The Condensation Phase Transition in Random Graph Coloring Felicia Rassmann Goethe University Frankfurt Joint work with Victor Bapst, Amin Coja-Oghlan, Samuel Hetterich and Dan Vilenchik arXiv:1404.5513 03. September 2014 1 / 19
What we want to do Cavity method : predictions on phase transitions in discrete structures 2 / 19
What we want to do Cavity method : predictions on phase transitions in discrete structures Phase transition called condensation shortly before the threshold for the existence of solutions [Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborov´ a, PNAS 2007] 2 / 19
What we want to do Cavity method : predictions on phase transitions in discrete structures Phase transition called condensation shortly before the threshold for the existence of solutions [Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborov´ a, PNAS 2007] Related to the difficulty of proving precise results 2 / 19
What we want to do Cavity method : predictions on phase transitions in discrete structures Phase transition called condensation shortly before the threshold for the existence of solutions [Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborov´ a, PNAS 2007] Related to the difficulty of proving precise results Random graph k -coloring: precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point 2 / 19
What we want to do Cavity method : predictions on phase transitions in discrete structures Phase transition called condensation shortly before the threshold for the existence of solutions [Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborov´ a, PNAS 2007] Related to the difficulty of proving precise results Random graph k -coloring: precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point Conjecture now proven for k ≥ k 0 2 / 19
Overview Model 1 Results 2 Outline of the proof 3 Conclusion 4 3 / 19
Model Model 1 Random graph coloring Phase transitions The physics picture The planted model Results 2 Outline of the proof 3 Conclusion 4 4 / 19
Model Random graph coloring Model of interest: Erd˝ os-R´ enyi random graph G ( n, p ) with vertex set V = { 1 , . . . , n } and p = d/n where d > 0 remains fixed as n → ∞ Z k ( G ( n, p )) =number of k -colorings of G ( n, p ) 5 / 19
Model Random graph coloring Model of interest: Erd˝ os-R´ enyi random graph G ( n, p ) with vertex set V = { 1 , . . . , n } and p = d/n where d > 0 remains fixed as n → ∞ Z k ( G ( n, p )) =number of k -colorings of G ( n, p ) Questions: With which probability is such a graph k -colorable? How many k -colorings are there typically? What does the structure of the solution space look like for different values of d ? 5 / 19
Model Random graph coloring Model of interest: Erd˝ os-R´ enyi random graph G ( n, p ) with vertex set V = { 1 , . . . , n } and p = d/n where d > 0 remains fixed as n → ∞ Z k ( G ( n, p )) =number of k -colorings of G ( n, p ) Questions: With which probability is such a graph k -colorable? How many k -colorings are there typically? What does the structure of the solution space look like for different values of d ? Quantity of interest: “typical value“ of Z k ( G ( n, d/n )) as n → ∞ : n →∞ E [ Z k ( G ( n, d/n )) 1 /n ] Φ k ( d ) = lim 5 / 19
Model Phase transitions Φ k ( d ) is not currently known to exist for all d, k . 6 / 19
Model Phase transitions Φ k ( d ) is not currently known to exist for all d, k . For a given k ≥ 3 we call d 0 ∈ (0 , ∞ ) smooth if there exists ε > 0 such that for any d ∈ ( d 0 − ε, d 0 + ε ) the limit Φ k ( d ) exists, and the map d ∈ ( d 0 − ε, d 0 + ε ) �→ Φ k ( d ) has an expansion as an absolutely convergent power series around d 0 . If d 0 fails to be smooth, we say that a phase transition occurs at d 0 . 6 / 19
Model Phase transitions Φ k ( d ) is not currently known to exist for all d, k . For a given k ≥ 3 we call d 0 ∈ (0 , ∞ ) smooth if there exists ε > 0 such that for any d ∈ ( d 0 − ε, d 0 + ε ) the limit Φ k ( d ) exists, and the map d ∈ ( d 0 − ε, d 0 + ε ) �→ Φ k ( d ) has an expansion as an absolutely convergent power series around d 0 . If d 0 fails to be smooth, we say that a phase transition occurs at d 0 . Example: If the conjectured sharp satisfiability threshold d k, col exists, it is a phase transition in the above sense. Here: Interested in the ”condensation phase transition“ d k, cond . 6 / 19
Model The physics picture Random graph coloring is an example of a “diluted mean-field model of a disordered system”. 7 / 19
