability distributions - population dynamics (Mezard,Parisi 01) technique for numerical solution

18 Temperature The phase diagram dynamical glass transition Kauzmann transition Average degree 5-coloring of E-R random graphs

19 Relation to structural glass transition

20 Structural Glass transition (from Debenedetti, Stillinger 01)

21 Angell s plot log(viscosity) η e T η e (T ) T T K inverse temperature

22 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 1/Σ(T ). Kauzmann transition (vanishing structural entropy) In finite dimension (Kauzmann 48), barriers grow with 1/Σ(T ) and diverge at T K (Adam, Gibbs 85 relation, Bouchaud, Biroli 04)

23 Applications Algorithmic hardness Lattice model of colloidal glass

24 Algorithmic hardness

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28 Where the REALLY hard problems are? (Cheeseman, Kanefsky, Taylor, 1991) probability of colorability T p SAT N = 100 p SAT N = 71 p SAT N = 50 comp. time N = 100 comp. time N = 71 comp. time N = time to decide average degree of the graph

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

30 Answer 2: Glassiness makes problems hard (Mezard, Parisi, Zecchina 02) BUT!

31 Answer 2: Glassiness makes problems hard (Mezard, Parisi, Zecchina 02) BUT! Many simple algorithms work in the glass phase.

32 Answer 2: Glassiness makes problems hard (Mezard, Parisi, Zecchina 02) BUT! Many simple algorithms work in the glass phase. Canyon dominated vs. Valley dominated Positive energy states Positive energy states Zero energy states Zero energy states

33 0.03 Valleys 3-XOR-SAT with L=3 solvable only by Gauss E(S) S Canyons 4-coloring of 9-regular random graphs solvable by reinforced belief propagation E(S) S

34 Model for colloidal glass

35 Model for colloidal glass Hard 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 (Cho, Yi, et all, 2005)

36 The lattice model Each site has c neighbors c = 4, l = 2 Occupied site can have up to l occupied neighbors (Biroli, Mezard 01) Attraction between nearest occupied neighbors Z(µ, β) = allowed {n} 1 Z(µ, β) e17µ+16β e µ P i n i+β P (ij) n in j

37 Motivated by patchy colloids l = 2 l = 3 l = 4 l = 6 Parameter c, graph degree, e.g. the kissing number, i.e. c=12 in 3d.

38 Exactly Solvable on Random Graphs

39 Attractive colloids, c=6, l=4 T c=6,l=4 transient percolation Liquid-gas coexistence 0.6 Glass-gas coexistence at low T T T LL spinodals T d T K Packing fraction! 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

40 Patchy colloids, c=10, l=3 T T c=10,l=3 Coexistence region shrinks, 0.6 transient percolation Kauzmann transition on at large density, 0.4 spinodals Bellow dynamical transition - ideal gel? T LL T d T K Packing fraction! 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

41 T Z(µ, β) = 1 T allowed {n} Re-entrance e µ P i n i+β P (ij) n in j s p P i δ(l P j i n j) Entropic penalty when max # of neighbors High T - particle run out of space - repulsive glass Low T - particle stick together - attractive glass c=4,l=2, s p =2 T d T K Packing fraction! 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

42 Conclusions Random graphs - solvable (tree-like!) Ideal glass phase transition Easy / Hard transition in optimization Models for structural glasses

43 References Potts glass / Coloring / Hardness Krzakala, Montanari, Ricci-Tersenghi, Semerjian, LZ, PNAS 104, (2007). LZ, Krzakala, PRE 76, (2007). F. Krkazala, LZ, EPL 81 (2008) LZ, PhD thesis, Acta Phys. Slov. 59, No.3, (2009). Colloidal Glass F. Krzakala, M. Tarzia, LZ; PRL 101, (2008).