Party Polarization in Congress: A Network Science Approach
Andrew Scott Waugh, Liuyi Pei, James H. Fowler, Peter J. Mucha, Mason A. Porter
PParty Polarization in Congress:A Network Science Approach
Andrew Scott Waugh,
University of California, San Diego
Liuyi Pei,
University of California, Irvine
James H. Fowler,
University of California, San Diego
Peter J. Mucha,
University of North Carolina at Chapel Hill
Mason A. Porter,
University of Oxford
July 26, 2011
We measure polarization in the United States Congress using the network science conceptof modularity . Modularity provides a conceptually-clear measure of polarization that revealsboth the number of relevant groups and the strength of inter-group divisions without makingrestrictive assumptions about the structure of the party system or the shape of legislator util-ities. We show that party influence on Congressional blocs varies widely throughout history,and that existing measures underestimate polarization in periods with weak party structures.We demonstrate that modularity is a significant predictor of changes in majority party andthat turnover is more prevalent at medium levels of modularity. We show that two variablesrelated to modularity, called ‘divisiveness’ and ‘solidarity,’ are significant predictors of re-election success for individual House members. Our results suggest that modularity can serveas an early warning of changing group dynamics, which are reflected only later by changesin party labels.
A great deal of recent research has been devoted to identifying and explaining polarizationin the United States. Indeed, there has been enough to fill the pages of two annual re-view articles in the past half-decade (Layman et al. 2006, Fiorina & Abrams 2008). At itscore, the polarization debate began with an observation about the nature of partisanshipin Congress. Starting in the late 1970s, researchers began to notice increases in intra-party1 a r X i v : . [ phy s i c s . s o c - ph ] J u l ohesion and decreases in inter-party cooperation on roll-call votes (McCarty et al. 2007).This finding puzzled scholars because it ran counter to empirical evidence for the weaken-ing of partisanship in the electorate (Coleman 1996) and the theoretical expectation thatinstitutional incentives should drive parties to adopt similar (i.e., median) policy positions(Downs 1957 a ).These puzzles form the fault lines of the two major conflicts in the literature on partisanpolarization. The first conflict attempts to locate the origins of partisan polarization eitherin the electorate (McCarty et al. 2007), among political elites (Fiorina & Abrams 2008),or in a combination of the two (Jacobson 2005, Jacobson 2006). The second attempts toexplain the relative influences of party and ideology on Congressional voting, and well-knownpublications have argued for party effects (Cox & McCubbins 1993, Cox & McCubbins2005, Smith 2007), ideological effects (Krehbiel 1991), and interactions between the two(Rohde 1991, Aldrich & Rohde 2001). In both instances, researchers have pondered hownon-median party positions result from institutions that in theory lend themselves towardsmoderation (Layman et al. 2006).We argue that the polarization debate has been limited by its overemphasis on ex-pectations derived from spatial models of ideology. By contrast, we define polarizationbehaviorally—rather than ideologically—based on the voting decisions made by the legisla-tors and the way those decisions divide them into distinct groups. To operationalize thisdefinition, we use the tools of network science. Suppose that each legislator is a ‘node’ in thenetwork and that the level of agreement between two legislators in roll-call voting indicatesthe strength of a ‘tie’ between them. In a highly-polarized legislature, we reason that groups As discussed in more detail in the online supplementary information, the ties between all pairs of legis-lators in one legislative body (Senate or House of Representatives) in a single Congress are described by an adjacency matrix A with elements A ij = 1 b ij (cid:88) k γ ijk , (1)where γ ijk equals 1 if legislators i and j voted the same on bill k and 0 otherwise, and b ij is the total numberof bills on which both legislators voted. modularity to first identify relevant ‘communities’(tightly-knit groups) in Congress and then quantify the severity of such a division.In section 2, we use insights from Downs’s (1957 a ) model of ideology to explain our cri-tique of spatial models as tools for measuring partisan polarization. In section 3, we explainthe intuition behind our use of modularity and discuss its advantages over existing measures.In sections 4 and 5, we present empirical evidence for the utility of modularity in predictingmajority-party switches Congress as well as the electoral fortunes of individual legislators.We conclude in section 6 by discussing the ongoing role for modularity in advancing our un-derstanding of party polarization. In online supplementary information, we provide furtherdetails of the network-science methodology, additional figures and tables, and a descriptiveexample from the 19th century. Political scientists have focused intently on Downs’s median voter theorem (Downs 1957 a )and of the spatial model at its core. However, although the median voter result is nor-matively appealing, it is also rather fragile. In the Congressional case, adding influentialactivists (Aldrich 1995) or rules favoring the majority party (Cox & McCubbins 1993, Cox& McCubbins 2005) to the model results in ideological divergence of parties. In the presiden-tial case, empirical investigations show that parties alter the composition of the electorate by3ppealing to core supporters, which allows them to maintain extreme positions (Holbrook &McClurg 2005). These results challenge the expectation of median outcomes while maintain-ing strong assumptions about the nature of ideology, especially as perceived by the electorate.We argue that these assumptions might not be appropriate to the study of Congressionalpolarization.Although most scholars assume the existence and structure of ideology, Downs arguedthat the creation of party ideologies, as well as their tendency to diverge, derives from imper-fect knowledge in the electorate (Downs 1957 b ). In a complex world, Downs reasoned, votersdemand a method to reduce information costs associated with making electoral decisions.Motivated by the desire to increase vote shares (Mayhew 1974), the parties supply ideologiesin response to this demand. Importantly, Downs argued that ideologies reduce informationcosts for voters primarily by highlighting differences between parties. He therefore concludedthat ‘parties cannot adopt identical ideologies, because they must create enough product dif-ferentiation to make their output distinguishable from that of their rivals, so as to enticevoters to the polls’ (Downs 1957b:142). Given incomplete information, it therefore seemsthat one should expect some level of polarization rather than convergence to the median.Additionally, Downs attributed the original position of party ideology to the interests ofthose present at that party’s founding but presumed that successful ideologies acquire powerthat is independent of any particular interest group (i.e., parties eventually lose control overthe ideologies that they create) (Downs 1957 b ). Indeed, parties have imperfect information about the nature of their own ideology. Downs also argued that party ideology must remainrelatively stable once established in order to retain legitimacy, and that subsequent politicalactions must offer ‘persistent correlation’ with that ideology in order for it to remain auseful cognitive shortcut (Downs 1957b:142). Moreover, if a party lacks control over its ownideologies, then it becomes more difficult both for it to ensure that its ideology is stable andfor it to craft policy that adheres to its ideology.4n this scenario, ideology is a coordination device between the electorate and the parties.The parties want to provide just enough information to appear stable and trustworthy tovoters, and voters want just enough information to make decisions between competing par-ties. If voters are unwilling or unable to demand or process more information (Popkin 1994),then parties will prefer vague ideologies. Ambiguity allows competing intra-party factionsto appear united to voters who lack the necessary information to expose their contradictingpositions, and it gives parties the opportunity to pursue potentially divisive or polarizingpolicies while maintaining the guise of ideological consistency. This helps explain, for ex-ample, why parties in Congress appear to be radically divided even when the general publicseems not to be so (Fiorina & Abrams 2008) and why partisans in the electorate continueto support their representatives even when the rest of the public has withdrawn its support(Jacobson 2005).This exposition of Downs (1957 b ) calls into question the assumptions that underly manystudies of ideology, partisanship, and polarization. Scholars traditionally assume that indi-viduals have complete, transitive, single-peaked preferences over a small number of indepen-dent ideological dimensions, which presumably represent correlative sets of issue positions(Poole & Rosenthal 1997). However, given that all actors have imperfect information notonly about the composition of these dimensions but also about the location of party andelectoral medians along them, one encounters the possibility of two actors holding radicallydifferent opinions about the structure of ideology but still thinking that they occupy thesame position. If ideology means different things to different people, then it makes littlesense to assume the opposite. Furthermore, parties