2.1 Projection onto a Unipartite Network Can Be Misleading

Aggregating over heterogeneity in bill- and senator-bill level data can lead to incorrect substantive takeaways. To see how this may be the case, revisit the stylized scenario presented in Figure 1. In it, two distinct bipartite networks (b and c) represent different collaborative environments—one with high productivity and cross-party collaboration, and the other with low productivity and limited cross-party work. Despite these differences, they both result in the same unipartite projection (a). In (b), the bipartisan-to-within-party cosponsorship ratio is 3:4, with a 0.43 probability of randomly selecting same-party cosponsors; in (c), this ratio is 1:2, and the probability is 0.84. These crucial differences, which speak to the degree of polarization in the underlying network, are obscured in the unipartite projection (a), which shows strong within-party ties regardless of the underlying bipartite structure.

Such distortions are palpable when considering the 107th Senate. Figure 3 shows the distribution of probabilities that a randomly selected pair of cosponsors belong to the same party, computed for each bill in the original bipartite cosponsorship network (left panel) and for each senator in the unipartite weighted projection of the same. While the strongly bimodal distribution associated with the bipartite network (Figure 3, left panel) suggests we are roughly as likely to find perfectly bipartisan bills as we are to find perfectly partisan ones, the distribution associated with the projected unipartite network (Figure 3, right panel) paints a completely different picture. With an average same-party cosponsorship probability of 0.75 and a left-skewed distribution, the projection suggests the majority of senators collaborate with copartisans only.

Figure 3 Probability of copartisan cosponsors during the 107th Senate.

Note: The left panel shows the probabilities that any two distinct cosponsors of a bill are from the same party, and the right panel shows the probabilities that a senator’s randomly chosen pair of cosponsors are copartisans. The bipartite network reveals substantial bipartisan cosponsorship, while the weighted unipartite network among senators indicates less cooperation.

Unfortunately, this kind of strong, artificial clustering present in both the real projected network of the 107th Senate and the simple example in Figure 1(a) is a common phenomenon (Latapy, Magnien, and Del Vecchio Reference Latapy, Magnien and Del Vecchio2008; Newman et al. Reference Newman, Strogatz and Watts2001; Tam Cho and Fowler Reference Tam Cho and Fowler2010), and can lead to incorrect conclusions about the extent and nature of polarization in Congress, as we show in Section 4 below. In general, only in the rare cases when degree and group composition are independent among nodes in the family being aggregated over can we expect the projection to have no effect on the conclusions that can be drawn from the projected network. To address these concerns, we introduce a new modeling strategy next.

3.1 Modeling Strategy

We represent an observed network as a bipartite graph, with two disjoint node sets (e.g., senators and bills) linked only by edges representing cosponsorships (no edges among legislators or bills).

The biMMSBM allows nodes to belong to one of several latent groups when interacting with each node of the other family. For any dyadic relationship between two nodes of different families, the latent group memberships of the nodes determine the likelihood of forming an edge. Thus, a senator may belong to different latent communities when deciding whether to co-sponsor different legislation. Similarly, bills can be sorted into separate latent groups across senator–bill dyads. For instance, John McCain (R-AZ) might have behaved similarly to other Republicans when deciding whether to cosponsor bills related to national security, but might have acted differently when considering bills related to campaign finance reform—a pattern that could help us understand his reputation as a party maverick.

To capture this, we define a probabilistic model to account for diverse latent community memberships. Figure 4 schematically depicts our model’s mixed membership structure. Pie charts show the probability of four senators belonging to two communities (blue and orange) and five bills belonging to three (red, green, and yellow). Node-level covariates (e.g., senator ideology and bill policy area) explain these mixed memberships.

Figure 4 Mixed-membership stochastic blockmodel for bipartite networks.

Note: The schematic depicts a 2 $\times $ 3 latent community model, where senators exhibit mixed memberships across two communities (blue and orange) represented as pie charts to indicate probabilities in each community summing up to 1, and bills exhibit mixed memberships across three communities (yellow, red, and green). Community affinities are encoded in the block model matrix (right), illustrated by edge thickness (left).

