A Statistical Model of Bipartite Networks: Application to Cosponsorship in the United States Senate

TL;DR

The paper tackles aggregation bias in bipartite networks by introducing biMMSBM, a mixed-membership stochastic blockmodel adapted for two node types that incorporates node- and dyad-level covariates. It provides a scalable estimation approach via collapsed posterior and variational inference, implemented in NetMix. Applying biMMSBM to the 107th U.S. Senate cosponsorship data reveals cross-party coalitions driven by junior power brokers, reciprocity, and shared committee experience, along with three latent bill groups that structure cosponsorship patterns. The approach preserves information lost in projection, enabling granular predictions and richer interpretation of coalition formation, with broad applicability to political science and other bipartite settings.

Abstract

Many networks in political and social research are bipartite, with edges connecting exclusively across two distinct types of nodes. A common example includes cosponsorship networks, in which legislators are connected indirectly through the bills they support. Yet most existing network models are designed for unipartite networks, where edges can arise between any pair of nodes. However, using a unipartite network model to analyze bipartite networks, as often done in practice, can result in aggregation bias and artificially high-clustering -- a particularly insidious problem when studying the role groups play in network formation. To address these methodological problems, we develop a statistical model of bipartite networks theorized to be generated through group interactions by extending the popular mixed-membership stochastic blockmodel. Our model allows researchers to identify the groups of nodes, within each node type in the bipartite structure, that share common patterns of edge formation. The model also incorporates both node and dyad-level covariates as the predictors of group membership and of observed dyadic relations. We develop an efficient computational algorithm for fitting the model, and apply it to cosponsorship data from the United States Senate. We show that legislators in a Senate that was perfectly split along party lines were able to remain productive and pass major legislation by forming non-partisan, power-brokering coalitions that found common ground through their collaboration on low-stakes bills. We also find evidence for norms of reciprocity, and uncover the substantial role played by policy expertise in the formation of cosponsorships between senators and legislation. We make an open-source software package available that makes it possible for other researchers to uncover similar insights from bipartite networks.

Figures (10)

• Figure 1: Example networks for bill cosponsorship in bipartite and unipartite forms. Panels (b) and (c) show different bipartite networks that project to the same unipartite network in panel (a). This projection loses information about bill types (triangle colors) and cosponsorship details (e.g., number of cosponsors, number of bills).
• Figure 2: Cosponsorship Networks among Senators in the 107th Congress. The figure shows two bipartite networks sampled from the 107th Congress, with all 100 senators sorted by ideology (most conservative senators at the top) and a sample of bills sorted by node degree. The left panel depicts a network where bills are predominantly cosponsored either by Republicans or Democrats, while the right panel shows bills with highly bipartisan cosponsorship compositions. These networks highlight significant heterogeneity in composition and degree across bill nodes in our dataset.
• Figure 3: Probability of copartisan cosponsors during the 107th Senate: The figure displays distributions of probabilities: left panel shows probabilities that any two distinct cosponsors of a bill are from the same party, and right panel shows probabilities that a senator's randomly chosen pair of cosponsors are copartisan. The bipartite network reveals substantial bipartisan cosponsorship, while the weighted unipartite network among senators indicates less cooperation.
• Figure 4: Mixed-Membership Stochastic Blockmodel for Bipartite Networks. The schematic depicts a 2-by-3 latent community model, where four senators exhibit mixed memberships across two communities (blue and orange), and five bills exhibit mixed memberships across three communities (yellow, red, green). Community affinities between senators and bills are encoded in the block model matrix on the right, illustrated by edge thickness in the left graph.
• Figure 5: Posterior predictive goodness-of-fit checks, out-of-sample. Vertical black rectangles represent the interquantile range across 50 replicate networks. The red line in each panel denotes the observed value in a network formed by a random 25% sample of cosponsorship decisions during the 107th Senate. The model generally replicates structural features well, shown by overlap between black bars and red lines. However, $k$-stars of bills are consistently underpredicted in the out-of-sample set.