Statistical Mechanics of maximal independent sets

TL;DR

This paper applies statistical physics methods to analyze the structure and complexity of maximal independent sets in random graphs, revealing insights into their solution landscape and implications for strategic interactions in networks.

Contribution

It introduces a novel application of cavity method and Monte Carlo simulations to study the entropy and structure of maximal independent sets, providing new theoretical insights.

Findings

Calculated entropy of maximal independent sets using replica symmetry frameworks

Revealed the solution landscape structure and potential algorithms

Linked maximal independent sets to Nash equilibria in network games

Abstract

The graph theoretic concept of maximal independent set arises in several practical problems in computer science as well as in game theory. A maximal independent set is defined by the set of occupied nodes that satisfy some packing and covering constraints. It is known that finding minimum and maximum-density maximal independent sets are hard optimization problems. In this paper, we use cavity method of statistical physics and Monte Carlo simulations to study the corresponding constraint satisfaction problem on random graphs. We obtain the entropy of maximal independent sets within the replica symmetric and one-step replica symmetry breaking frameworks, shedding light on the metric structure of the landscape of solutions and suggesting a class of possible algorithms. This is of particular relevance for the application to the study of strategic interactions in social and economic networks, where maximal independent sets correspond to pure Nash equilibria of a graphical game of public goods allocation.