have the incentive to provide—and vot-ers have the incentive to consume—only the minimal amount of information necessary todistinguish between candidates for office, and this undermines the assumption that actorshave complete and well-ordered preferences over any particular ideological space.Finally, nearly all rational-choice models assume that actors are able to make better5olitical decisions when given more information (Poole 2005). Indeed, spatial models arerather extreme in this respect: they assume that actors make perfect decisions when theyhave perfect information (Krehbiel 1991). For politicians, one typically assumes that perfectdecisions are those that maximize reelection chances (Mayhew 1974). For voters, the story isdifferent. Though an incomplete-information setting encourages voters to rely on partisan,ideological, or other cues (Popkin 1994), voters in a complete-information setting are capablein theory of associating policies with outcomes (rendering such cues unnecessary). In acomplete-information setting, reelection rates should hinge on the ability of politicians toeffectively generate good public policy, ostensibly by processing the maximal amount ofpolicy information (Krehbiel 1991). This assumption is fundamental to dominant theoriesof committee organization in Congress (Shepsle & Weingast 1987). However, empiricalinvestigations suggest that committees use information polemically as a way of defendingexisting partisan positions while appealing to foundational assumptions among voters aboutthe utility of rational and scientific methods (Shulock 1998). In this case, even analysesof scientific policy serve as low-information cues about the validity of what are essentiallyideological or partisan positions.These critiques imply a world in which ideology is a necessary—but frustratingly imprecise—tool of party competition. Nearly all studies of Congressional polarization treat it as ideolog-ical and then measure polarization using traditional assumptions. Perhaps the most popularof these measures, defined by McCarty, Poole, and Rosenthal (MPR) (2007), gauges polar-ization by measuring the Cartesian distance between the mean DW-NOMINATE (which wehereafter call ‘DW-NOM’) scores of political parties (Poole & Rosenthal 1997). Calculat-ing these polarization scores requires restrictive assumptions about the nature of ideology inCongres. DW-NOMINATE assumes the existence of a low-dimensional space with consistentideological dimensions over time. This assumption is made in in order to estimate dynamicideology scores. W-NOMINATE relaxes this restriction while maintaining the spatial model-6ng assumptions (Poole 2005). In both cases, measuring the distance between political partymeans, furthermore, requires the researcher to assume a particular party system structure.We compare the modularity measure to polarization scores calculated using both DW- andW-NOMINATE (see section 4 for more discussion).While we acknowledge the utility of NOMINATE-based measures for fitting individualroll-call decisions, we question the value of aggregating ideal-point estimates into measuresof system-wide polarization. In these situations, we reason, it is prudent to also employmeasures that hew more closely to observed behaviors without imposing assumptions abouttheir rationality or spatial structure. We argue that the polarization debate should bemore concerned with the identification of relevant political groups and the evaluation ofthe divisions between them. By moving to a network framework, we can use the tools ofcommunity detection and the diagnostic known as modularity to perform both of these tasksusing more plausible assumptions. When studying a network, it is often useful or convenient for analysis to partition it intogroups. Network scientists have recently developed a measure called modularity (Newman& Girvan 2004, Newman 2006 a ) that uses information about the ties between each pair ofindividuals in a network to compare the total strength of ties lying within each group to thetotal tie strength between individuals from different groups. Previous work has used modu-larity to study cohesive groups (typically called communities ) in legislation cosponsorshipsnetworks in Congress (Zhang et al. 2008), committee membership networks in the Houseof Representatives (Porter et al. 2005, Porter et al. 2007), and a large variety of otherreal-world and computer-generated networks (Porter et al. 2009, Fortunato 2010). Otherapplications of network analysis have also flowered in the political science literature (see,7.g., Huckfeldt 1987, Fowler 2006a, Fowler 2006b, McClurg 2006, Baldassarri & Bearman2007, Koger et al. 2009, Park et al. 2009, Lazer 2011, Ward et al. 2011).‘Modular’ networks contain groups that have many ties within them but few betweenthem. Network scientists call such groups ‘communities’ because they form strongly con-nected subnetworks that, in the extreme, can be nearly separate from other parts of thenetwork (Porter et al. 2009, Fortunato 2010). Networks with stronger ties within groupsand weaker ties between groups are thereby more modular. Conceptually, this is exactly whatone means when claiming that groups are polarized. This operationalization of polarizationin roll-call votes allows us to quantify the number of cohesive groups (i.e., communities) in alegislature, quantify the strength of division between such blocs, identify which individualsare likely to belong to each cohesive group, and quantify the position of individuals withintheir groups.We employ multiple community-detection algorithms to identify groups that maximizemodularity for each roll-call network for both the Senate and the House of Representativesin the 1st–109th Congresses. Using regression analyses, we find that maximum modularityis a significant predictor of future majority party changes in the House (and approachessignificance in the Senate). Additionally, we find several periods in American history—mostnotably, during the 75th–95th Congresses from 1937 to 1979—in which a large discrepancyexists between formal party divisions and real voting coalitions. We hypothesize that suchdiscrepancies, and the corresponding changes in maximum modularity, might serve as anearly warning signal for changes in the partisan composition of Congress (perhaps due to afailure of parties to coordinate with voters on the definition of ideology). As a preliminary testof this hypothesis, we use modularity values in Congress t to predict changes in the majorityparty for Congress t + 1. We find a non-monotonic relationship between modularity and thestability of the majority party in both chambers of Congress. At low levels of modularity, We briefly discuss these procedures in section 7.2 of the online supplementary information.
We begin with a common assumption about the nature of roll-call votes: Congressmen whovote with one another are more similar than those whose votes conflict. We further assumethat two Congressmen are more similar when they agree on more roll-call decisions. Theseassumptions underly all investigations of roll-call voting blocs (Anderson et al. 1966) fromRice’s (1927) identification of blocs in small political bodies to Truman’s (1959) case study ofthe 81st Congress, early investigations of policy dimensions by MacRae (1958) and Clausen(1973), and more quantitative analyses by Poole and Rosenthal (1997) and others (Clintonet al. 2004).The method of identifying communities that we employ is philosophically similar to the9luster analyses employed by Rice and Truman, but those authors had limited computingpower at their disposal and lacked an objective method for evaluating the quality of thecommunities that they obtain. However, because of a wealth of conceptual and algorithmicadvances from the past decades (and especially from the past 10 years; Porter et al. 2009,Fortunato 2010), we have not only an appropriate measure (modularity) but also goodcomputational algorithms to partition networks into communities in order to maximize it.In the remainder of this section, we define modularity and describe the methodology thatwe use to generate modularity scores. Using roll-call data compiled by Poole and Rosenthal (1997, 2011), we generate a networkin the form of an adjacency matrix (Wasserman & Faust 1994) that describes voting similar-ities among legislators in a single Congress of the House of Representatives or Senate. Thisis done in similar fashion to the assembly of agreement score matrices in Poole (2005). Westudy the 1st–109th Senates and Houses, so we consider 218 networks in total. We representeach of these networks by an n × n matrix A , where n equals the number of legislators inthe body and each element A ij gives the proportion of votes on which two legislators agreed.The value of A ij represents the weighted strength of connection between legislators. Havinggenerated the adjacency matrices, we can calculate modularity values for any given partitionof these roll-call networks into specified, non-overlapping communities (Porter et al. 2009).Modularity relies on the intuitive notion that communities in networks should consistof nodes with more intra-community than extra-community ties (Newman & Girvan 2004,Porter et al. 2009). This mirrors our conceptualization of polarization described in section2. For a given partition of the network into communities, the modularity Q represents thefraction of total tie strength contained within the specified communities minus the expectedtotal strength of such ties. The expected strength depends on an assumed null model. Here We provide more details of this process in section 7.2 of the online supplementary information. Defined in footnote 1 and discussed in detail in section 7.1 of the online supplementary information.
