A modelling and computational study of the frustration index in signed networks (1611.09030v4)

Abstract: Computing the frustration index of a signed graph is a key step toward solving problems in many fields including social networks, political science, physics, chemistry, and biology. The frustration index determines the distance of a network from a state of total structural balance. Although the definition of the frustration index goes back to the 1950's, its exact algorithmic computation, which is closely related to classic NP-hard graph problems, has only become a focus in recent years. We develop three new binary linear programming models to compute the frustration index exactly and efficiently as the solution to a global optimisation problem. Solving the models with prioritised branching and valid inequalities in Gurobi, we can compute the frustration index of real signed networks with over 15000 edges in less than a minute on inexpensive hardware. We provide extensive performance analysis for both random and real signed networks and show that our models outperform all existing approaches by large factors. Based on solve time, algorithm output, and effective branching factor we highlight the superiority of our models to both exact and heuristic methods in the literature.

• The paper introduces three binary linear programming models (AND, XOR, ABS) designed to efficiently compute the frustration index in signed networks.
• The proposed methods achieve 2 to 9 times faster solve times on large datasets, significantly outperforming previous exact and heuristic approaches.
• The study’s findings have practical implications for fields like biology, finance, and political science, while also paving the way for extensions to more complex network problems.

Analysis of Computational Techniques for the Frustration Index in Signed Networks

This paper presents the development and analysis of efficient computational methods for the exact calculation of the frustration index within signed networks. The frustration index serves as an important measure to determine how far a network is from a state of complete structural balance. Despite its origins in the 1950s, its computational treatment has only recently garnered serious attention due to its NP-hard nature, which aligns it with other classic graph problems.

The research introduces three distinctive binary linear programming models aimed at computing the frustration index through global optimization efficiently. These models differ in their construction: the AND, XOR, and ABS models each employ variations in representing node colors and the frustration state of edges to minimize the frustration index effectively. The paper asserts a significant improvement in solve times compared to existing exact and heuristic methods, capable of processing graphs with more than 15,000 edges in under a minute on standard computing resources.

Numerical Results and Performance Analysis

The authors subjected their models to both random and real network datasets, including biological networks and international relations datasets, demonstrating their superior performance over prior methods. Extensive numerical experimentation indicates that their models are not only faster but also achieve better accuracy in solving instances that were previously infeasible under older models. Specifically, the solve times for their worst-performing model were found to be 2 to 9 times faster than the best-performing previously available approaches.

Theoretical and Practical Implications

Practically, these findings have direct implications in fields such as biology, finance, political science, and international relations, wherever the frustration index can be applied to convert signed network problems into solvable models. The paper not only provides a detailed computational paper but also extends the analysis to include speed-up techniques to further enhance the branch and bound algorithm's performance.

Theoretically, the introduction of binary optimization models paves the way for leveraging powerful solvers like Gurobi to dissect NP-hard problems efficiently. Furthermore, the paper also reformulates the problem as a constraint satisfaction problem, setting a stage for potential advancements using techniques from the CSP domain.

Prospect for Future Research

Looking forward, the paper acknowledges potential exploration into further model extensions, including the weighted and multi-color optimization problems. These extensions would tackle more complex problems by considering edge weights and partitioning with additional colors, broadening the applicability of the frustration index to more nuanced network structures.

In conclusion, the research significantly aids in the advancement of graph optimization techniques, offering valuable insights and tools for accurately computing the frustration index in large, complex networks. While providing robust experimental support, the paper also charts out future directions that could refine and extend the capability of existing computational strategies in graph theory and network science.