The $2 \times 3$ matrix on the right of Figure 4 shows the blockmodel, indicating probabilities of cosponsorship between senator and bill communities. Certain community pairs (e.g., orange senators and red bills) show higher cosponsorship probabilities than others (e.g., blue senators and green bills), reflecting diverse coalitional strategies among senators toward legislation.

Cosponsorship networks often exhibit this stochastic equivalence, where coalitions of senators support similar legislative classes (e.g., Bratton and Rouse Reference Bratton and Rouse2011). Similar group-based dynamics are found in other networks, like economic trade between countries and co-occurrence of words in documents. We now proceed with a formal presentation of our full model.

3.2 The Bipartite Mixed-Membership Stochastic Blockmodel

Formally, let $G(V_1,V_2,E)$ represent a bipartite graph, where ( $V_1$ , $V_2$ ) denote the two disjoint families of nodes ( $V_1 \cap V_2 = \emptyset $ ), and E represents the undirected edge set, or node pairs of different families. Suppose that family 1 has $N_1 = |V_1|$ total nodes, and that family 2 has $N_2=|V_2|$ . For each dyad, let $z_{pq}\in \{1,\ldots ,K_1\}$ denote the latent group, to which node $p \in V_1$ of family 1 belongs when interacting with node $q \in V_2$ of family 2, whose latent group membership is denoted by $u_{pq}\in \{1, \ldots , K_2\}$ . Generally, we allow $K_1\neq K_2$ . Further, we use $y_{pq} = 1$ to denote the existence of an edge between node pair $\{p,q\}\in E$ while $y_{pq} = 0$ indicates its absence.

We assume edge formation probability is a function of dyadic predictors $\boldsymbol {d}_{pq}$ and a blockmodel $\boldsymbol {B}$ , which is a $K_1\times K_2$ matrix representing the log odds of edge formation between members of any two latent groups (Figure 4),

(1) $$ \begin{align} y_{pq} \mid z_{pq},u_{pq}, \boldsymbol{B}, \boldsymbol {\unicode{x3b3} } \ {\stackrel{\text{indep.}}{\sim}} \ \text{Bernoulli}\left(\text{logit}^{-1}(B_{z_{pq},u_{pq}} + \boldsymbol{d}_{pq}^\top \boldsymbol {\unicode{x3b3} })\right), \end{align} $$

where $\boldsymbol {\unicode{x3b3} }$ is a dyad-level regression coefficient vector. Our dyadic predictors allow for varying edge formation probabilities even within the same latent group pairs, relaxing the stochastic equivalence assumption standard to stochastic blockmodels (SBMs). Substantively, this allows for scenarios where senator–bill dyads whose respective nodes sort into the same pairs of latent communities to be further differentiated by characteristics pertinent to their particular dyad (e.g., a senator’s history with a bill’s author).

As is common in mixed-membership SBMs, we define a categorical sampling model for the dyad-specific group memberships, $z_{pq}$ and $u_{pq}$ , so that

(2) $$ \begin{align} z_{pq} \mid \boldsymbol{\unicode{x3c0}}_p \sim \text{Categorical}(\boldsymbol{\unicode{x3c0}}_p), \quad u_{pq} \mid \boldsymbol{\unicode{x3c8}}_q \sim \text{Categorical}(\boldsymbol{\unicode{x3c8}}_q), \end{align} $$

where the probability that family 1 node p (family 2 node q) belongs to a latent group on any possible interaction is given by $\boldsymbol {\unicode{x3c0} }_p$ ( $\boldsymbol {\unicode{x3c8} }_q$ )—a $K_1$ -dimensional ( $K_2$ -dimensional) probability vector usually known as the mixed-membership vector (represented as pie charts in Figure 4).