10e use the standard Newman-Girvan null model that posits a hypothetical network with thesame expected degree distribution as the observed network (Newman 2006 a , Newman 2006 b ).This standard null model implies that modularity is given by the formula Q = 12 m (cid:88) ij (cid:20) A ij − k i k j m (cid:21) δ ( g i , g j ) ≡ m (cid:88) ij B ij δ ( g i , g j ) , (2)where m = (cid:80) i k i is the total strength of ties in the network, k i = (cid:80) j A ij is the weighteddegree (i.e., the strength) of the i th node, g i is the community to which i is assigned (andsimilarly for g j ), and δ ( g i , g j ) = 1 if i and j belong to the same community and 0 if they donot. In equation (2) we have defined a modularity matrix B with components B ij = A ij − k i k j m .Modularity evaluates the quality of community partitions, implying that partitions withhigher modularity are, by our conceptualization, more polarized. However, it remains forus to determine the community partition that maximizes modularity for each Congress.We call this the ‘maximum-modularity partition’, though strictly speaking no partition canever be proven to be the global optimum without computationally-prohibitive exhaustiveenumeration (modularity maximization is an NP-hard problem; Brandes et al. 2008). Asdiscussed in the online supplementary information, we consider a variety of computationalheuristics in our optimization of modularity. We also consider individual-level diagnostics associated with modularity: divisiveness can beused to identify the extent to which individual legislators potentially contribute to system-wide modularity (polarization), and solidarity can be used to measure the alignment ofindividual legislators to their communities. Calculating divisiveness and solidarity allow usto explore hypotheses about the relationship between the behavior of individual legislatorsand outcomes of interest (such as reelection rates).11athematically, the divisiveness | x i | of legislator i is obtained from (Newman 2006 a ) | x i | = p (cid:88) j =1 ( (cid:112) λ j U ij ) , (3)where p is the number of positive eigenvalues λ j of the modularity matrix B and the matrixelement U ij is the i th component of the j th (normalized) eigenvector. That is, x i is a p-dimensional vector (with j th element equal to (cid:112) λ j U ij ) and the magnitude | x i | of this nodevector measures the potential positive impact on aggregate modularity from legislator i . The divisiveness measure uses the roll-call adjacency matrices to estimate the potentialeffect that each individual legislator has on the aggregate polarization of his/her legislature,but it need not say anything about the alignment of that legislator’s voting behavior withthat of his or her own group. Estimating alignment requires us to compare the divisivenessmeasure with the associated community vector X k = (cid:80) i ∈ c k x i , where we have summed overall node vectors corresponding to legislators assigned to the k th community c k . We can thencalculate the solidarity cos θ ik , where θ ik is the angle between the node vector x i and thecommunity vector X k . When the solidarity is close to 1, the legislator and community arein strong alignment; when the solidarity is close to 0, however, the legislator is not stronglyaligned with his or her community (Newman 2006 a ).p Let us first consider the number of communities revealed by the modularity procedure.Theoretical models suggest that single-member districts should yield a two-party system(Duverger 1954, Cox 1997), so we expect most Congresses to achieve their maximum modu-larity when partitioned into two communities. However, we find three or more communities The divisiveness | x i | is known as ‘community centrality’ in the networks literature (Newman 2006 a ). From a networks perspective, one might wish to use the name ‘community alignment’ for the solidarity.
12n 35 of 109 Houses and in 67 of 109 Senates. We tended to obtain more communities whenmaximum modularity is low. Maximum modularities in Congresses partitioned into threeor more communities are on average 0 .
045 lower in the House and 0 .
066 lower in the Senatethan maximum modularities in two-community Congresses. These differences are both sig-nificant ( p < . t -tests. When we constrain our focus to those Houses andSenates in which the third-largest community is larger than the size difference between thetwo largest communities, we find three or more communities in 11 Houses and 31 Senates.We provide descriptive statistics for these Congresses in section 7.3 of the online supplementand a historical example from the 19th century in section 7.4.In figure 1, we compare maximum-modularity values to the NOMINATE-based measureused by McCarty, Poole, and Rosenthal (2007) and described in section 2. Following theadvice of Aldrich et al. (2004), we calculate this measure using two dimensions of DW-NOM. For reference, we include a calculation of the MPR measure using two-dimensions ofW-NOMINATE (which we hereafter call ‘W-NOM’) as well. For comparability, we rescaleboth modularity and the NOMINATE-based measures to lie in the interval [0 , < FIGURE 1 ABOUT HERE > The modularity values in figure 1 are consistent with several stylized facts about po-larization. Most notably, they capture the spike in polarization associated with the end ofReconstruction as well as the well-documented modern spike (McCarty et al. 2007). Theyalso show a lull, corresponding to the era of party decline during the 75th–95th Congresses(1937–1979) (Coleman 1996). Interestingly, the DW-NOM version of the MPR measuresuggests a much lower level of polarization over this period than does modularity or theW-NOM version. Further, the DW-NOM-based measure derives much of its visual impactfrom its limitation to post-Reconstruction Congresses. The modularity and W-NOM-basedmeasures show that modern-day polarization is high but not to a greater extent than what13eems to be the case in many other periods. The low-modularity period of the 75th–95thCongresses appears to be the exception rather than the rule.Another difference between modularity and the MPR measure is the year-to-year varia-tion. DW-NOM assumes that legislators always remain in the same voting bloc and allowstheir ideology to move in only one direction over time, resulting in a time series that issmoother than that for the modularity or W-NOM measures. As we discuss below, webelieve that allowing such year-to-year variation is informative.In addition to maximum modularity Q , we calculate party modularity P , which is themodularity obtained from the network partitioned so that legislators are assigned to groupsthat contain only members of the same party. In figure 2, we report P/Q , which representsthe relative contribution of formal party divisions to total polarization. In periods in whichpolarization is predominantly partisan, one finds that P ≈ Q . When P is substantiallyless than Q , however, community divisions other than party better explain polarization.The party partition captures the vast majority of the maximum modularity in all modernHouses, with the 85th–95th Congress period (1957–1979) serving as a notable exception.Party importance varies more in the Senate, where it oscillates from one Congress to thenext between the 67th (1921) and 75th (1937) Congresses and is again a smaller fraction ofmodularity during the 85th–95th Congresses. < FIGURE 2 ABOUT HERE > Although the DW-NOM MPR measure suggests a substantially lower polarization levelfor the 75th–95th Congresses than does the modularity measure, party modularity nearlyequals maximum modularity for the 75th–84th Houses before dropping off during the 85th–95th, suggesting that the importance of party did not start to wane until the 85th Congress The W-NOM version also suggests higher levels of polarization than maximum modularity for mostCongresses. We have no theory to explain this finding but stress that the W-NOM measure is not a significantpredictor of majority-party switches (see table 1) The existence of a disparity between party allegiance and voting behavior is unsurprising.Many studies show that parties have reorganized throughout history (Merrill et al. 2008).These realignments represent changes in the formal allegiances of members of Congress. Itis reasonable to assume that such changes are costly to politicians (Downs 1957 b , Cox &McCubbins 2005), so they are unlikely to be undertaken without substantial prior effort tosalvage the existing party order. As party bonds disintegrate, we reason that some legislatorsseek to preserve alliances while other (opportunistic) legislators seek new alliances that reflect(or perhaps help create) a different order (Riker 1986).The existing measures of polarization based on DW-NOM are ill-equipped to identifythese shifts for three reasons. First, they assume a party-system structure to orient theirlegislators in space, and this assumption might mask the importance of intra-party commu-nities. Second, DW-NOM is weighted dynamically, which constrains the spatial movement oflegislators over time to a single direction. This restriction allows one to identify ideal pointson a consistent spatial metric over time (Poole 2005). Cox & Poole defend this constraint bynoting many legislators have changed parties over their careers but that none have changedback (Cox & Poole 2002). This defense is justifiable if exogenously-defined groups, suchas parties, sufficiently capture group dynamics (Poole & Rosenthal 1997), but our resultssuggest otherwise. Third, these measures rely on strict assumptions about the nature ofideology (Downs 1957 b ) that might be more appropriate for fitting individual roll-call votesthan for examining the effect of group dynamics on formal party divisions. Regression results, which we present in section 4, provide some support for this finding in the House butnot in the Senate. .4 Changes in Group Dynamics One clear indicator of a formal power shift in Congress is a change in the majority party.When a new majority party is elected, one can normally point to major policy failure onthe part of the previous majority. One important way for a majority party to remaineffective—and for its brand to remain strong—is for its caucus members to coordinate ona policy agenda. The House and Senate majorities resolve their coordination problemsthrough various institutional means, such as the delegation of agenda control to leaders andthe appointment of party whips (Rohde 1991, Cox & McCubbins 2005). Willingness tocoordinate depends on a variety of forces, including electoral pressure, ideological cohesion,and career ambition (Aldrich & Rohde 2001).When party membership poses electoral risk, members hedge their bets by seeking extra-partisan coalitions. Modularity captures this dynamic, showing the emergence of third (non-party) communities and the evolution of party-dominated communities into more heteroge-nous groups. The less that communities in Congress reflect party labels, the more likelythat interest groups, party organizations, and ultimately voters notice the gap. If these po-litical actors support the party, then the legislator might be replaced and the party systemmight thereby be preserved. However, if they support the legislator, then he/she might ei-ther switch parties or attempt to bring his/her party closer to his/her district’s preferences.When party positions shift, we reason, it becomes more difficult for parties and voters tocoordinate on ideology. Thus, electoral volatility increases and changes in formal groups,such as majority party switches, become more likely.In order to test the ability of maximum modularity to predict changes in the majority-party in Congress, we examine the values of modularity in section 4. We then conductindividual-level analyses in section 5 to examine the ability of divisiveness and solidarity(which we defined in section 3.2) to predict the electoral fates of House members. In bothcases, we compare regressions using the modularity measures to similar specifications that16se NOMINATE-based measures as their independent variables.
