The condensation phase transition in random graph coloring

Transcription

1 The condensation phase transition in random graph coloring Victor Bapst Goethe University, Frankfurt Joint work with Amin Coja-Oghlan, Samuel Hetterich, Felicia Rassmann and Dan Vilenchik arxiv: Tuesday, May 6, 2014

2 Outline Overview The model Clustering and condensation Rigorous results Outline of the proof Using the planted model Identifying the frozen vertices The remaining: a problem over finite trees Conclusions 2 / 31

3 Overview Outline Overview The model Clustering and condensation Rigorous results Outline of the proof Using the planted model Identifying the frozen vertices The remaining: a problem over finite trees Conclusions 3 / 31

4 Overview The model Outline Overview The model Clustering and condensation Rigorous results Outline of the proof Using the planted model Identifying the frozen vertices The remaining: a problem over finite trees Conclusions 4 / 31

5 Overview The model Random graph coloring Draw a random graph on N vertices by connecting any two vertices with probability d/n at random. Is this graph k-colorable? How many k-colorings can we find? For a given graph: Z(G). In general: either zero or exponentially many. Taking the average over the choice of the graph and N : Average number of colorings: [EZ(G)] h 1/N i k(1 1/k) d/2. Typical number of colorings: E Z(G) 1/N? Φ k (d) =??. 5 / 31

6 Overview The model Phase transitions Phase transition (informel): discontinuity in some macroscopic quantity describing a problem. For instance: the size of the largest connected component for Erdős-Rényi random graphs, upon increasing the average degree, the density when freezing water, the derivative of the magnetization when heating a magnet. Phase transition (here): non analyticity of Φ k (d). For instance it is conjectured that there exists d col (k) such that: for d < d col (k), Φ k (d) > 0, and lim dրdcol (k) Φ k (d) > 0. for d > d col (k), Φ k (d) = 0. Here we look at another phase transition that happens for d < d col (k). 6 / 31

7 Overview Clustering and condensation Outline Overview The model Clustering and condensation Rigorous results Outline of the proof Using the planted model Identifying the frozen vertices The remaining: a problem over finite trees Conclusions 7 / 31

8 Overview Clustering and condensation The physics picture A powerful tool to study random optimization problems: the cavity method. Introduced by Mézard and Parisi in General overview for random optimization problems: Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborová in PNAS Application to coloring: Krzakala, Pagnani, Weigt, Zdeborová... Upon increasing d, solutions tend to group into clusters. C(G, σ) = {colorings τ that can be reached from σ by altering at most N/(k log k) vertices at a time} (Proofs: [Achlioptas - Coja-Oghlan 2008, Molloy 2012]) 8 / 31 d col (k) d

9 Overview Clustering and condensation The physics picture [Zdeborová - Krzakala 2007] Compare the cluster size with the total number of colorings. Φ k (d) C(G, σ) 1/N d cond (k) d Number of clusters: Φ k (d) C(G, σ) 1/N. What happens for d > d cond (k)? 9 / 31

10 Overview Clustering and condensation Interlude: a broader view of condensation. A similar phenomenon appears when cooling too fast some liquids. This is the famous Kauzmann paradox:. 10 / 31

11 Overview Clustering and condensation The physics picture [Zdeborová - Krzakala 2007] Physics prediction: d cond (k) marks a phase transition: - for d < d cond (k), C(G, σ) 1/N < Φ k (d) = k(1 1/k) d/2, - for d > d cond (k), C(G, σ) 1/N = Φ k (d) < k(1 1/k) d/2, - the second derivative of Φ k (d) is discontinuous at d cond (k). k(1 1/k) d/2 Φ k (d) C(G, σ) 1/N d cond (k) d col (k) d 11 / 31

12 Overview Clustering and condensation The physics picture [Zdeborová - Krzakala 2007] Upon increasing d, the geometry of the set of solutions dramatically changes. C(G, σ) = {colorings τ that can be reached from σ by altering at most N/(k logk) vertices at a time} d cond (k) d col (k) d Condensation: when the number of clusters becomes sub-exponential. when the cluster size C(G, σ) 1/N equals Φ k (d). 12 / 31

13 Overview Rigorous results Outline Overview The model Clustering and condensation Rigorous results Outline of the proof Using the planted model Identifying the frozen vertices The remaining: a problem over finite trees Conclusions 13 / 31

14 Overview Rigorous results A first transition: the satisfiability transition The number of colorings is easily understood when d < 1. Φ k (d) k(1 1/k) d/2 0 1 d 14 / 31

15 Overview Rigorous results Upper bounds Upper bound on the typical number of colorings: first moment method. Can be improved from the naive result [Coja-Oghlan 2013]. Φ k (d) k(1 1/k) d/2 d k = (2k 1)lnk 0 1 d k 1 d 15 / 31

16 Overview Rigorous results Lower bounds Lower bound on the typical number of colorings: second moment method [Achlioptas - Naor 2005, Coja-Oghlan - Vilenchik 2010]. Φ k (d) k(1 1/k) d/2 d k = (2k 1)lnk 0 1 d k 2 ln2 d k 1 d 16 / 31

