The network source location problem: ground state energy, entropy and effects of freezing

TL;DR

This paper evaluates the ground state entropy of the network source location problem using advanced cavity methods, compares algorithms for optimal source placement, and analyzes the effects of entropy and freezing in complex networks.

Contribution

It introduces entropic cavity methods at different replica symmetry levels and develops algorithms inspired by entropic message passing for source placement in networks.

Findings

Entropic message passing yields lower ground state energy in the glassy phase.

The study compares belief propagation and entropic methods, showing the latter's advantages.

Extremal optimization provides detailed statistics on entropy and frozen hubs.

Abstract

Ground state entropy of the network source location problem is evaluated at both the replica symmetric level and one-step replica symmetry breaking level using the entropic cavity method. The regime that is a focus of this study, is closely related to the vertex cover problem with randomly quenched covered nodes. The resulting entropic message passing inspired decimation and reinforcement algorithms are used to identify the optimal location of sources in single instances of transportation networks. The conventional belief propagation without taking the entropic effect into account is also compared. We find that in the glassy phase the entropic message passing inspired decimation yields a lower ground state energy compared to the belief propagation without taking the entropic effect. Using the extremal optimization algorithm, we study the ground state energy and the fraction of frozen hubs, and extend the algorithm to collect statistics of the entropy. The theoretical results are compared with the extremal optimization results.