We begin to explore the relationship between maximum modularity in Congress t and a ma-jority party switch in the next Congress ( t + 1) using locally weighted polynomial regression(LOESS) (Loader 1999). Our analyses demonstrate that changes in majority-party switchesare most common when polarization is moderate, and they are relatively uncommon whenpolarization is low or high. This non-monotonic relationship suggests that it is appropriateto include both linear and squared modularity scores in multivariate regressions.
We compiled a time-series data set that covers each of the 4th–109th Congresses. Thedata set contains both the key independent variable ( maximum modularity ) and the keydependent variable ( majority-party switches ) for each House and Senate. ‘Majority-partyswitches’ is a binary variable that takes the value 1 if a switch occurred as a result of theprevious election and a 0 if it did not. Using information provided in Kernell et al. (2009),we identified 27 switches in the House and 26 switches in the Senate. We control for economic indicators such as gross domestic product (GDP), consumerprice index (CPI), and national debt (as a percentage of GDP) (Historical Statistics of theUnited States, 2009). We also include indicator variables for divided government, midtermCongress, and Republican or Democratic majorities. We included the first two variables tocontrol for the impacts of presidential races on Congressional electoral outcomes and the The medium-modularity range is approximately [0 . , .
30] for the House, and [0 . , .
23] for the Senate(see figure 3). We show associated plots in section 7.5.1 of the online supplementary information. The accompanying economic data that we gathered were not available for the first three Congressionalsessions. We include a table of these switches in section 7.5 of the online supplementary information. t and retested the model for majority-party switches in the next Congress ( t + 1).To aid comparison with NOMINATE-based measures, we report similar specifications usingboth W- and DW-NOM-based MPR measures of polarization. MPR analyses are necessarilylimited to Congresses 46-109 (the time period over which DW-NOM is calculated), so weconducted modularity regressions over this time period as well. We present our results intable 1. < TABLE 1 ABOUT HERE > The regression results show a clear relationship between modularity and majority-partyshifts in the House. In all four specifications, modularity is significant and positive ( p < . p < . p < .
1) in only two of the four models, and the squaredterm approaches significance in only one. Neither the DW- nor W-NOM MPR measures aresignificant in any specification in either chamber.The existence of a non-monotonic relationship between modularity and House majorityswitches has important implications for the study of legislative organization and party dy-namics. With some caution, we offer some preliminary explanations for these results in thefollowing section. We refer to these results and explanations as our ‘partial polarization’hypothesis. 18 .2 Discussion
We believe that the instability of partially-polarized Houses might be driven by the strategicbehavior of legislators, candidates, and other partisans as they attempt to coordinate withlow-information voters. These dynamics are most easily explained by dividing Congressesinto three categories: those with low, medium, and high modularities. We observe that low-and high-modularity Congresses tend to have stable majorities, whereas the majorities inmedium-modularity Congresses are less stable.In low-modularity Congresses, communities tend to be weak and are presumably lessinformative to political elites and the general electorate. Recall from section 3.3 that Con-gresses with more than two communities tend to have lower modularity than those with twocommunities. If one imagines Congressional activity as a coordination problem, then low-modularity Congresses are those in which coordination takes place between different coali-tions on different issues and in which mechanisms to aggregate preferences within groupshave little power. In such an environment, coordination costs are likely to be high, andindividual legislators might see little benefit in group alliances (Olson 1965), which couldresult in committee rule (Shepsle & Weingast 1987) or gridlock (Binder 1999). Electoralinstitutions (Herrnson 2004) also give Congressmen the incentive to pursue particular bene-fits for their districts in order to win reelection (regardless of collective impact). In such anenvironment, coordination likely occurs through logrolling and the exchange of district-levelbenefits.High-modularity Congresses have the opposite problem: communities are well-definedand usually party-oriented. Presumably, legislators have solved their coordination problemby coalescing into voting blocs that reduce the costs of governing and improve the valueof ideological signals to the electorate. Such efficiency comes with a corresponding lossin voting freedom, as the electoral costs of defecting from a community increase. Donors,lobbyists, activists, and elites who have invested in the existing community structure might19e less willing to support defectors, which impairs a Congressman’s ability to fundraise anddecreases his/her chance of reelection. Defection might also muddle the ideological signal tovoters, which could in turn decrease turnout or encourage the consideration of challengers.Consequently, pressure to conform might explain the dearth of majority switches in theseHouses.Medium-modularity Congresses reveal environments that are subject to potential flux.Such Congresses might represent a highly modular environment that is in the process ofbreaking down or a poorly-structured environment in the process of consolidating. Whengroup structures exist but are not well-established, politicians have a strategic incentiveto develop and control stable communities that will convey more effective signals to voters.The strategic behavior of legislators, in turn, causes communities to fracture and reassemble;meanwhile, voters attempt to make sense of the more complex environment.We investigate the strategic incentives of individual legislators in the following section byusing divisiveness and solidarity. These concepts connect modularity and group dynamicsdirectly to the electoral fortunes of individual legislators.
We begin with insights that arise from our analyses in section 4.2 on the effect of themaximum-modularity value on changes in the majority party in Congress. Our ‘partialpolarization’ hypothesis suggests that medium levels of maximum modularity might leadto instability in Congressional blocs, with some alliances breaking down and others beingforged. At the individual-level, this instability should be reflected in the electoral successesand failures of legislators. Here we explore the relationship between maximum-modularity,its associated individual-level quantities (namely, divisiveness and solidarity), and reelectionoutcomes in the House. We start this exploration by conducting a series of two-dimensional20OESS regressions. In our first regression, we find evidence for an interactive effect between modularity anddivisiveness as they impact reelection. Divisive legislators in medium-modularity Houseshave the highest rates of electoral failure but are more successful when modularity is lowor high. In low-modularity Houses, we suspect that this arises because group solidarity isa less valuable cue for voters when group structures are weak. In high-modularity Houses,we suspect that divisiveness is only successful in combination with strong group solidarity,as groups are highly salient in these Congresses and members are likely to be penalized fordefection. In medium-modularity Congresses, the legislative environment appears to be morecomplex, as both Congressmen and voters have poorer information about the structure andsalience of communities. This results in coordination failures between Congressmen, parties,and voters (Downs 1957 b ), which in turn leads to lower reelection rates.We conduct a second regression to illuminate the impact of solidarity and divisivenesson reelection. We find that highly divisive Congressmen suffer in their electoral prospectsunless they also have high group solidarity, providing further tentative support for our ‘partialpolarization’ hypothesis. Significant numbers of legislators with both high divisiveness andhigh solidarity in a Congress are associated with high modularities (that is, they have ahigh potential contribution to modularity and that contribution is realized by their strongalignment with identified groups; see also the additional regressions in section 7.6 of the onlinesupplement). In the absence of such strong solidarity, the aggregate maximum modularitymay only reach the medium-modularity level; legislators in such Congresses possibly usevotes to form coalitions (yielding high divisiveness) but are not always successful in formingcohesive groups in line with the full body of their votes (i.e., low solidarity). Electoral failurein this case could stem from the loss of activist support (Aldrich 1995) or a coordination We show associated plots in section 7.6 of the online supplementary information. To aid interpretation, we report correlations between divisiveness and solidarity in section 7.6 of theonline supplementary information. We also report examples of the various legislator types. b ).From these regressions, we derive three testable hypotheses that lend support to ourbroader ‘partial polarization’ hypothesis. First, increasing divisiveness causes a decrease inreelection probability. Second, increasing solidarity increases reelection chances. Finally,the impaired electoral chances of highly-divisive legislators can be mitigated if their divisivebehavior is consistent with the voting behavior of their community. In other words, we expecta positive association between electoral success and an interaction between divisiveness andsolidarity (divisiveness × solidarity). We test these three hypotheses in the next subsection. In this section, we test our three hypotheses concerning reelection to the House of Represen-tatives. Our dependent variable is reelection to the House (1 for success, and 0 for defeat).We exclude legislators who do not participate in the general election. Our explanatory vari-ables are divisiveness , solidarity , and their interaction. We have rescaled divisiveness to theinterval [0 ,
1] to make the regression results easier to interpret (solidarity lies in this intervalby definition).We also include several control variables in our specifications. At the Congress level,we control for presidential election years and divided government using indicator variables.At the individual level, we control for party (indicators for ‘Democrat’ and ‘Republican’),preference extremity (absolute value of first-dimension DW-NOM), seniority (number ofCongresses served), party unity (as defined in Poole 2005), previous margin of victory (per-centage of votes), and district-level partisanship. We pool data for the 56th—103rd Houses(1899—1995) for our analyses. We require a model that can estimate appropriate standard errors for our explanatory District-level partisanship is estimated by multiplying the most recent Democratic Presidential vote(percentage) for a district by the party indicator variables. Covariate data were compiled by Keith Poole in collaboration with Andrew Scott Waugh.
