Warning Propagation: stability and subcriticality

TL;DR

This paper provides a comprehensive analysis of Warning Propagation, a message passing algorithm, demonstrating its rapid convergence and stability on multi-type random graphs, and generalizing previous heuristic results for k-core analysis.

Contribution

It offers a general theoretical framework for analyzing Warning Propagation's convergence and stability on complex random graph models, extending prior heuristic insights.

Findings

Warning Propagation converges rapidly under mild conditions.

Analysis reduces to studying the process on a multi-type Galton-Watson tree.

Generalizes heuristic results for k-core and related algorithms.

Abstract

Warning Propagation is a combinatorial message passing algorithm that unifies and generalises a wide variety of recursive combinatorial procedures. Special cases include the Unit Clause Propagation and Pure Literal algorithms for satisfiability as well as the peeling process for identifying the $k$-core of a random graph. Here we analyse Warning Propagation in full generality on a very general class of multi-type random graphs. We prove that under mild assumptions on the random graph model and the stability of the the message limit, Warning Propagation converges rapidly. In effect, the analysis of the fixed point of the message passing process on a random graph reduces to analysing the process on a multi-type Galton-Watson tree. This result corroborates and generalises a heuristic first put forward by Pittel, Spencer and Wormald in their seminal $k$-core paper (JCTB 1996).