Computing the Line Index of Balance Using Integer Programming Optimisation
Abstract
An important measure of signed graphs is the line index of balance which has several applications in many fields. However, this graphtheoretic measure was underused for decades because of the inherent complexity in its computation which is closely related to solving NPhard 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 realworld and synthetic datasets. The synthetic data involves ErdősRényi graphs, BarabásiAlbert 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 realworld social networks and some biological networks have small line index of balance which indicates that they are close to balanced.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 arXiv:
 arXiv:1710.09876
 Bibcode:
 2017arXiv171009876A
 Keywords:

 Computer Science  Social and Information Networks;
 Mathematics  Optimization and Control;
 05C22;
 15C22;
 90C09;
 90C11;
 90C90;
 90C35
 EPrint:
 Accepted author copy, 20 pages, 4 tables and 3 figures. This work is followed up in another study with more focus on Operations Research aspects of the topic that can be found in arXiv:1611.09030