P r ( y i = 1) = logit − ( α legislator i + α Congress i + β fixed i + (cid:15) i ) . (4)We evaluate four mixed-effects logistic regression specifications and report our results intable 2. In the first specification, we regress divisiveness and solidarity against the reelectionindicator. We find that divisiveness has a significant negative impact on reelection chancesand that group solidarity has a significant positive impact. In the second specification, wealso include the interaction of solidarity and divisiveness (Divisiveness × Solidarity). Thismodel maintains the finding that divisiveness is associated with decreased reelection proba-bility, and it also suggests that the combination of divisiveness and solidarity has a significantpositive impact on reelection. Although the sign of solidarity flips from positive to negative,the aggregate effect of solidarity must include the interaction term. At even moderatelylow levels of divisiveness (i.e., rescaled values of 0.2 or lower), the effect of the interactionterm exceeds the main solidarity effect, suggesting that most legislators benefit from in-creased solidarity with their community. The third and fourth specifications include theaforementioned Congress-level and individual-level controls, and they yield similar results. < TABLE 2 ABOUT HERE > The interaction variable projects the node vector onto the group vector of its associated community,thereby indicating the actual contribution to aggregate modularity that is made by assigning the legislatorto that specific group (Newman 2006 a ). .2 Discussion The negative influence of divisiveness suggests that legislators suffer a penalty for polarizingvotes unless they also exhibit high community solidarity, and that this is particularly truein high-modularity environments. We hypothesize that divisiveness without solidarity notonly disconnects legislators from needed elite support but also complicates the decisions ofrationally-ignorant voters. Conversely, legislators cannot be highly divisive while simulta-neously maintaining independence from a coalition. Only by appropriately balancing groupsolidarity and individual divisiveness do Congressmen maximize their chances at reelection.Our individual-level results support our Congress-level findings and show significant dif-ferences in the value of communities across different levels of polarization. In partially-polarized (i.e., medium-modularity) Congresses, legislators face complex environments inwhich their alliance choices are subject both to greater error and to greater risk. In such en-vironments, they must balance community cohesion with concerns for constituents, activists,and others who impact their electoral fates.
Researchers have long sought to separate the effects of party on voting behavior from elec-toral, interest-group, and other pressures. Prior studies have assumed the existence andstructure of parties (Poole & Rosenthal 1997)—or of alternate mechanisms such as com-mittees (Shepsle & Weingast 1987) or institutional veto players (Tsebelis 2002)—to deriveimplications for the structure of roll-call voting. These studies have tended to consider theirorganizational mechanisms in isolation, which our results suggest might be a mistake.In this paper, we have used the network-science concept of modularity to provide anovel measure of polarization in Congress. Using roll-call data, we calculated the commu-nity structures that maximize modularity for the 1st–109th Houses of Representatives and24enates. Such structure includes the membership of each community and also (via modu-larity) indicates the cohesiveness of communities. We argue that modularity offers a clearerand more parsimonious measure of polarization than existing measures that are based onspatial-modeling assumptions. The introduction of modularity and related measures to theanalysis of Congressional behavior has the potential to fundamentally alter the study ofgroup dynamics and partisanship in legislatures.We demonstrate the value of this modularity by demonstrating that there exists a non-monotonic relationship between modularity and majority party switches in the House, whichsuggests that ‘partially-polarized’ Congresses are more unstable than ones with either lowor high levels of polarization. Similar uses of NOMINATE-based polarization measures failto replicate this result. We further investigate the ‘partial polarization’ hypothesis usingdivisiveness and solidarity, which capture the individual-level impacts of legislative alliances,and we find that these measures are significant predictors of reelections in the House.Modularity provides a valuable and parsimonious benchmark measure of polarizationagainst which to compare alternate legislative orderings. By comparing the maximum mod-ularity of a Congress to modularities calculated either using party divisions or using anyother exogenously-determined partition, one might be able to identify the conditions underwhich particular structural arrangements succeed and fail. This, in turn, might help todisentangle the complex interplay of environmental, ideological, and institutional pressuresthat impact the structure of Congressional voting. Our results also suggest that communitystructure in Congress strongly influences the strategic incentives of political elites to preserveor subvert existing order, and that the value in pursuing a new order depends on the presenceof community structures that are neither too strong to break nor too weak to identify.25 cknowledgements
We thank Keith Poole for providing roll-call data via voteview.com (Poole 2011). Wethank Dan Fenn, Santo Fortunato, A. J. Friend, Wojciech Gryc, Eric Kelsic, Keith Poole,Amanda Traud, Doug White, and Yan Zhang for useful discussions. We also thank KevinMacon, Stephen Reid, Thomas Richardson, and Amanda Traud for use of their code. ASW,JHF, and MAP gratefully acknowledge a research award ( upplementary Information
In this section, we describe in detail the process by which we calculated adjacency matricesusing House and Senate roll-call data (which we obtained from voteview.com ) . For eachlegislative body, roll calls for a two-year Congress are encoded in an n × b matrix M . Eachmatrix element M ik equals 1 if legislator i voted yea on bill k , − n × n adjacency matrix A whose elements A ij ∈ [0 ,
1] represent the extent of votingagreement between legislators i and j . We define these matrix elements by A ij = 1 b ij (cid:88) k γ ijk , (5)where γ ijk equals 1 if legislators i and j voted the same on bill k and 0 otherwise, and b ij isthe total number of bills on which both legislators voted. Because perfect similarity betweena legislator and him/herself provides no information, we set all diagonal elements to be zero(i.e., A ii = 0). The matrix A thereby encodes a network of weighted ties between legislators,and the weights are determined by the similarity of their roll-call records in a single two-yearCongress.Following the guidelines of Poole & Rosenthal (1997) and Anderson et al. (1966), weconsider only ‘non-unanimous’ roll-call votes. A roll-call vote is classified as ‘non-unanimous’if at least 3% of legislators are in the minority. For modern Congresses, this implies that aroll-call minority must contain at least 4 Senators or at least 13 Representatives to yield a‘non-unanimous’ vote. This ensures that our data sets mirror those used by McCarty, Poole,and Rosenthal (2007), permitting more explicit comparison of our polarization measure with27heirs. In this section, we list and provide brief descriptions of the community-detection heuristicsthat we used to calculate maximum modularity partitions of the Senate and House networks.We also indicate references that provide complete specifications of the algorithms. We notethat it is important to consider several such algorithms when studying community structureby optimizing a quality function such as modularity (Fortunato & Barth´elemy 2007, Porter,Onnela & Mucha 2009, Fortunato 2010, Good, de Montjoye & Clauset 2010). In the results ofour investigation, we partition each network using various community-detection algorithms,and we use in each case the highest-modularity partition that we obtained. < TABLE 3 ABOUT HERE >< TABLE 4 ABOUT HERE > Heuristics 1–5 are all spectral methods of modularity optimization. Spectral methodsuse eigenvectors of the modularity matrix B that are associated to B ’s largest positiveeigenvalues. Heuristics 1–3 consist of