Algebraic characterization of binary graphs

TL;DR

This paper provides a logic-algebraic characterization of the grandcanonical ensemble of binary graphs, offering a new analytical perspective on a fundamental concept in network theory and statistical mechanics.

Contribution

It introduces a novel logic-algebraic framework for understanding the structure of the grandcanonical ensemble of binary graphs, which has been primarily used for analytical calculations.

Findings

Provides a formal algebraic description of the grandcanonical ensemble

Reveals the rich structure of the ensemble beyond average calculations

Lays groundwork for further algebraic analysis of network ensembles

Abstract

One of the fundamental concepts in the statistical mechanics field is that of ensemble. Ensembles of graphs are collections of graphs, defined according to certain rules. The two most used ensembles in network theory are the microcanonical and the grandcanonical (whose definitions mimick the classical ones, originally proposed by Boltzmann and Gibbs), even if the latter is far more used than the former to carry on the analytical calculations. For binary (undirected or directed) networks, the grandcanonical ensemble is defined by considering all the graphs with the same number of vertices and a variable number of links, ranging from 0 to the maximum: N(N-1)/2 for binary, undirected graphs and N(N-1) for binary, directed graphs. Even if it is commonly used almost exclusively as a tool to calculate the average of some topological quantity of interest, its structure is so rich to deserve an analysis on its own. In this paper a logic-algebraic characterization of the grandcanonical ensemble of binary graphs is provided.