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215 appendix b Exponential Random Graph Models The ERGM technology is designed to account for the various types of autocorrelation that are present in network models. In this analysis, we explain the network of relationships created by legislators’ decisions to cosponsor bills in part as a function of their relationships in the caucus network. We treat the N × N matrix of legislators in the 109th and 110th Congresses as dichotomous directed adjacency matrices in which each cell (i.j) represents cosponsorships by legislator i on bills sponsored by legislator j. We also generate a 1-mode projection of the 2-mode incidence of legislators’ membership in caucuses in the 109th and 110th Congresses. This creates two N × N adjacency matrices, which we dichotomize so that the matrix indicates pairs of legislators who share some common caucus connections. ERGMs are a statistical approach that is specifically designed for network data and models the inherent autocorrelation and dependencies in the data (see Cranmer and Desmarais 2011). This approach accounts for interdependency in the data by defining a probability distribution for the set of all possible networks for a set of nodes. In the ERGM environment, one can have the highest level of confidence that network dependence has been not only accounted for but explicitly modeled. As Cranmer and Desmarais put it, Researchers can proceed with ERGM analysis based on hypotheses similar to those that would produce regression specifications (i.e., covariate x is expected affect the outcome y), and as much network structure (dependence ) as they see fit. Moreover, the ERGM is poised for widespread application in political science and is implemented in a number of software packages. The ERGM is, as we will show, widely applicable to network analysis in political science and is remarkably flexible in its ability to model relational interdependence. (2011, 67) 216 appendixes We control for a number of endogenous and exogenous covariates that we assume affect network structure. For the endogenous structure, we control for the overall number of edges (or ties) in the network. This term acts like a constant in a typical regression analysis. We also control for the number of “mixed two-stars,” or “two-paths.” This term adds one statistic to the model equal to the number of pairs that share distinct edges. This is a directed path of length 2. We also control for the number of triangles (any set of three connected edges) in the model. The ERGMs are estimated in the R package ERGM (Handcock et al., 2003, 2011; see also Butts et al. 2011). We dichotomize the networks at the 0 threshold level for the dependent variables , meaning that any pair of legislators who share any tie receives a 1, otherwise, they receive a 0. The dichotomization is necessary for the ERGM analysis. As discussed in the text, we also include a series of exogenous covariates that we assume affect legislators’ propensity to be in the same LMOs: caucus memberships (our key variable of interest), party, shared committee, shared state, gender, shared leadership, shared African American ethnicity, seniority , and the percentage by which members won their last election. ...

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