Recovery thresholds in the sparse planted matching problem

TL;DR

This paper investigates the conditions under which it is possible to recover a hidden perfect matching in a weighted random graph, providing a generalized criterion for phase transitions in recovery success.

Contribution

It extends previous results by establishing a general criterion for phase transitions in matching recovery for various weight distributions and graph sparsity levels.

Findings

Derived a criterion for the phase transition location in generic weight distributions.

Extended analysis to possibly sparse graphs.

Provided a detailed description of the critical regime around the phase transition.

Abstract

We consider the statistical inference problem of recovering an unknown perfect matching, hidden in a weighted random graph, by exploiting the information arising from the use of two different distributions for the weights on the edges inside and outside the planted matching. A recent work has demonstrated the existence of a phase transition, in the large size limit, between a full and a partial recovery phase for a specific form of the weights distribution on fully connected graphs. We generalize and extend this result in two directions: we obtain a criterion for the location of the phase transition for generic weights distributions and possibly sparse graphs, exploiting a technical connection with branching random walk processes, as well as a quantitatively more precise description of the critical regime around the phase transition.