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Charting the Replica Symmetric Phase

Authors Amin Coja-Oghlan, Charilaos Efthymiou, Nor Jaafari, Mihyun Kang, Tobias Kapetanopoulos

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Amin Coja-Oghlan
Charilaos Efthymiou
Nor Jaafari
Mihyun Kang
Tobias Kapetanopoulos

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Amin Coja-Oghlan, Charilaos Efthymiou, Nor Jaafari, Mihyun Kang, and Tobias Kapetanopoulos. Charting the Replica Symmetric Phase. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 40:1-40:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2017.40

Abstract

Random graph models and associated inference problems such as the stochastic block model play an eminent role  in computer science, discrete mathematics and statistics. Based on non-rigorous arguments physicists predicted the existence of a generic phase transition  that separates a "replica symmetric phase" where statistical inference is impossible  from a phase where the detection of the "ground truth" is information-theoretically possible.

In this paper we  prove a contiguity result that shows that detectability is indeed impossible within the replica-symmetric phase for a broad class of models. In particular, this implies the detectability conjecture for the disassortative stochastic block model from [Decelle et al.: Phys Rev E 2011]. Additionally, we investigate key features of the replica symmetric phase such as the nature of point-to-set correlations (`reconstruction').

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Keywords
  • Random factor graph
  • bounds for condensation phase transition
  • Potts antiferromagnet
  • diluted k-spin model
  • stochastic block model

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