three different implementations in which we use only theleading eigenvector (i.e., the one associated with the largest eigenvalue), so the final partitionis obtained from recursive steps involving partitions of some portion of the network into twosmaller pieces (Newman 2006 b , Newman 2006 a , Fortunato 2010). The difference betweenthe heuristics 1–3 arises from the use of different tie-breaking and fine-tuning procedures,which attempt to improve partitions between the recursive spectral partitioning steps. Inheuristics 4 and 5, we use the two eigenvectors associated with the two largest eigenvaluesof B (Richardson, Mucha & Porter 2009): Heuristic 4 allows only bi-partitioning steps,whereas heuristic 5 allows both bi-partitioning and tri-partitioning steps. Heuristic 6, known28s the ‘Louvain’ method, is a locally greedy modularity-optimization technique (Blondel,Guillaume, Lambiotte & Lefebvre 2008, Fortunato 2010). Heuristic 7 employs simulatedannealing to maximize modularity (Reichardt & Bornholdt 2006, Fortunato 2010).All of the results from heuristics 1–7 have been improved by subsequent application of aKernighan-Lin algorithm (Kernighan & Lin 1970, Porter, Onnela & Mucha 2009, Fortunato2010) (via the specification described in Newman 2006 a , Richardson, Mucha & Porter 2009).This algorithm takes the community partitions generated by the heuristics and conducts aseries of node swaps—moving nodes from one community to another—in order to find highermodularity values. This is a fine-tuning procedure that can be applied to the partitionobtained from any other method.Heuristic 8 is the walktrap algorithm (Pons & Latapy 2005). This algorithm starts bypartitioning the network into n communities that each contain a single node (i.e., a singlelegislator). It calculates a distance between each pair of communities and then begins merg-ing groups by taking random walks between them. After each merging step, one calculatesthe modularity score for the current partition. The algorithm finishes after n − k -means clustering pro-cedure (which divides a network into precisely k communities), but is more robust: oneordinarily needs to specify k in advance—which is inappropriate for our investigation—butthe PAM method allows one to determine an optimum number of communities based onmean silhouette width (Kaufman & Rousseeuw 1990).In tables 3 and 4, we provide summary statistics about the community-detection heuris-tics we used. Note that the modularity values obtained using different modularity-optimization29euristics vary little, especially in the House. Additionally, neither the cluster-analysis tech-nique nor the walktrap algorithm ever obtain the best result. Moreover, by using manydifferent computational heuristics, we more confidently sample the complicated modularitylandscape to find higher-modularity partitions to employ in our subsequent analysis. Ashas been discussed in a recent paper on modularity-optimization in practical contexts (i.e.,situations that consider real-world networks) (Good, de Montjoye & Clauset 2010), the useof multiple different optimization heuristics is an important protocol to follow. In table 5, we give descriptive statistics for Congresses in which community-detection identi-fies three or more communities. In the table, we list all Congresses in which the third-largestcommunity has at least as many legislators as the number in the largest community minusthe number in the second-largest community. For each community in such Congresses, wegive the size (i.e., number of legislators), mean divisiveness of legislators, and mean solidarityof legislators. < TABLE 5 ABOUT HERE > As we discussed in Section 3.3, modularity and its associated individual-level quantities canbe used to investigate party polarization even during periods of history that include morethan two dominant parties. In this section, we demonstrate the utility of modularity usingan illustrative 19th century example.In the early 19th century, the fledgling party system of the United States was go-ing through a transitional period. The existing party system, which pitted the dominant30emocratic-Republicans against a dying Federalist party, finally broke down in the 18thCongress (1823–1825) as the Democratic-Republicans broke ranks based on their affiliationswith national leaders (most notably, John Quincy Adams and Andrew Jackson). This re-sulted in a new period, reflected by partisan conflict between supporters of Adams andsupporters of Jackson, which lasted until the emergence of the Whigs and the Democraticparty in the 25th Congress (1837–1839) (Kernell, Jacobson & Kousser 2009). The timeseries of maximum modularity (table 1) captures this transition nicely and provides ev-idence that group structures began to change as early as the 14th Congress—before theDemocratic-Republicans divided into the aforementioned camps. The Adams-Jackson partysystem finally emerged in the 19th Congress, representing a majority-party switch in bothchambers.One can see using modularity that the breakdown of the Democratic-Republican Partyfirst becomes apparent in the transition from the 13th to 14th Congress (i.e., with the 1814election). The second largest negative shift in maximum modularity over the last 200 yearsoccurs during this transition in both chambers ( − .
152 in the House and − .
085 in theSenate). This decline is particularly interesting given that the country was experiencing aunified Democratic-Republican government and that the Democratic-Republicans held hugemajorities in both chambers. Some of this decline is likely due to the end of the War of 1812during the 13th Congress. With the war over, the Democratic-Republicans no longer neededto maintain a united front, which freed legislators to pursue alternate agendas.This breakdown yielded a maximum-modularity partition with four communities in theSenate (containing 14, 13, 12, and 5 Senators), and three communities in the House (con-taining 80, 71 and 44 Representatives), suggesting that Democratic-Republicans in the bothchambers were already beginning to explore alternate alliance structures. In table 5 we seethat the mean solidarity scores for the 14th Congress communities are 0.70, 076, 0.66, and0.82 in the Senate and 0.5, 0.58, and 0.57 in the House. Compare these to the 13th Congress,31n which both chambers have two communities, with sizes of 26 and 20 in the Senate and 117and 78 in the House, and mean solidarity scores of 0.63 and 0.74 in the Senate, and 0.78 and0.87 in the House. While mean solidarity appears not to vary across the 13th-14th Senates,mean solidarity in the House appear substantially lower in the 14th Congress than the 13th.The decrease in solidarity, coupled with the increase in the number of communities, suggestsa weakening of party control over the House in this period.In the House, this transition becomes further apparent in the 17th Congress (1821–1823), where we again identify three communities when the Democratic-Republican partyis nominally whole and maintaining large majorities in both chambers of Congress. Thesecommunities contain 78, 65, and 56 legislators (see table 5). By the 18th Congress, thedivisions within the Democratic-Republican party become formally acknowledged, as threecamps emerge behind the leadership of Adams, Jackson, and William Crawford. This formalrecognition results in a dramatic increase in the maximum modularity of the House comparedto the previous Congress, demonstrating the impact that formal party divisions can have onlegislator behavior.The same basic groups had emerged during the 17th House but had not yet formallyconsolidated into well-defined camps, as evidenced by their mean solidarity scores (0.39,0.40, and 0.49). After this consolidation, however, the cost to coordinate had become lower,with an accompanying increase in maximum modularity and the elimination of the thirdmajor community in the modularity-maximizing structure. In this case, the two largestcommunities contain 114 and 104 legislators. The elimination of the third communitydespite the emergence of a third party is especially interesting, as it suggests that two ofthe parties saw the institutional value in coordinating on roll-call votes in order to pursuetheir own agendas in a majority-rule institution. This is corroborated by the mean solidarity We technically observe 4 communities in the 18th House, but the third and fourth communities containonly 2 and 1 legislators respectively.