17 Overview Rigorous results The condensation transition Theorem (1/2): for k large enough there exists d cond (k) such that: there is a phase transition at d cond (k), for d < d cond (k) : Φ k (d) = k(1 1/k) d/2, for d > d cond (k) : Φ k (d) < k(1 1/k) d/2 (or does not exist). Φ k (d) k(1 1/k) d/2 d k = (2k 1)lnk d cond (k) 0 d k 2 ln2 d k 1 d 17 / 31

18 Overview Rigorous results The condensation transition Theorem (2/2): d cond (k) is given by the formula predicted by the cavity method [Zdeborová - Krzakala 2007]. That is: - Ω = {probability distributions on {1,...,k}}, - f : γ 0 Ωγ Ω, γ j=1 f (µ 1,..., µ γ )(i) = 1 µ j(i) γ h [k] j=1 1 µ j(h). - P = {probability distributions on Ω}, - F k,d : P P X γ d Z " kx # exp( d) γy F k,d (π) = 1 µ j(h) δ f [µ1,...,µ γ! Z γ] γ=0 γ(π) Ω γ h=1 j=1 where Z γ (π) = k ( h=1 1 Ω µ(h)dπ(µ)) γ γo dπ(µ j). - Σ k,d : P R ( Complexity ). Σ k,d (π) =... - d cond (k) is the unique solution of Σ k,d (π k,d ) = 0 in [d k 2, d k ], where π k,d is a particular fixed point of F k,d. 18 / 31 j=1

19 Overview Rigorous results Conjectures: the satisfiability transition Conjecture 1: Φ k (d) exists for all d. There exist a colorability threshold d col (k). There is a phase transition at d col (k). Conjecture 2: d cond (k) < d col (k). There are exactly two phase transitions. Φ k (d) k(1 1/k) d/2 d k = (2k 1)lnk d cond (k) d col (k) 0 d k 2 ln2 d k 1 d 19 / 31

20 Outline of the proof Outline Overview The model Clustering and condensation Rigorous results Outline of the proof Using the planted model Identifying the frozen vertices The remaining: a problem over finite trees Conclusions 20 / 31

21 Outline of the proof Using the planted model Outline Overview The model Clustering and condensation Rigorous results Outline of the proof Using the planted model Identifying the frozen vertices The remaining: a problem over finite trees Conclusions 21 / 31

22 Outline of the proof Using the planted model The planted model The condensation corresponds to the point where the cluster size C(G, σ) 1/N equals (w.h.p.) k(1 1/k) d/2. However it is hard to compute the cluster size: given a random graph, how do we even find a coloring? Planting: first pick a configuration σ at random. Then generate a graph G by adding edges independently and uniformly at random such that: G has as many vertices as G (in average), σ is a coloring of this graph. Generating the pair (G, σ ) is easy. The cluster size C(G, σ ) 1/N is also easier to compute. 22 / 31

23 Outline of the proof Using the planted model Condensation and clusters sizes k(1 1/k) d/2 Φ k (d) C(G, σ) 1/N C(G, σ ) 1/N d cond (k) d Physics intuition: [Krzakala - Zdeborová 2009] if d < d cond (k), C(G, σ) 1/N = C(G, σ ) 1/N < Φ k (d) = k(1 1/k) d/2, if d > d cond (k), C(G, σ) 1/N = Φ k (d) < k(1 1/k) d/2 < C(G, σ ) 1/N. 23 / 31

24 Outline of the proof Using the planted model Condensation and clusters sizes k(1 1/k) d/2 C(G, σ ) 1/N d cond (k) d We use the following result: ǫ > 0 [Coja-Oghlan - Vilenchik 2010] if C(G, σ ) 1/N < k(1 1/k) d/2 ǫ, then d < d cond (k), if C(G, σ ) 1/N > k(1 1/k) d/2 + ǫ, then d > d cond (k). Therefore it is enough to understand the cluster size in the planted model. 24 / 31

25 Outline of the proof Identifying the frozen vertices Outline Overview The model Clustering and condensation Rigorous results Outline of the proof Using the planted model Identifying the frozen vertices The remaining: a problem over finite trees Conclusions 25 / 31

26 Outline of the proof Identifying the frozen vertices Frozen vertices We need to compute C(G, σ ) 1/N. Remember that we defined C(G, σ ) = {colorings τ that can be reached from σ by altering at most N/(k log k) vertices at a time}. Close to d cond (k) most of the vertices are frozen : they take the same value for all τ C(G, σ ). Most of : all but a fraction 1/k. We need to identify the frozen vertices. 26 / 31

27 Outline of the proof Identifying the frozen vertices Frozen vertices Close to d cond (k) most of the vertices are frozen: they take the same value for all τ C(G, σ ). Intuition for that: a vertex v typically has many neighbors of each color. v 27 / 31