Critically, our model incorporates node-level information (e.g., senator and bill level predictors) into the definition of the mixed-membership probabilities of latent groups. These covariates themselves predict the likelihood of an edge (e.g., cosponsorship) through the resulting instantiated element of the blockmodel. Specifically, we assume that the mixed-membership probability vectors are generated by a Dirichlet distribution with concentration parameters that are a function of node covariates,Footnote ³

(3) $$ \begin{align} \boldsymbol{\unicode{x3c0}}_{p} \mid \boldsymbol{\unicode{x3b2}}_1 \sim \text{Dirichlet}\left(\left\{\exp(\boldsymbol{x}_{p}^\top\boldsymbol{\unicode{x3b2}}_{1g})\right\}_{g=1}^{K_1}\right), \quad \boldsymbol{\unicode{x3c8}}_{q} \mid \boldsymbol{\unicode{x3b2}}_2 \sim \text{Dirichlet}\left(\left\{\exp(\boldsymbol{w}_{q}^\top\boldsymbol{\unicode{x3b2}}_{2h})\right\}_{h=1}^{K_2}\right), \end{align} $$

where hyper-parameter vectors $\boldsymbol {\unicode{x3b2} }_{1g}$ and $\boldsymbol {\unicode{x3b2} }_{2h}$ contain regression coefficients associated with the gth and hth groups of vertex families 1 and 2, respectively.

Putting it all together, the full joint distribution of data and latent variables is given by,

(4) $$ \begin{align} f(\boldsymbol{Y},\boldsymbol{Z},\boldsymbol{U},\boldsymbol{\Pi},\boldsymbol{\Psi}, \mid \boldsymbol{B}, \boldsymbol{\unicode{x3b2}}, \boldsymbol {\unicode{x3b3} }) & = \prod_{p,q \in V_1\times V_2} f(y_{pq}\mid z_{pq}, u_{qp}, \boldsymbol{B}, \boldsymbol {\unicode{x3b3} }) f(z_{pq}\mid \boldsymbol{\unicode{x3c0}}_{p}) f(u_{pq} \mid \boldsymbol{\unicode{x3c8}}_{q}) \notag\\ &\quad\times \prod_{p\in V_{1}} f(\boldsymbol{\unicode{x3c0}}_{p}\mid \boldsymbol{\unicode{x3b2}}_1) \prod_{q\in V_{2}} f(\boldsymbol{\unicode{x3c8}}_{q}\mid \boldsymbol{\unicode{x3b2}}_2). \end{align} $$

This specification allows us to more formally describe the potential issues raised by aggregation illustrated informally in Section 2.1. A common strategy simply sums the number of connections to a member of family $V_2$ shared by two members of family $V_1$ , forming an aggregated sociomatrix $\widetilde {\boldsymbol {Y}}=\boldsymbol {Y}\boldsymbol {Y}^\top $ . Under this strategy, and in the absence of dyadic covariates, the model in Equation (4) implies

(5) $$ \begin{align} \mathbb{E}[\widetilde{\boldsymbol{Y}}] = \mathbb{E}[\boldsymbol{Y}\boldsymbol{Y}^\top]> \boldsymbol{\Pi}\left[\boldsymbol{B}\boldsymbol{\Psi}^\top\boldsymbol{\Psi}\boldsymbol{B}^\top \right]\boldsymbol{\Pi}^\top, \end{align} $$

where $\boldsymbol {\Pi }$ is an $N_1\times K_1$ matrix that stacks mixed memberships $\boldsymbol {\unicode{x3c0} }_p$ for all $p\in V_1$ , and similarly for $\boldsymbol{\Psi}$ . The issue arises because the bracketed terms in Equation (5) (i.e., blockmodel and Family $V_2$ mixed-memberships) cannot be separately identified from the aggregated sociomatrix $\widetilde {\boldsymbol {Y}}$ , leading to observational equivalence like the one illustrated in Figure 1. This can lead to misconstrued relationships among members of Family 1 when relying on aggregated data. Our model avoids this by directly modeling the bipartite network without the need to aggregate.

3.3 Estimation

With the thousands of vertices and millions of potential edges involved in an application such as bill cosponsorships, sampling directly from the posterior distribution given in Equation (4) is computationally prohibitive. To obtain estimates of quantities of interest in a reasonable amount of time, we follow the computational strategy of Olivella et al. (Reference Olivella, Pratt and Imai2022) by first marginalizing latent variables and then defining a stochastic variational approximation to the full posterior. We briefly summarize these computational strategies here.