In this section, we present table 6, which summarizes the changes in majority party in theUnited States Congress from 1788-2002. These switches are used to generate the dependentvariable for our Congress-level regressions in section 4 of the main text. < TABLE 6 ABOUT HERE > In this section, we present plots of our Congress-level LOESS regressions (Loader 1999) tosupplement section 4 of the text. Observe the non-monotonic relationship between maximum33odularity and majority party switches. < FIGURE 3 ABOUT HERE > The plots and tables in this section supplement section 5 of the main text. Note in figure 5and table 7 that divisiveness, solidarity, and their interaction (divisiveness × solidarity) arepositively correlated but clearly provide different pieces of information. In addition to the LOESS regressions discussed in section 5, here we also examine the rela-tionship between divisiveness, solidarity, and the maximum-modularity level of Congress. Asexpected from the definitions of these quantities, we find that high-modularity Congressesare characterized by high levels of divisiveness and solidarity, as they are composed of com-munities that are highly structured and partisan. When either divisiveness or solidarityvalues dip, however, then medium-modularity Congresses become more likely. Instabilityin medium-modularity cases appears to be driven either by divisiveness without solidarityor vice versa. Intuitively, a legislator who is divisive but not solidary holds highly divisivepositions while nevertheless being assigned to a large community (i.e., low solidarity). Alegislator who is solidary but not divisive tends to side with his/her community on mostissues but holds broadly popular positions on other issues. < FIGURE 4 ABOUT HERE > < TABLE 7 ABOUT HERE >< FIGURE 5 ABOUT HERE > .6.3 Divisiveness and Solidarity Summary Statistics In this section, we provide additional details on the divisiveness and solidarity of individuallegislators. Table 8 gives summary statistics for legislators who are divisive (90th percentileor more) but not solidary (10th percentile or less), solidary but not divisive, both solidaryand divisive, and neither solidary nor divisive. In table 9, we give examples of legislatorswho fall into each of these four categories. < TABLE 8 ABOUT HERE >< TABLE 9 ABOUT HERE > anel A: House . . . . . . Election Year/Congress P o l a r i z a t i on M ea s u r e Maximum Modularity
MPR Polarization (DW-NOM, 2D)MPR Polarization (W-NOM, 2D)
Panel B: Senate . . . . . . Election Year/Congress P o l a r i z a t i on M ea s u r e Maximum Modularity
MPR Polarization (DW-NOM, 2D)MPR Polarization (W-NOM, 2D)
Figure 1: [Color online] Longitudinal comparison of modularity and MPR measures in theHouse (Panel A) and Senate (Panel B). Each measure has been rescaled to [0 ,
1] for vi-sual convenience. Maximum modularity lies in the interval [0 . , . . , . . . . . . . Election Year/Congress P a r t y M odu l a r i t y ÷ M a x M odu l a r i t y House
Senate
Figure 2: [Color online] Longitudinal plot of party modularity divided by maximum mod-ularity for both the House and Senate. The contribution of party to maximum modularityvaries considerably over time, particularly in the Senate, suggesting that polarization inCongress is usually, but not always, driven by formal party divisions.37 anel A: House . . . . . . Maximum Modularity P r obab ili t y o f M a j o r i t y P a r t y S w i t c h Panel B: Senate . . . . . . Maximum Modularity P r obab ili t y o f M a j o r i t y P a r t y S w i t c h Figure 3: [Color online] LOESS plot of maximum modularity in Congress t versus majorityparty change in Congress t + 1 for the House and Senate. Majority party changes are mostprobable during medium-modularity Congresses.38 anel A: Divisiveness, Modularity, Reelection . . . . . . Divisiveness M odu l a r i t y High
Reelection
Rate
Low
Rate
Panel B: Divisiveness, Solidarity, Modularity . . . . . . Divisiveness S o li da r i t y High
Modularity
Low
Modularity
Panel C: Divisiveness, Solidarity, Reelection . . . . . . Divisiveness S o li da r i t y High
Reelection
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Figure 4: [Color online] Two-Dimensional LOESS plots indicating the relationship betweenmodularity, divisiveness, solidarity, and reelection rates in the House of Representatives.39 anel A: Divisiveness and Solidarity
Divisiveness S o li da r i t y . . . . . . Panel B: Divisiveness and Interaction
Divisiveness I n t e r a c t i on . . . . . . Panel C: Solidarity and Interaction
Solidarity I n t e r a c t i on . . . . . . Figure 5: [Color online] Scatter plots illustrating the relationship between divisiveness, soli-darity, and their interaction for the House of Representatives. All variables are scaled to liein the interval [0 , a n e l A : H o u s e R eg r e ss i o n s M a x i m u m M o du l a r i t y - D i m e n s i o n a l W - N O MM a x i m u m M o du l a r i t y ( : ) - D i m e n s i o n a l D W - N O M ( : ) M a j o r i t y C h a n g e . ( . ) . ( . ) . ( . ) . ( . ) . . ( . ) . ( . ) . ( . ) . ( . ) M a x M o du l a r i t y . ( . ) *61 . ( . ) *98 . ( . ) *160 . ( . ) ** [ M a x M o du l a r i t y ] − . ( . ) * − . ( . ) * − . ( . ) * − . ( . ) * > C o mm un i t i e s . ( . ) . . ( . ) . . ( . ) *1 . ( . ) . W - N O M - D i m − . ( . ) − . ( . ) [ W - N O M - D i m ] . ( . ) − . ( . ) D W - N O M - D i m − . ( . ) . ( . ) [ D W - N O M - D i m ] . ( . ) − . ( . ) D i v i d e d G o v e r n m e n t − . ( . ) − . ( . ) − . ( . ) − . ( . ) M i d t e r m C o n g r e ss − . ( . ) − . ( . ) − . ( . ) − . ( . ) - y e a r ∆ G D P − . ( ) − . ( ) − . ( ) * − . ( ) - y e a r ∆ C P I . ( . ) . ( . ) . ( . ) . ( . ) - y e a r ∆ D e b t( % G D P ) ( I n t e r ce p t) − . ( . ) *** − . ( . ) ** − . ( . ) − . ( . ) − . ( . ) ** − . ( . ) **0 . ( . ) − . ( . ) O b s e r v a t i o n s P a n e l B : S e n a t e R eg r e ss i o n s M a x i m u m M o du l a r i t y - D i m e n s i o n a l W - N O MM a x i m u m M o du l a r i t y ( : ) - D i m e n s i o n a l D W - N O M ( : ) M a j o r i t y C h a n g e . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) . ( . ) M a x M o du l a r i t y . ( . ) . . ( . ) . . ( . ) . ( . ) [ M a x M o du l a r i t y ] − . ( . ) . − . ( . ) − . ( . ) . ( . ) > C o mm un i t i e s − . ( . ) − . ( . ) . ( . ) . ( . ) W - N O M - D i m − . ( . ) − . ( . ) [ W - N O M - D i m ] . ( . ) . ( . ) D W - N O M - D i m . ( . ) . ( . ) [ D W - N O M - D i m ] − . ( . ) − . ( . ) D i v i d e d G o v e r n m e n t . ( . ) . ( . ) . ( . ) − . ( . ) M i d t e r m C o n g r e ss . ( . ) . ( . ) . ( . ) . ( . ) - y e a r ∆ G D P − . ( ) − . ( ) − . ( ) − . ( ) - y e a r ∆ C P I . ( . ) . ( . ) . ( . ) . ( . ) - y e a r ∆ D e b t( % G D P ) ( I n t e r ce p t) − . ( . ) * − . ( . ) *5 . ( ) . ( . ) − . ( . ) − . ( . ) − . ( . ) − . ( . ) O b s e r v a t i o n s S t a nd a r d e rr o r s i np a r e n t h e s e s . S i g n i fi c a n cec o d e s ( p < ) : ***0 . , **0 . , *0 . ,. . T a b l e : L og i s t i c R e g r e ss i o n R e s u l t s : P a n e l A i s t h e H o u s e ; P a n e l B i s t h e S e n a t e . anel A: Fixed Effects (1) (2) (3) (4) Divisiveness − − − − − − − × Solidarity 10.030 (0.992) *** 10.874 (1.002) *** 10.990 (1.003) ***Presidential Year − − − − | Nominate (1st dim.) | − − − − − − − − × Dem. Pres. Vote 0.005 (0.004)Republican × Dem. Pres. Vote − p < ): *** 0.001, ** 0.01, * 0.05, . 0.1. N = 16891 Panel B: Random Effects (1) (2) (3) (4)
Legislator (ICPSR) 0.933 0.970 0.633 0.633(Number of Legislators = 3867) (0.966) (0.985) (0.796) (0.796)Congress (
Table 2: Mixed-Effects Logistic Regression Results for the 56th–103rd Houses. The depen-dent variable is reelection to the House. The key independent variables are divisiveness,solidarity, and their interaction. Note that divisiveness and solidarity individually have anegative impact on electoral prospects but that the interaction has a positive impact. Thissuggests that divisiveness might only be sustainable for Congressmen who are also strongmembers of a community. 42