28 Outline of the proof Identifying the frozen vertices Frozen vertices Close to d cond (k) most of the vertices are frozen: they take the same value for all τ C(G, σ ). Intuition for that: a vertex v typically has many neighbors of each color. v If in addition to that, most of the neighbors of v are frozen, then so is v. Technically: Warning Propagation + existence a priori of a large set of frozen vertices + convergence of local neighborhoods to trees. 27 / 31

29 Outline of the proof The remaining: a problem over finite trees Outline Overview The model Clustering and condensation Rigorous results Outline of the proof Using the planted model Identifying the frozen vertices The remaining: a problem over finite trees Conclusions 28 / 31

30 Outline of the proof The remaining: a problem over finite trees Outcome of the previous analysis: coloring a graph where the color of some vertices is fixed w v 29 / 31

31 Outline of the proof The remaining: a problem over finite trees Outcome of the previous analysis: coloring a graph where each vertex v can take colors in L v [k] fixed. w v 29 / 31

32 Outline of the proof The remaining: a problem over finite trees Outcome of the previous analysis: coloring a graph where each vertex v can take colors in L v [k] fixed. w v 29 / 31

33 Outline of the proof The remaining: a problem over finite trees Outcome of the previous analysis: coloring a graph where each vertex v can take colors in L v [k] fixed. w v If L v = 1, we can remove v. 29 / 31

34 Outline of the proof The remaining: a problem over finite trees Outcome of the previous analysis: coloring a graph where each vertex v can take colors in L v [k] fixed. w v If L v = 1, we can remove v. If L v L w = { }, we can disconnect v and w problem over finite trees: easy. 29 / 31

35 Conclusions Outline Overview The model Clustering and condensation Rigorous results Outline of the proof Using the planted model Identifying the frozen vertices The remaining: a problem over finite trees Conclusions 30 / 31

36 Conclusions Conclusions Rigorous proof of the existence of the condensation transition for (Erdős-Rényi) random graphs coloring. The transition point is a number (does not depend on N). Confirms the prediction of the cavity method. Condensation for a model with fluctuating degrees. Some directions for future work: what about the colorability threshold? finite temperature? models where the non-condensed phase is non-trivial? 31 / 31

37 Conclusions Warning propagation Consider the following process: associate to each pair of vertices (v, w) connected by an edge a sequence µ v w (i [k], t 0) {0, 1} defined by: µ v w(i, t = 0) = 1 iff v has color i under σ, a b c v w µ a v µ b v µ c v µ v w 32 / 31

38 Conclusions Warning propagation Consider the following process: associate to each pair of vertices (v, w) connected by an edge a sequence µ v w (i [k], t 0) {0, 1} defined by: µ v w(i, t = 0) = 1 iff v has color i under σ, t = 0 a b c v w µ a v µ b v µ c v µ v w / 31

39 Conclusions Warning propagation Consider the following process: associate to each pair of vertices (v, w) connected by an edge a sequence µ v w (i [k], t 0) {0, 1} defined by: µ v w(i, t = 0) = 1 iff v has color i under σ, µ v w(i, t + 1) = 1 iff for all j i, there is u v \ {w} such that µ u v(j, t) = 1 ( u warns v that it cannot take color j ). t = 1 a b c v w µ a v µ b v µ c v µ v w / 31

40 Conclusions Warning propagation Consider the following process: associate to each pair of vertices (v, w) connected by an edge a sequence µ v w (i [k], t 0) {0, 1} defined by: µ v w(i, t = 0) = 1 iff v has color i under σ, µ v w(i, t + 1) = 1 iff for all j i, there is u v \ {w} such that µ u v(j, t) = 1 ( u warns v that it cannot take color j ). t = 2 a b c v w µ a v µ b v µ c v µ v w / 31

41 Conclusions Warning propagation The process is decreasing and converges. Define L(v) = {i [k], u v, µ u v (i, t = ) = 0} ( colors that v is allowed to take ). Let Z(G, σ ) be the number of colorings of G such that σ(v) L(v). Then C(G, σ ) 1/N = Z(G, σ ) 1/N w.h.p. 36 / 31

42 Conclusions Warning propagation (2d version) Consider the following process: associate to each pair of vertices (v, w) connected by an edge a sequence µ v w (i [k], t 0) {0, 1} defined by: µ v w(i, t = 0) = 1 iff v has color i under σ and v is in the core, µ v w(i, t + 1) = 1 iff for all j i, there is u v \ {w} such that µ u v(j, t) = 1 ( u warns v that it cannot take color j ). The process is increasing and converges. Define L 2 (v) = {i [k], u v, µ u v (i, t = ) = 0} ( colors that v is allowed to take ). Let Z 2 (G, σ ) be the number of colorings of G such that σ(v) L 2 (v). Then Z 2 (G, σ ) is an upper bound on the cluster size (w.h.p). W.h.p. lnz 1 (G, σ ) = ln Z 2 (G, σ ) + o(n). 37 / 31