3.3.1 Marginalization

To reduce complexity, we collapse the full posterior over the mixed-membership vectors (i.e., $\boldsymbol {\Pi }$ and $\boldsymbol{\Psi}$ ):

(6)

where $\Gamma (\cdot )$ is the Gamma function; $\unicode{x3b1} _{pg} = \exp (\boldsymbol {x}_{p}^\top \boldsymbol {\unicode{x3b2} }_{1g})$ , $\xi _{p} = \sum _{g=1}^{K_1} \unicode{x3b1} _{pg}$ (and similarly for $\unicode{x3b1} _{qh}$ and $\xi _{q}$ ); is a count representing the number of times node p instantiates group g across its interactions with nodes in family 2 (and similarly for $C_{qh}$ ); and $\unicode{x3b8} _{pq, z_{pq}, u_{qp}} = \text {logit}^{-1}(B_{z_{pq}, u_{qp}} + \boldsymbol {d}_{pq}^\top \boldsymbol {\unicode{x3b3} })$ is the probability of a tie between the vertices in dyad $p,q$ .

3.4 Methodological Contributions

While the biMMSBM model is an extension of the unipartite MMSBM (Airoldi et al. Reference Airoldi, Blei, Fienberg and Xing2008) and its structural variant (Olivella et al. Reference Olivella, Pratt and Imai2022), we believe it makes three methodological contributions. First, while the MMSBM is a popular modeling framework for network data across disciplines, there is no version of it that can be applied to bipartite network data, which are common in political science. The model we propose can take full advantage of information about both kinds of vertices involved in bipartite networks, without the need to aggregate and ignore either. Second, by avoiding projections that are common in practice, our model allows researchers to avoid biased results (such as artificially higher clustering) related to either of the types of vertices under study. Finally, and particularly by incorporating node-level predictors, our model allows researchers to make predictions about specific pairs of actors (e.g., which senators will support which specific bills). Such granular predictions are not possible when working with the projected network, which most prior models forced researchers to do.Footnote ⁶

4.1 Model Specification and Fit

Our goal, then, is to understand the structural and contextual features that made collaboration possible during the 107th Senate. A rich literature on collaboration in Congress suggests that legislators make cosponsorship decisions based on partisanship, seniority, gender, and personal political history (Bratton and Rouse Reference Bratton and Rouse2011; Holman and Mahoney Reference Holman and Mahoney2018; Rippere Reference Rippere2016). Therefore, our model includes each senator’s party, ideology, seniority, and gender as predictors of community membership.

Harward and Moffett (Reference Harward and Moffett2010) articulate that senators are more likely to cosponsor bills when they share closer preferences with the sponsor of the bill, and when they are more connected to their colleagues. To capture this, we model legislation groups as a function of their corresponding sponsors’ party, ideology, seniority, and gender (self-sponsorship dyads are excluded).

Lastly, senators tend to cosponsor bills within specific policy domains (Harward and Moffett Reference Harward and Moffett2010) and may opt into bipartisan cosponsorships based on legislative bill topics (Harbridge Reference Harbridge2015). This inclination cannot be modeled in a senator-only unipartite network, but can be directly accounted for when modeling the bipartite structure. We address this by including the substantive topic as a bill covariate.Footnote ⁸

To capture the described shifts in the temporal context in which bills are introduced, we also include a bill-level covariate indicating whether a bill was presented in the first phase of the Congress (lasting only several months prior to Vermont Senator Jeffords leaving the Republican party in May 2001), in the second phase (post Jeffords leaving and prior to 9/11) or in the third phase (after 9/11). This temporal context would be lost in a unipartite network analysis (Kirkland and Gross Reference Kirkland and Gross2014).

As we indicated earlier, we also pay close attention to two additional forces that can be expected to affect the likelihood of cosponsorship. First, we aim to capture reciprocity behaviors, or favor-trading on the Senate floor (Brandenberger Reference Brandenberger2018; Harbridge-Yong et al. Reference Harbridge-Yong, Volden and Wiseman2023). The model includes a dyadic predictor: the log-transformed proportion of times a bill’s sponsor reciprocated cosponsorship in the previous Congress. As this proportion of reciprocity is heavily skewed and contains many zeros, we use the log transformation of non-zero values and an indicator variable for the cases of zeros.