Method House Senate1 Leading-Eigenvector Spectral 1 (Newman 2006 b , Newman 2006 a ) 76 (76) 49 (49)2 Leading-Eigenvector Spectral 2 (Newman 2006 b , Newman 2006 a ) 75 (1) 48 (4)3 Leading-Eigenvector Spectral 3 (Newman 2006 b , Newman 2006 a ) 75 (1) 45 (0)4 Two-Vector Bi-partitioning (Richardson, Mucha & Porter 2009) 81 (5) 63 (9)5 Two-Vector Bi/Tri-partitioning (Richardson, Mucha & Porter 2009) 90 (9) 69 (3)6 Louvain (Blondel et al. 2008) 91 (14) 88 (39)7 Simulated Annealing (Reichardt & Bornholdt 2006) 83 (3) 66 (5)8 Walktrap (Pons & Latapy 2005) 0 (0) 0 (0)9 PAM Cluster Analysis (Kaufman & Rousseeuw 1990) 0 (0) 0 (0)Mean Modularity Interval 0.0041 0.0166Mean Identical to ‘Maximum-Modularity’ Partition (minimum 1) 5.8807 4.6239 Table 3: Summary Statistics for Community-Detection Heuristics. In this table, we comparethe partitions that we obtained using the eight modularity-optimization heuristics and theone that we obtained using a standard cluster-analysis technique. Rows 1–9 give the numberof Congresses (out of 109) for which each measure finds the ‘maximum-modularity’ partition.In parentheses, we report the number of Congresses for which we subsequently used the re-sults of each method in our analyses. Row 10 gives the mean modularity interval for the ninemethods. Row 11 gives the mean number of heuristics that find the ‘maximum-modularity’community partition. These results suggest that community partitions in Congress are fairlyrobust to different heuristics for optimizing modularity.43
Method House Senate1 Leading-Eigenvector Spectral 1 (Newman 2006 b , Newman 2006 a ) 3–13, 16, 23–26, 28, 30–36,39, 41–43, 45–49, 51, 53–60,62–72, 74, 76–78, 82–84, 87–89, 93, 94, 97–99, 100, 102,104–109 2, 4, 7, 11–13, 16, 19, 22–24,26–28, 32, 36, 43–45, 47, 48,50–53, 55, 57–64, 66, 67, 82,84, 87, 90, 93, 94, 103–1092 Leading-Eigenvector Spectral 2 (Newman 2006 b , Newman 2006 a ) 79 1, 21, 33, 373 Leading-Eigenvector Spectral 3 (Newman 2006 b , Newman 2006 a ) 40 none4 Two-Vector Bi-partitioning (Richardson, Mucha & Porter 2009) 20, 22, 44, 61, 80 15, 42, 56, 75, 86, 95, 96, 100,1025 Two-Vector Bi/Tri-partitioning (Richardson, Mucha & Porter 2009) 2, 15, 17, 18, 21, 27, 37, 38,91 29, 39, 656 Louvain (Blondel et al. 2008) 14, 29, 50, 52, 73, 75, 81, 85,90, 92, 95, 96, 101, 103 3, 5, 6, 8, 10, 14, 17, 18, 20,25, 30, 31, 34, 35, 38, 40, 41,46, 49, 68-74, 77, 79–81, 83,88, 89, 91, 92, 97, 97–1017 Simulated Annealing (Reichardt & Bornholdt 2006) 1, 19, 86 9, 54, 76, 78, 858 Walktrap (Pons & Latapy 2005) none none9 PAM Cluster Analysis (Kaufman & Rousseeuw 1990) none none Table 4: This table lists the specific congresses for which each community-detection heuristicyielded the ‘maximum-modularity’ partition. These partitions were subsequently used forindividual-level analyses in section 5. 44 a n e l A : H o u s e C o n g r e ss C o mm un i t y C o mm un i t y C o mm un i t y C o mm un i t y S i z e D i v i s i ve n e ss S o l i da r i t y S i z e D i v i s i ve n e ss S o l i da r i t y S i z e D i v i s i ve n e ss S o l i da r i t y S i z e D i v i s i ve n e ss S o l i da r i t y ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( e - ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( e - ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . ( . ) . ( . ) ( . ) . 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W e r e p o rtt h e s i ze ( nu m b e r o f l e g i s l a t o r s ) o f e a c h c o mm un i t y w i t h t h e p e r ce n t ag e o f t o t a l l e g i s l a t o r s i np a r e n t h e s e s ) . W e a l s o r e p o rt m e a nd i v i s i v e n e ss a nd m e a n s o li d a r i t y w i t h s t a nd a r dd e v i a t i o n s i np a r e n t h e s e s . anel A: House Year Congress Old New
Panel B: Senate
Year Congress Old New
Table 6: Majority Party Switches in the U.S. Congress (1788–2002).46 ivisiveness Solidarity InteractionDivisiveness 1Solidarity 0.253 1Interaction 0.866 0.653 1
Table 7: Pearson correlations between Divisiveness, Solidarity, and their Interaction47 U n i q u e N % D e m% R e p % I n M a j o r i t y M e a n P a r t y U n i t y ( s . d . ) M e a nS e n i o r i t y ( s . d . ) M e a n V i c t . M a r g i n ( s . d . ) M e a n E x t r e m i t y ( s . d . ) % R ee l ec t e d D i v i s i v e n o t S o li d a r y . . . . ( . ) . ( . ) . ( . ) . ( . ) . S o li d a r y n o t D i v i s i v e . . . . ( . ) . ( . ) . ( . ) . ( . ) . S o li d a r y a nd D i v i s i v e . . . . ( . ) . ( . ) . ( . ) . ( . ) . N e i t h e r D i v . n o r S o l. . . . . ( . ) . ( . ) . ( . ) . ( . ) . A ll L e g i s l a t o r s . . . . ( . ) . ( . ) . ( . ) . ( . ) . T a b l e : Su mm a r y s t a t i s t i c s f o r l e g i s l a t o r s w h oa r e d i v i s i v e bu t n o t s o li d a r y , s o li d a r y bu t n o t d i v i s i v e , s o li d a r y a nd d i v i s i v e , a ndn e i t h e r s o li d a r y n o r d i v i s i v e . N i s t h e nu m b e r o f l e g i s l a t o r s i n t h ec a t e go r y , a nd ‘ un i q u e N ’i s t h e nu m b e r o f un i q u e l e g i s l a t o r s . A l s og i v e n a r e t h e p e r ce n t ag e o f D e m o c r a t s a nd R e pub li c a n s i n e a c h c a t e go r y , t h e p e r ce n t ag e w h o a r e i n t h e m a j o r i t y p a rt y , m e a np a rt y un i t y , m e a n s e n i o r i t y ( nu m b e r o f c o n g r e ss e ss e r v e d ) , m e a n i d e o l og i c a l e x tr e m i t y ( a b s o l u t e v a l u e o f s t D i m e n s i o n D W - N O M s c o r e ) , a nd t h e p e r ce n t w h o w e r ee l ec t e d t o t h e f o ll o w i n g H o u s e . ongress Name State District Party Divisiveness Solidarity Interaction Divisive but not Solidary (10 least solidary listed)63 LOFT G.W. NY 13 Dem 0.8264 0.0179 0.014877 WHIITEN MS 2 Dem 0.6724 0.0639 0.042966 SULLIVAN NY 13 Dem 0.5915 0.0881 0.052161 BROUSSARD LA 3 Dem 0.5748 0.0905 0.052082 IKARD TX 13 Dem 0.5422 0.1087 0.058975 CONNERY MA 7 Dem 0.6076 0.1233 0.074971 OCONNELL RI 3 Dem 0.5892 0.1260 0.074289 THOMPSON LA 7 Dem 0.5133 0.1331 0.068359 ADAMS H.C WI 2 Rep 0.8049 0.1333 0.107359 PRINCE IL 10 Rep 0.5092 0.1496 0.0762
Solidary but not Divisive (10 least divisive listed)95 LUKEN OH 1 Dem 0.0059 0.9502 0.0056101 CARR MI 6 Dem 0.0846 0.9488 0.0803101 CARPER T DE 1 Dem 0.1106 0.9531 0.105497 SMITH N. IA 5 Dem 0.1152 0.9518 0.1097101 ENGLISH OK 6 Dem 0.1154 0.9761 0.112794 VAN DEERLI CA 37 Dem 0.1193 0.9577 0.114395 SPELLMAN MD 5 Dem 0.1204 0.9462 0.1139101 SARPALIUS TX 13 Dem 0.1229 0.9828 0.120898 MCNULTY J AZ 5 Dem 0.1270 0.9788 0.1243100 CARR MI 6 Dem 0.1279 0.9759 0.1248
Solidary and Divisive (10 highest interactions listed)67 KITCHIN C NC 2 Dem 0.7742 0.9782 0.757356 ELLIOTT SC 7 Dem 0.7272 0.9477 0.689259 HOWARD W. GA 8 Dem 0.7055 0.9549 0.673758 POU E.W. NC 4 Dem 0.6931 0.9541 0.661358 RANDELL C TX 5 Dem 0.6944 0.9495 0.659459 MCCLAIN MS 6 Dem 0.6932 0.9504 0.658860 SMITH W.R TX 16 Dem 0.6901 0.9514 0.656659 CANDLER E MS 1 Dem 0.6701 0.9699 0.649959 ROBINSON AR 6 Dem 0.6851 0.9466 0.648558 WALLACE R AR 7 Dem 0.6691 0.9689 0.6482
Neither Solidary nor Divisive (10 lowest interactions listed)98 RINALDO W NJ 12 Rep 0.0863 0.0005 0.000095 RINALDO W NJ 12 Rep 0.1406 0.0008 0.000197 WHITLEY NC 3 Dem 0.1292 0.0017 0.0002102 RAY R GA 3 Dem 0.1435 0.00416 0.000694 DERRICK SC 3 Dem 0.0110 0.0533 0.000693 STUCKEY GA 8 Dem 0.0799 0.0084 0.000798 BENNETT C FL 2 Dem 0.0459 0.017308 0.000896 NELSON C FL 9 Dem 0.0488 0.0213 0.001095 GIBBONS FL 10 Dem 0.0515 0.0265 0.001497 BENNETT C FL 2 Dem 0.0430 0.0369 0.0016
Table 9: Examples of House members who are divisive but not solidary, solidary but notdivisive, solidary and divisive, and neither solidary nor divisive.49 eferences
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