Reconstruction for Models on Random Graphs

Transcription

1 1 Stanford University, 2 ENS Paris October 21, 2007

2 Outline 1 The Reconstruction Problem 2 Related work and applications 3 Results

3 The Reconstruction Problem: A Story

4 Alice and Bob

5 Alice, Bob and G root

6 Exit Bob root

7 Alice samples a proper coloring (uniformly)... root

8 ... and hides a ball B(root, t) root

9 Bob... root?

10 ... guesses right! root!

11 The problem Does Bob have a chance?

12 The problem Does Bob have a bigger chance than when everything is hidden?

13 Formally X = {X i : i V } uniformly random proper coloring. P U { G} distribution of X U {X i : i U V } B(r, t) = {i V : d(i, r) t} Definition The reconstruction problem is (t, ε) solvable for the rooted graph G if P r,b(r,t) {, G} P r { G}P B(r,t) { G} TV ε.

14 Formally X = {X i : i V } uniformly random proper coloring. P U { G} distribution of X U {X i : i U V } B(r, t) = {i V : d(i, r) t} Definition The reconstruction problem is (t, ε) solvable for the rooted graph G if P r,b(r,t) {, G} P r { G}P B(r,t) { G} TV ε.

15 Formally X = {X i : i V } uniformly random proper coloring. P U { G} distribution of X U {X i : i U V } B(r, t) = {i V : d(i, r) t} Definition The reconstruction problem is (t, ε) solvable for the rooted graph G if P r,b(r,t) {, G} P r { G}P B(r,t) { G} TV ε.

16 Formally X = {X i : i V } uniformly random proper coloring. P U { G} distribution of X U {X i : i U V } B(r, t) = {i V : d(i, r) t} Definition The reconstruction problem is (t, ε) solvable for the rooted graph G if P r,b(r,t) {, G} P r { G}P B(r,t) { G} TV ε.

17 Formally X = {X i : i V } uniformly random proper coloring. P U { G} distribution of X U {X i : i U V } B(r, t) = {i V : d(i, r) t} Definition The reconstruction problem is solvable for the sequence of rooted graphs G N = (V N = [N], E N ) if for some ε > 0 and all t 0 P r,b(r,t) {, G N } P r { G N }P B(r,t) { G N } TV ε.

18 Formally X = {X i : i V } uniformly random proper coloring. P U { G} distribution of X U {X i : i U V } B(r, t) = {i V : d(i, r) t} Definition The reconstruction problem is solvable for the sequence of random rooted graphs G N = (V N = [N], E N ) if for some ε > 0, P r,b(r,t) {, G N } P r { G N }P B(r,t) { G N } TV ε, with positive probability.

19 More generally: Markov distributions on G X = {X i : i V } discrete radom variables, X i {1,..., q} ψ ij : {1,..., q} {1,..., q} R + P{X G, ψ} = 1 Z G (ij) E ψ ij (x i, x j ).

20 Examples 1. (ɛ-proper) Colorings/Potts model: X i {1,..., q} { 1 if xi x ψ ij (x i, x j ) = j, ɛ otherwise. 2. Ferromagnetic Ising model: X i {+1, 1} { 1 if xi = x ψ ij (x i, x j ) = j, ɛ otherwise.

21 Examples 1. (ɛ-proper) Colorings/Potts model: X i {1,..., q} { 1 if xi x ψ ij (x i, x j ) = j, ɛ otherwise. 2. Ferromagnetic Ising model: X i {+1, 1} { 1 if xi = x ψ ij (x i, x j ) = j, ɛ otherwise.

22 Examples (continued) 3. Ising spin glass: X i {+1, 1} Label (i, j) E with J ij {+, } independently { 1 if xi = x ψ ij (x i, x j ) = j, ɛ otherwise. if J ij = +. ψ ij (x i, x j ) = { ɛ if xi = x j, 1 otherwise. if J ij =.

23 Examples (continued) 3. Ising spin glass: X i {+1, 1} Label (i, j) E with J ij {+, } independently { 1 if xi = x ψ ij (x i, x j ) = j, ɛ otherwise. if J ij = +. ψ ij (x i, x j ) = { ɛ if xi = x j, 1 otherwise. if J ij =.

24 Examples (continued) 3. Ising spin glass: X i {+1, 1} Label (i, j) E with J ij {+, } independently { 1 if xi = x ψ ij (x i, x j ) = j, ɛ otherwise. if J ij = +. ψ ij (x i, x j ) = { ɛ if xi = x j, 1 otherwise. if J ij =.

25 Related work and applications

26 When G =Tree Bleher, Ruiz, Zagrebenov (1995): Ising model on b-ary trees Solvable iff b(1 2ɛ) 2 > 1. Evans, Kenyon, Peres, Schulman (2000): Ising on general trees Solvable iff br(t )(1 2ɛ) 2 > 1. Mossel, Peres (2003): Non binary variables Brightwell, Winkler (2004), Martin (2004): Independent sets. Chayes et al. (2006): Asymmetric Ising.