Second, our dyadic model includes the number of committees shared by a senator and a piece of legislation. A greater number of shared committees indicates a higher chance that the senator has overseen the development of a bill and holds relevant substantive expertise. While the roles of committees have been studied previously (Cirone and Van Coppenolle Reference Cirone and Van Coppenolle2018; Porter et al. Reference Porter, Mucha, Newman and Warmbrand2005), our analysis directly examines how overlap in committees between legislator and legislation relates to cosponsorship. Relatedly, Gross and Kirkland (Reference Gross and Kirkland2019) find evidence of strong predictive power of shared committee leadership among the subset of ranking legislators when exploring cosponsorship decisions.

With predictors at the monadic and dyadic levels in place, we determine the number of latent groups for senators and bills; we first randomly select 25% of data as a test set, and compare models with a range of possible latent group-size pairings through the area under the ROC curve (AUROC) values for the out-of-sample edges. We select group sizes offering the best fit according to this criterion, resulting in three groups each for legislators and bills, i.e., $K_1=K_2=3$ .

In Section S.3.3 of the Supplementary Material, we establish that the model generally fits the data well even out of sample (on posterior predictive goodness-of-fit checks and comparisons of network-level statistics); we obtain the estimates of all parameters and hyper-parameters in Equation (4) for this ${K_1=K_2=3}$ model fitted to the entire bipartite cosponsorship network. More specifically, we compute various quantities of interest in the form of predicted probabilities of block interactions and block memberships. As our discussion hinges on these derived quantities, we present all estimated values in Tables S.4 and S.5 in the Supplementary Material.

4.2 Pathways to Legislative Collaboration

What kinds of coalitions are at play when it comes to making cosponsorship decisions, and how do these coalitions interact when considering different types of legislation? Figure 5 presents the 107th Senate estimated blockmodel, showing cosponsorship probabilities between senator and bill groups. Node size reflects group frequency; edge shading, cosponsorship likelihood. Ideological distributions of senator and bill groups are also presented. The density for each senator group represents the distribution of ideal points of its members, while the density for a bill group is that of its members’ sponsors.

Figure 5 Blockmodel of senator and legislation latent group connection probabilities.

Note: Block size is proportional to the number of nodes expected to instantiate the corresponding latent group, and connections between them are shaded denoting cosponsorship probabilities between group members (darker shades indicate higher connection likelihoods). Senator groups tend to engage more with an “Uncontroversial” legislation group but less with a larger “Contentious” one. Next to each block, we also present the density of ideological positions of member senators (top row) and bill sponsors (bottom row), revealing that while ideology can help distinguish across types of senator coalitions, it cannot discriminate across relevant types of legislation.

As expected, Figure 5 shows senator groups aligning with party lines, including seasoned Republicans (e.g., Strom Thurmond [R-SC] and Jesse Helms [R-NC]) and Democrats (like Robert Byrd [D-WV] and Edward Kennedy [D-MA]). In addition, however, our model identifies a distinct senator group (depicted in purple) who stand out as having different cosponsorship patterns than their more partisan counterparts. Exemplars of this group, whom we call the junior power brokers, include Jon Corzine (D-NJ), Tom Carper (D-DE), Susan Collins (R-ME), Bill Frist (R-TN), Zell Miller (D-GA), and Hillary Clinton (D-NY)—all junior Senators at the time. Figure 6 presents the estimated mixed memberships of all Senators (i.e., their probability of acting as part of any of the discovered latent groups), highlighting a few of the most notable legislators of the session. Table S.2 in the Supplementary Material presents the top 10 members of each senator latent group by mixed membership probability.

Figure 6 Ternary plot of senator latent group membership probabilities.

Note: For clarity of presentation, example senators are colored by party. Senators in group 1 (top corner) are more likely to be Democrats, while senators in group 2 (right corner) are more likely to be Republicans; Group 3 (left corner) senators hail from both sides of the aisle and are likely to be junior and involved in cross-partisan bill sharing.

