The Bethe lattice spin glass revisited

Mézard, Marc; Parisi, Giorgio

doi:10.1007/pl00011099

Cited by 781 publications

(1,736 citation statements)

References 34 publications

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“…We use a population dynamics method, as introduced in [26,27], and model the distribution P i→j (ψ i→j ) by a population of N vectors ψ i→j . To compute P i→j (ψ i→j ) knowing the P k→i (ψ k→i ) for all incoming k we perform the 1RSB recursion in eq.…”

Section: Appendix D: Numerical Methodsmentioning

confidence: 99%

“…We start by reviewing the replica symmetric (RS) version of the cavity method [26,27] and its implications for the coloring problem. In the last part of the section we show when, and why, the RS approach fails.…”

Section: The Cavity Formalism At the Replica Symmetric Levelmentioning

confidence: 99%

“…Solving this equation for a general ensemble of random graphs and a general parameter m is a numerically quite tedious problem. In the population dynamics algorithm [26,27] we need to deal with a population of populations of q-components fields. It is much more convenient to look at the ensemble of random regular graphs where a factorized solution P[P (ψ)] = δ(P (ψ) − P 0 (ψ)) must exists.…”

Section: A Analyzing the 1rsb Equationsmentioning

confidence: 99%

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Phase transitions in the coloring of random graphs

2007

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We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study of the space of proper colorings (solutions).We show that for a fixed number of colors and as the average vertex degree (number of constraints) increases, the set of solutions undergoes several phase transitions similar to those observed in the mean field theory of glasses. First, at the clustering transition, the entropically dominant part of the phase space decomposes into an exponential number of pure states so that beyond this transition a uniform sampling of solutions becomes hard. Afterward, the space of solutions condenses over a finite number of the largest states and consequently the total entropy of solutions becomes smaller than the annealed one. Another transition takes place when in all the entropically dominant states a finite fraction of nodes freezes so that each of these nodes is allowed a single color in all the solutions inside the state. Eventually, above the coloring threshold, no more solutions are available. We compute all the critical connectivities for Erdős-Rényi and regular random graphs and determine their asymptotic values for large number of colors.Finally, we discuss the algorithmic consequences of our findings. We argue that the onset of computational hardness is not associated with the clustering transition and we suggest instead that the freezing transition might be the relevant phenomenon. We also discuss the performance of a simple local Walk-COL algorithm and of the belief propagation algorithm in the light of our results.

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Section: Appendix D: Numerical Methodsmentioning

confidence: 99%

Section: The Cavity Formalism At the Replica Symmetric Levelmentioning

confidence: 99%

Section: A Analyzing the 1rsb Equationsmentioning

confidence: 99%

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Phase transitions in the coloring of random graphs

2007

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“…while the sequence (u γ 1 ,...,γr ) γr≥1 has distribution Ξ mr . Therefore, writing for simplicity 6) where ∼ means equality in distribution and where we introduced the notation…”

Section: Theorem 3 Suppose That (12) (13) and (14) Hold And H ′ Nmentioning

confidence: 99%

“…To write ϕ(0) in terms of these functions B and u, we observe that 6) using that Ek = αp in the last line. In the case when H ′ N (σ) = 0, it was proved in [9] (see Chapter 7, [10]) that for α small enough, lim N →∞ F N = Φ(ζ α ) = inf ζ Φ(ζ) where ζ α is the unique solution of the equation…”

Section: Introductionmentioning

confidence: 99%

Bounds for diluted mean-fields spin glass models

Panchenko

Talagrand

2004

Probab. Theory Relat. Fields

124

169

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Functional inference from probabilistic protein interaction networks

Bader

2005

Encyclopedia of Genetics, Genomics, Proteomics and Bioinformatics

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The ability to obtain complete genome sequence for human and model organisms has shifted the grand challenge of genomics to describing the functional units of a genome and their interactions at the systems level. Protein‐coding genes are the dominant class of functional units, and protein–protein interactions provide the framework for understanding manifold biological processes. Recent years have seen exponential growth in the quantity of protein–protein interaction data available for inferring protein networks. Inference remains challenging, however, because of high false‐positive and false‐negative rates in the high‐throughput studies generating the majority of protein interaction data. Experimental studies now report confidence metrics together with interaction data, and similar methods have been applied retrospectively to analyze legacy data sets. Thus, a convenient level of abstraction for protein interaction data is a weighted graph in which vertices represent proteins, perhaps labeled by biological function, and edge weights represent confidence scores or likelihoods for pairwise interactions. Graph‐based models permit intuitive inference of protein function based on the function of neighboring proteins. This review provides a critical view of recent advances in algorithms for protein functional inference from probabilistic protein interaction networks. The types of algorithms considered, in order of increasing statistical rigor and computational cost, include nearest‐neighbor approaches, adaptive neighborhoods and tree‐building methods, energy minimization by simulated annealing and free energy minimization by message passing methods, and stochastic methods based on Gibbs sampling. The review explores the trade‐off between computational cost, predictive accuracy, and generalization error.

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The Bethe lattice spin glass revisited

Cited by 781 publications

References 34 publications

Phase transitions in the coloring of random graphs

Phase transitions in the coloring of random graphs

Bounds for diluted mean-fields spin glass models

Functional inference from probabilistic protein interaction networks

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