Conditional Random Fields, Planted Constraint Satisfaction, and Entropy Concentration

TL;DR

This paper analyzes probabilistic graph models including planted CSPs and community clustering, showing entropy concentration and thresholds that connect coding, clustering, and satisfiability problems.

Contribution

It establishes entropy concentration results and thresholds for a broad class of models, linking coding, clustering, and satisfiability theories.

Findings

Entropy of node variables concentrates around a deterministic threshold.

Number of solutions in planted CSPs concentrates.

Threshold function exists for disassortative stochastic block model.

Abstract

This paper studies a class of probabilistic models on graphs, where edge variables depend on incident node variables through a fixed probability kernel. The class includes planted con- straint satisfaction problems (CSPs), as well as more general structures motivated by coding and community clustering problems. It is shown that under mild assumptions on the kernel and for sparse random graphs, the conditional entropy of the node variables given the edge variables concentrates around a deterministic threshold. This implies in particular the concentration of the number of solutions in a broad class of planted CSPs, the existence of a threshold function for the disassortative stochastic block model, and the proof of a conjecture on parity check codes. It also establishes new connections among coding, clustering and satisfiability.