This third bipartisan group is likely to be formed by senators who have little experience in the Senate coming from all over the ideological spectrum, as evidenced by the distribution of ideological positions depicted over the corresponding group in Figure 5.Footnote ⁹ We explore this in the left-most panel of Figure 7, depicting how the probability of group membership changes as a function of ideology. While junior power brokers (depicted in purple) is primarily predicted to be composed of left-leaners, positions along the second ideological dimension (seen in the central panel of Figure 7)—often interpreted as capturing cross-cutting salient issues of the day (Poole and Rosenthal Reference Poole and Rosenthal2017)—distinguish this group of senators from their staunch Democratic counterparts.

Figure 7 Predicted mixed memberships of senator predictors.

Note: The y-axes plot average predicted mixed memberships across the three possible senator latent groups, given each shift in the value of a senator predictor in the x-axes; for instance at low values of Ideology (dimension) 1, the average predicted memberships for being in group 1 (Seasoned Democrats) and group 3 (power-brokers) are highest; as Ideology 1 values increase (corresponding to increase in the conservative direction), average predicted group 2 membership (Seasoned Republicans) increases and supplants group 1 entirely.

Many of these junior power brokers would become leaders within their parties. For instance, during the latter part of the 107th Congress, Republican Conference leader Trent Lott resigned and was swiftly replaced by Bill Frist (R-TN)—a top member of the junior power brokers identified by our model.

Similarly, many of them were pivotal “last” votes in large contentious bills that required just an extra nudge for passage. For example, consider the Farm Bill, designed to repeal the Freedom to Farm Act of 1996. While politics over agriculture had historically been regional rather than ideological, the Freedom to Farm Act was a significant deviation from that norm. Veteran senators Tom Daschle (D-SD) and Agriculture Committee Chairman Tom Harkin (D-IA) collaborated to bring the Farm Bill together, and negotiations began to generate the necessary support—including that of small dairy farmers affected by the bill. In the end, the largely Democratic set of supporters was complemented by key support from Republicans Susan Collins (R-ME) and Jeff Sessions (R-AL)—again identified by our model as likely members of the power brokers group. This role as brokers is further supported by analyses of the betweenness centrality of Senators who are likely to instantiate this group, which tends to be higher than that of Senators likely to instantiate other groups (see Table S.6 in the Supplementary Material).

The model is also able to identify the types of legislation which these groups of senators are likely to cosponsor. Specifically, the model uncovers three broad classes of bills and resolutions (depicted in the bottom row of circles in Figure 5), and the corresponding probabilities that members of any of the three senator groups will cosponsor them. While ideology plays an important role in defining the latent senator groups that structure cosponsorship (with right-skewed, left-skewed, and bimodal distributions characterizing membership into the three groups at the top of Figure 5), no such differences in the ideology of sponsors can help distinguish across the groups of legislation uncovered by our model (as indicated by the similarly bimodal densities of sponsor ideology across all three groups in the bottom of Figure 5).

We next show that investigating this nuance in bill composition can help us understand how collaborations took place during this nominally partisan Congress.

4.3 Legislation Types That Facilitate Cosponsorship

The largest type of legislation uncovered by our model is also the least likely to be supported by members of any senator group, suggesting that the bulk of legislation introduced in the Senate received little support from Senators other than the original sponsor. This latent class of bills, which we labeled “Contentious Bills” in Figure 5, consists of high-stakes bills on controversial economic issues and social programs, including those that handle the allocation of public funds for such programs. For example, the Senior Self-Sufficiency Act (SN 107 2842), Bioterrorism Awareness Act (SN 107 1548), and the Nationwide Health Tracking Act of 2002 (SN 107 2054) belong to this group. Table S.3 in the Supplementary Material presents details of legislation with the top ten mixed membership probabilities in each of the three latent groups.

