Two Algorithms for Relaxed Structural Balance Partitioning: Linking Theory, Models, and Data to Understand Social Network Phenomena

Understanding social phenomena with the help of mathematical models requires a coherent combination of theory, models, and data together with using valid data analytic methods. The study of social networks through the use of mathematical models is no exception. The intuitions of structural balance were formalized and led to a pair of remarkable theorems giving the nature of partition structures for balanced signed networks. Algorithms for partitioning signed networks, informed by these formal results, were developed and applied empirically. More recently, "structural balance'' was generalized to "relaxed structural balance,'' and a modified partitioning algorithm was proposed. Given the critical interplay of theory, models, and data, it is important that methods for the partitioning of signed networks in terms of relaxed structural balance model are appropriate. The authors consider two algorithms for establishing partitions of signed networks in terms of relaxed structural balance. One is an older heuristic relocation algorithm, and the other is a new exact solution procedure. The former can be used both inductively and deductively. When used deductively, this requires some prespecification incorporating substantive insights. The new branch-and-bound algorithm is used inductively and requires no prespecification of an image matrix in terms of ideal blocks. Both procedures are demonstrated using several examples from the literature, and their contributions are discussed. Together, the two algorithms provide a sound foundation for partitioning signed networks and yield optimal partitions. Issues of network size and density are considered in terms of their consequences for algorithm performance.