## Balance and clusterability in signed networks of political collaboration in US Congress

*Samin Aref, Zachary Neal*

A signed network is balanced if its set of vertices can be partitioned into two subsets (allowing empty set) such that each negative edge joins vertices belonging to different subsets and each positive edge joins vertices belonging to the same subset. If a signed network satisfies the same condition when partitioned into k subsets, it is called clusterable (k-balanced). These two definitions can be also expressed in terms of the classic definition of balance theory (attributed to the works of Heider in 40’s and that of Cartwright and Harary in 50’s) and the generalized definition of weak balance (by Davis in 60’s); which consider different types of triangles to be balanced.
We analyze signed networks of US Congress legislators in the Senate and House of Representatives based on their clusterability for different number of subsets. To be more precise, we compute the minimum number of edges who removal leads to a k-balanced network under different values of k. Substantively, this analysis allows examining the US Congress similar to a parliamentary body, where coalitions form to achieve a majority voting bloc.
Previous studies on the same data show that signed networks of US Congress are close to being balanced according to optimal partitioning of networks into k=2 groups. We use more general mathematical models for (1) specific pre-defined values of k and (2) general value of k* which compute the clusterability index defined as the minimum number of intra-group negative and inter-group positive edges under all possible partitions.
Our numerical results based on exact optimization models show that the clusterability indices of US Congress signed networks initially decrease for k>2 which suggest that the signed ties between legislators in the US Congress are more consistent with a parliamentary-style set of coalitions than with a more conventional two-party categorization. We also obtain the globally minimum number of groups which minimize the clusterability index of each network by grouping legislators into a relatively large number of clusters k*>>2.
The results demonstrate that signed network of US Congress can be partitioned into coalitions exhibiting generalized balance and that the coalitions are not strictly related to party membership. The initial decline of clusterability index when k is gradually increased shows that the networks are more consistent with generalized balance than classic balance theory. However, clusterability index increases if the number of pre-defined groups is set to a value larger than k* which seems to suggest that political collaborations prevents legislators from forming too many opposing sides.
Our observations show that legislators in US Congress seem to act more like a parliamentary body with more than two coalitions, but that these coalitions still mirror broad liberal-conservative tendencies. In this presentation, we provide our results on clusterability of US Congress network and the dynamics of its major coalitions over 1979-2018.← Schedule