Detecting coalitions by optimally partitioning signed networks of political collaboration

Abstract

We propose new mathematical programming models for optimal partitioning of a signed graph into cohesive groups. To demonstrate the approach's utility, we apply it to identify coalitions in US Congress since 1979 and examine the impact of polarized coalitions on the effectiveness of passing bills. Our models produce a globally optimal solution to the NP-hard problem of minimizing the total number of intra-group negative and inter-group positive edges. We tackle the intensive computations of dense signed networks by providing upper and lower bounds, then solving an optimization model which closes the gap between the two bounds and returns the optimal partitioning of vertices. our substantive findings suggest that the dominance of an ideologically homogeneous coalition (i.e. partisan polarization) can be a protective factor that enhances legislative effectiveness.

Multilevel structural evaluation of signed directed social networks based on balance theory

“…Several studies have compared triadic level balance with estimates for the line index of balance, and found that the line index (as a measure of unbalance) correlates with other measures of unbalance, such as the proportion of unbalanced triads 60,61 . Using normalized versions of triadic balance and the line index, high correlation is observed for networks with high density 8,18 , while measurements for sparse networks usually do not match 8 . However, the literature does not clarify whether such correlations are due to similarities in measurement or structural mechanisms in networks that yield similar balance values at different levels.…”

“…These edges can be thought of as sources of tension in this approach. While the historical roots of the frustration index go back to the 1950's 5,50 , this approach only started to receive major attention in recent years 8,9,18,21,51 . This might be due to the computational complexity of obtaining this index exactly, which is an NP-hard problem 52 .…”

“…To the best of our knowledge, the frustration index has not been previously applied to directed graphs. As a technical contribution, we have developed the first exact method for computing this index for directed signed graphs by building on recently proposed optimization models 18,45 .…”

“…We solve the optimization model that produces using Gurobi solver 53 (version 9.0) in Python . For large networks, we follow the two-step method presented in 18 , which involves computing a lower bound for the frustration index before solving the optimization model.…”

“…Structural balance has primarily been studied for undirected signed networks 8 , 11 – 13 as opposed to directed signed networks 14 – 16 , although directed signed network data have been available since the early days of balance theory 17 . Using undirected network models for balance assessment could be justified when the modeled relationships are truly undirected, such as collaborations 18 , or inherently reciprocal, such as bi-lateral alliances 9 . However, many real-world relationships are intrinsically directed, such as social preferences 17 , 19 , and not necessarily reciprocated, such as a friendship 20 .…”