Inference and Optimization of Real Edges on Sparse Graphs - A Statistical Physics Perspective

TL;DR

This paper applies statistical physics methods to infer and optimize real-valued edge variables in sparse graphs, developing algorithms and analyzing their scaling, with applications to network resource allocation.

Contribution

It introduces a novel approach combining Bethe approximation and replica method for edge inference, along with efficient distributed algorithms for resource allocation.

Findings

Theoretical equilibrium states derived for various energy functions.

Distributed algorithms perform well in simulations, matching theoretical predictions.

Scaling laws relate network connectivity and resource levels to inference accuracy.

Abstract

Inference and optimization of real-value edge variables in sparse graphs are studied using the Bethe approximation and replica method of statistical physics. Equilibrium states of general energy functions involving a large set of real edge-variables that interact at the network nodes are obtained in various cases. When applied to the representative problem of network resource allocation, efficient distributed algorithms are also devised. Scaling properties with respect to the network connectivity and the resource availability are found, and links to probabilistic Bayesian approximation methods are established. Different cost measures are considered and algorithmic solutions in the various cases are devised and examined numerically. Simulation results are in full agreement with the theory.