Model The physics picture Random graph coloring is an example of a “diluted mean-field model of a disordered system”. Upon increasing d , the geometry of the set of solutions dramatically changes. The solution space begins to cluster. r sat r uni r re r fr r con Figure: ”Gibbs uniqueness”, ”reconstruction”, ”freezing”, ”condensation”, ”satisfiability” 7 / 19
Model The physics picture Random graph coloring is an example of a “diluted mean-field model of a disordered system”. Upon increasing d , the geometry of the set of solutions dramatically changes. The solution space begins to cluster. r sat r uni r re r fr r con Figure: ”Gibbs uniqueness”, ”reconstruction”, ”freezing”, ”condensation”, ”satisfiability” Conjecture: d k, col = (2 k − 1) ln k − 1 + η k , where η k → 0 as k → ∞ Prediction: d k, cond = (2 k − 1) ln k − 2 ln 2 + ε k , where ε k → 0 as k → ∞ 7 / 19
Model The physics picture Random graph coloring is an example of a “diluted mean-field model of a disordered system”. Upon increasing d , the geometry of the set of solutions dramatically changes. The solution space begins to cluster. r sat r uni r re r fr r con Figure: ”Gibbs uniqueness”, ”reconstruction”, ”freezing”, ”condensation”, ”satisfiability” Conjecture: d k, col = (2 k − 1) ln k − 1 + η k , where η k → 0 as k → ∞ Prediction: d k, cond = (2 k − 1) ln k − 2 ln 2 + ε k , where ε k → 0 as k → ∞ As the density tends to d k, cond , the typical cluster size approaches the total expected number of k -colorings. 7 / 19
Model The planted model Cluster of σ in G : C ( G, σ ) = { τ : τ is a k -coloring of G and ρ ii ( σ, τ ) ≥ 0 . 51 /k for all i ∈ [ k ] } For our purposes equivalent to “colorings that can be reached from σ by iteratively altering the colors of o ( n ) vertices at a time”. 8 / 19
Model The planted model Cluster of σ in G : C ( G, σ ) = { τ : τ is a k -coloring of G and ρ ii ( σ, τ ) ≥ 0 . 51 /k for all i ∈ [ k ] } For our purposes equivalent to “colorings that can be reached from σ by iteratively altering the colors of o ( n ) vertices at a time”. How can we sample a random k -coloring σ of a random graph G ( n, p ) ? 8 / 19
Model The planted model Cluster of σ in G : C ( G, σ ) = { τ : τ is a k -coloring of G and ρ ii ( σ, τ ) ≥ 0 . 51 /k for all i ∈ [ k ] } For our purposes equivalent to “colorings that can be reached from σ by iteratively altering the colors of o ( n ) vertices at a time”. How can we sample a random k -coloring σ of a random graph G ( n, p ) ? The planting trick: Choose a map σ : [ n ] → [ k ] uniformly at random. 8 / 19
Model The planted model Cluster of σ in G : C ( G, σ ) = { τ : τ is a k -coloring of G and ρ ii ( σ, τ ) ≥ 0 . 51 /k for all i ∈ [ k ] } For our purposes equivalent to “colorings that can be reached from σ by iteratively altering the colors of o ( n ) vertices at a time”. How can we sample a random k -coloring σ of a random graph G ( n, p ) ? The planting trick: Choose a map σ : [ n ] → [ k ] uniformly at random. Generate a graph G = G ( n, p ′ , σ ) by connecting v, w ∈ [ n ] such that σ ( v ) � = σ ( w ) with probability p ′ independently. 8 / 19
Model The planted model Cluster of σ in G : C ( G, σ ) = { τ : τ is a k -coloring of G and ρ ii ( σ, τ ) ≥ 0 . 51 /k for all i ∈ [ k ] } For our purposes equivalent to “colorings that can be reached from σ by iteratively altering the colors of o ( n ) vertices at a time”. How can we sample a random k -coloring σ of a random graph G ( n, p ) ? The planting trick: Choose a map σ : [ n ] → [ k ] uniformly at random. Generate a graph G = G ( n, p ′ , σ ) by connecting v, w ∈ [ n ] such that σ ( v ) � = σ ( w ) with probability p ′ independently. Good approximation if p ′ = dk/n ( k − 1) and Φ k ( d ) = k (1 − 1 /k ) d/ 2 . 8 / 19
Results Model 1 Results 2 Ingredients Theorem Outline of the proof 3 Conclusion 4 9 / 19
Results Ingredients The set Ω of probability measures µ : [ k ] → [0 , 1] 10 / 19
Results Ingredients The set Ω of probability measures µ : [ k ] → [0 , 1] The Belief Propagation operator: � γ j =1 1 − µ j ( i ) B [ µ 1 , . . . , µ γ ]( i ) = � � γ j =1 1 − µ j ( h ) h ∈ [ k ] for any i ∈ [ k ] 10 / 19
Results Ingredients The set Ω of probability measures µ : [ k ] → [0 , 1] The Belief Propagation operator: � γ j =1 1 − µ j ( i ) B [ µ 1 , . . . , µ γ ]( i ) = � � γ j =1 1 − µ j ( h ) h ∈ [ k ] for any i ∈ [ k ] The set P of all probability measures on Ω . π ∈ P frozen if π ( { δ 1 , ..., δ k } ) ≥ 2 / 3 . 10 / 19
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