An exact method for computing the frustration index in signed networks using binary programming

Computing the frustration index of a signed graph is a key to solving problems in different fields of research including social networks, physics, material science, and biology. In social networks the frustration index determines network distance from a state of structural balance. Although the definition of frustration index goes back to 1960, an exact algorithmic computation method has not yet been proposed. The main reason seems to be the complexity of computing the frustration index which is closely related to well-known NP-hard problems such as MAXCUT. New quadratic and linear binary programming models are developed to compute the frustration index exactly. We introduce several speed-up techniques involving prioritised branching, local search heuristics, and valid inequalities inferred from graph structural properties. The computational improvements achieved by implementing the speed-up techniques allow us to calculate the exact values of the frustration index by running the optimisation models in Gurobi solver. The speed-up techniques make our models capable of processing graphs with thousands of nodes and edges in seconds on inexpensive hardware. The solve time and solution quality comparison against the literature shows the superiority of our models in both random and real signed networks.