Balance and frustration in signed networks

The frustration index is a key measure for analysing signed networks, which has been underused due to its computational complexity. We use an exact optimisation-based method to analyse frustration as a global structural property of signed networks coming from diverse application areas. In the classic friend-enemy interpretation of balance theory, a by-product of computing the frustration index is the partitioning of nodes into two internally solidary but mutually hostile groups. The main purpose of this paper is to present general methodology for answering questions related to partial balance in signed networks, and apply it to a range of representative examples that are now analysable because of advances in computational methods. We provide exact numerical results on social and biological signed networks, networks of formal alliances and antagonisms between countries, and financial portfolio networks. Molecular graphs of carbon and Ising models are also considered. The purpose served by exploring several problems in this paper is to propose a single general methodology for studying signed networks and to demonstrate its relevance to applications. We point out several mistakes in the signed networks literature caused by inaccurate computation, implementation errors or inappropriate measures.