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Computing the Line Index of Balance Using Integer Programming Optimisation

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Optimization Problems in Graph Theory

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 139))

Abstract

An important measure of signed graphs is the line index of balance which has 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 Erdős-Rényi graphs, Barabási-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 a small line index of balance which indicates that they are close to balanced.

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Acknowledgements

The authors are grateful for the extremely valuable comments of the anonymous reviewers that have prevented incorrect attributions in the literature review section and helped improve the discussions in this chapter.

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  1. Department of Computer Science, University of Auckland, Auckland, New Zealand

    Samin Aref & Mark C. Wilson

  2. Department of Engineering Science, University of Auckland, Auckland, New Zealand

    Andrew J. Mason

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  1. Samin Aref
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Correspondence to Samin Aref .

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  1. Department of Information Systems and Decision Science, Merrick School of Business, University of Baltimore, Baltimore, MD, USA

    Boris Goldengorin

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Aref, S., Mason, A.J., Wilson, M.C. (2018). Computing the Line Index of Balance Using Integer Programming Optimisation. In: Goldengorin, B. (eds) Optimization Problems in Graph Theory. Springer Optimization and Its Applications, vol 139. Springer, Cham. https://doi.org/10.1007/978-3-319-94830-0_3

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