27 This talk G = Random and Sparse Examples Uniformly random with M = Nγ edges. Uniformly random of degree b + 1.

28 Application 1: MCMC with local moves Kenyon,Mossel,Peres (2001) Reconstruction is solvable Slow mixing. Martinelli,Sinclair,Weitz (2005) On trees: Reconstruction is solvable Slow mixing. [The connection is likely to be more general]

29 Application 1: MCMC with local moves Kenyon,Mossel,Peres (2001) Reconstruction is solvable Slow mixing. Martinelli,Sinclair,Weitz (2005) On trees: Reconstruction is solvable Slow mixing. [The connection is likely to be more general]

30 Message passing algorithms, k-sat, etc. Problems: 1. Find x such that P{x G} > Sample from P{X G}. Basic philosophy Do computations on G as if you were on a tree. Extremely sucessful when G is random and sparse (eg random k-sat, survey propagation, Mézard, Zecchina 2003)

31 Message passing algorithms, k-sat, etc. Problems: 1. Find x such that P{x G} > Sample from P{X G}. Basic philosophy Do computations on G as if you were on a tree. Extremely sucessful when G is random and sparse (eg random k-sat, survey propagation, Mézard, Zecchina 2003)

32 Message passing algorithms, k-sat, etc (continued) Non-reconstructibility Local structure (tree) decouples from global (loops)

33 Results

34 Naive guess Random sparse graphs converge locally to trees hence reconstruction on G is solvable iff it is solvale on the associated tree.

35 Naive guess Random sparse graphs converge locally to trees hence reconstruction on G is solvable iff it is solvale on the associated tree.

36 A counterexample Theorem Consider the ferromagnetic Ising model on a random (b + 1)-regular graph. Then reconstruction is solvable iff b(1 2ɛ) 1 > 1. [Other counterexample in Mossel, Weitz, Wormald (2006)]

37 A general sufficient condition Theorem If P{X G} is roughly spherical then Graph solvable Tree solvable. If P{X G} is not roughly spherical then Graph reconstruction is solvable

38 A general sufficient condition Theorem If P{X G} is roughly spherical then Graph solvable Tree solvable. If P{X G} is not roughly spherical then Graph reconstruction is solvable

39 A general sufficient condition Theorem If P{X G} is roughly spherical then Graph solvable Tree solvable. If P{X G} is not roughly spherical then Graph reconstruction is solvable

40 Roughly spherical??? X i {0, 1}. X (1) = {X (1) i }, X (2) = {X (2) i } independent with distribution P{ G N } X (1) X (2) P{ G N } is roughly spherical if d(x (1), X (2) ) N/2 with high probability.

41 Roughly spherical??? X i {0, 1}. X (1) = {X (1) i }, X (2) = {X (2) i } independent with distribution P{ G N } X (1) X (2) P{ G N } is roughly spherical if d(x (1), X (2) ) N/2 with high probability.

42 Can you check this condition? Theorem Let G N be a random graph with M = Nγ edges, and P{ G N } the measure over ɛ-proper colorings. If γ < (q 1) log(q 1) then Graph solvable Tree solvable. Proof: Uses second moment calculation in Achlioptas, Naor (2004). Theorem Let G N be a random graph with M = Nγ edges, and P{ G N } the Boltzmann measure for Ising spin-glasses. Then reconstruction is solvabel if and only if 2γ(1 2ɛ) 2 > 1. Proof: Uses Large deviation result in Guerra, Toninelli (2003).

43 Can you check this condition? Theorem Let G N be a random graph with M = Nγ edges, and P{ G N } the measure over ɛ-proper colorings. If γ < (q 1) log(q 1) then Graph solvable Tree solvable. Proof: Uses second moment calculation in Achlioptas, Naor (2004). Theorem Let G N be a random graph with M = Nγ edges, and P{ G N } the Boltzmann measure for Ising spin-glasses. Then reconstruction is solvabel if and only if 2γ(1 2ɛ) 2 > 1. Proof: Uses Large deviation result in Guerra, Toninelli (2003).

44 Can you check this condition? Theorem Let G N be a random graph with M = Nγ edges, and P{ G N } the measure over ɛ-proper colorings. If γ < (q 1) log(q 1) then Graph solvable Tree solvable. Proof: Uses second moment calculation in Achlioptas, Naor (2004). Theorem Let G N be a random graph with M = Nγ edges, and P{ G N } the Boltzmann measure for Ising spin-glasses. Then reconstruction is solvabel if and only if 2γ(1 2ɛ) 2 > 1. Proof: Uses Large deviation result in Guerra, Toninelli (2003).

45 One trick from the proof Consider X i {+1, 1}. Approximate sphericity X (1) X (2) 0 0 E{(X (1) X (2) ) 2 } = i,j E{X i X j } 2

46 Open problem Prove that approximate counting is easy for models on random graphs in the non-reconstructibility regime.