Computing the Line Index of Balance Using Integer Programming Optimisation

TL;DR

This paper introduces new integer programming models to efficiently compute the line index of balance in large signed graphs, enabling exact analysis of complex real-world social and biological networks.

Contribution

The authors develop quadratic and linear programming models that allow exact computation of the line index of balance, overcoming previous computational challenges.

Findings

Exact line index of balance can be computed efficiently for large graphs.

Most real-world social and biological networks are close to balanced.

The models work on diverse datasets including social, biological, and synthetic graphs.

Abstract

An important measure of signed graphs is the line index of balance which has several applications in many fields. However, this graph-theoretic measure was underused for decades because of the inherent complexity in its computation which is closely related to solving NP-hard graph optimisation problems like MAXCUT. We develop new quadratic and linear programming models to compute the line index of balance exactly. Using the Gurobi integer programming optimisation solver, we evaluate the line index of balance on real-world and synthetic datasets. The synthetic data involves Erd\H{o}s-R\'{e}nyi graphs, Barab\'{a}si-Albert graphs, and specially structured random graphs. We also use well known datasets from the sociology literature, such as signed graphs inferred from students' choice and rejection as well as datasets from the biology literature including gene regulatory networks. The results show that exact values of the line index of balance in relatively large signed graphs can be efficiently computed using our suggested optimisation models. We find that most real-world social networks and some biological networks have small line index of balance which indicates that they are close to balanced.