Optimally partitioning signed networks based on generalized balance.

Signed networks, which contain both positive and negative edges, are now studied in many disciplines. A fundamental challenge in their analysis is partitioning the nodes into internally cohesive and mutually divisive clusters based on generalized balance theory. However, identifying optimal partitions is NP-hard. We introduce a binary linear programming model, and reformulate an existing one, to solve this challenge by minimizing the number of "frustrated" (intra-cluster negative or inter-cluster positive) edges. These models make it possible to partition a signed network into exactly $k$ clusters that minimize the number of frustrated edges, and to identify the smallest number of clusters that minimizes the number of frustrated edges across all possible partitions. They guarantee an optimal solution, and can be practically applied to signed networks containing up to 30,000 edges, and thus offer a robust method for uncovering the structure of signed networks. We demonstrate the practicality and utility of these models by using them to identify optimal partitions of signed networks representing the collaborations and oppositions among legislators in the US House of Representatives between 1981 and 2018. We show that an optimal partition into three coalitions better describes these networks than a partition based on political party. The optimal 3-partition reveals a large liberal coalition, a large conservative coalition, and a previously obscured third coalition. This hidden third coalition is noteworthy because its median ideology changes over time, but its members are consistently more effective at passing legislation than their colleagues in either of the dominant coalitions.