The size of the “Contentious Bills” group grew during the last phase of the 107th Senate, after the 9/11 attacks. This is easily seen in Figure 8, which presents radar plots of predicted legislation memberships by phase of the Congress (panels from left to right present bills from the pre-Jeffords’ split phase, post-Jeffords’ split second phase, and post 9/11 phase). Each radar graph positions the six observed substantive topics along spokes of a wheel, and plots the predicted number of bills on that topic as a point along the corresponding spoke: the farther away from the wheel center, the more bills are predicted to be on that topic. Doing this for each of the three latent groups results in the three shaded polygons presented in each panel of the figure.Footnote ¹⁰ The dominance of bills in the “Contentious Bills” group in the third phase, depicted in orange, is readily apparent.

Figure 8 Radar graphs of predicted legislation by topic within each phase of Congress, by bill latent group.

Note: Panels are phases 1 (pre-Jeffords split), 2 (post-Jeffords split), and 3 (post 9/11) in the Congress, from left to right. Each radar plot includes bill topics as poles, with the estimated number of bills in the topic plotted against each pole, by latent group. Phase 2 produces the fewest pieces of legislation, while Phase 3 produces the most. Over time, the predicted number of bills in the “Contentious Bills” group (orange polygon) increases, especially in domains related to social public programs and the economy. The number of bills in the “Bipartisan Resolutions” group grew more slowly than that in the “Contentious Bills” block (green polygon), but has similarly favored social/public programs and the economy. Finally, the number of bills in the “Popular & Uncontroversial” (yellow polygon) changed the least throughout the session.

The composition of the other two latent bill groups uncovered by our model—the groups we have labeled “Bipartisan Resolutions” and “Uncontroversial Bills” in both Figures 5 and 8—provides valuable clues for understanding how cross-party collaboration took place. Specifically, the topical composition of the “Bipartisan Resolutions” almost mirrors that of the “Contentious Bills” (i.e., it is composed of pieces of legislation that deal with controversial public social programs and economic issues, as indicated by the similarly-proportioned shapes of green and orange polygons in Figure 8), but it is mainly composed of concurrent and simple resolutions, rather than bills. As they do not result in codified law (unlike continuing resolutions), such resolution offer low-staked opportunities to build bridges across partisan divides. Table S.3 in the Supplementary Material contains the top pieces of legislation in the group.

In turn, legislation in the comparatively smaller “Uncontroversial” group (shown in yellow in Figures 5 and 8) also draws consistent cosponsorship support from all senator groups and across the aisle, as pieces in it tend to be either uncontroversial resolutions or bills on popular social programs. For instance, the Senate joint resolution over the September 11 attacks (SJ 107 22) has the second-highest mixed membership probability in this group, followed closely by bills, such as the Railroad Retirement and Survivors’ Improvement Act of 2001 (SN 107 697). Such legislation forms a small but steady core that supplements low-stakes efforts (such as those in the “Bipartisan Resolutions” block), and that can nevertheless result in substantial legislation, such as the Family Opportunity Act of 2002 (SN 107 321).

The importance of this meaningful cooperation mechanism revealed by the blockmodel is particularly notable, as the model was able to identify it net of two important drivers of cosponsorship: quid pro quo behaviors, measured as the coefficient on the (log) proportion of “reciprocity” (Log Reciprocity), and the shared committee experience of a given senator–bill dyad (Shared Committee). For the former, our model suggests that a 1% increase in the reciprocity (i.e., the proportion of times the sponsor of a piece of legislation acted as a cosponsor for a given senator’s bill in the previous Congress) is associated with a roughly 2% increase in the odds of cosponsorship. In the case of the latter, we find that sharing a committee is significantly and positively associated with collaboration, making cosponsorship about 3.7 times more likely. These results, which are fully explored in Table S.4 in the Supplementary Material, are consistent with previous research on the determinants of legislative collaboration.

In sum, Senators appear to have leveraged a mix of low-stakes resolutions over potentially contentious issues and a small but important set of bills for which there was bipartisan support. This enabled them to build cross-partisan bridges and keep the 107th term from devolving into stalemate. Our model identified these novel patterns of cooperation after accounting for other, more traditional forces predictive of collaboration and cosponsorship. Our application offers clues on which kinds of legislators likely to collaborate, but also about which kinds of legislation make such collaborations possible. These insights would be lost when analyzing data aggregated over bills and their characteristics.Footnote ¹¹