1 Introduction
The increase in partisan polarization over the last five decades is one of the most consequential developments in American politics. Despite continued scholarly debate on the degree of policy polarization, if any, among the public [1, 9], there is little doubt today that the public is polarized in their affections towards the parties [10, 16, 4, 14, 19, 12]. Affective polarization expects Republicans and Democrats to view their own parties favorably and other parties with increasing disdain [15]. Accordingly, it seems, we have witnessed an increase in party loyalty and straightticket voting [3], as well as animosity towards candidates [8], not to mention a decrease in ambivalence, indecision and floating in elections [28].
However, a peculiar facet of affective polarization theory has recently emerged. A historic rise in individuals identifying as independents and a more recent preference among swaths of the public to avoid politics altogether as well as hold negative evaluations of partisanship in general are not easily explained by group attachment [18, 17, 6]. Instead, the movement in affective polarization has largely been in terms of increasingly negative evaluations of the other party, while the public’s feelings toward their own party has remained fairly stable [2, 8]. Such dynamics suggest an increased role of negative partisanship. Originally noted in studies of multiparty voting [22, 7, 20], the idea is that negative feelings about the other party (i.e., negative partisanship)—as opposed to positive feelings about one’s own (i.e., positive partisanship)—are dominant in political behavior and opinions.
While many prior studies investigate the role of positive/negative partisanship in affective polarization through surveybased measures using selfreports on feeling thermometers [13, 3] and multiitem scales [5], our approach provides a new method for measuring polarization with network data. Moreover, our novel model allows us to uncover and quantify affective polarization, as well as disentangle positive and negative partisanship within and between groups in social network interactions. This is an important contribution because the apparent distance between opposing political parties may be driven by different combinations of positive and negative forces within and between them. A hypothesis of negative partisanship, for examples, requires an understanding of the strength of identity to the ingroup relative to the outgroup. Using our approach, we can untangle the direction and context of partisanship to shed light on their relative strengths in social network behavior.
In this article, we use two different longitudinal social networks, where nodes represent individual social actors and edges represent the presence of a specified interaction between actors from the social media platforms Twitter and Reddit, to investigate the relative contributions of positive (attractive) and negative (repulsive) partisanship in the elite and the public, respectively. To do so, we develop a class of coevolving latent space network with attractors (CLSNA) models in which both the links between nodes and certain characteristics (or attributes) of nodes evolve over time, each in a way that impacts the other, for characterizing coevolutionary phenomenon in social behaviors such as flocking and polarization. This class of models falls within the subclass of latent space models, of which static versions are now especially welldeveloped, and progress on dynamic versions has been made in recent years. Existing dynamic latent space network models [23, 25, 26, 27, 24]
dictate simply that latent characteristics evolve in time in a Markov fashion and then links between nodes exist with probabilities driven by node distance in the underlying latent space. In contrast, in our framework the temporal evolution of latent characteristics can depend on network connectivity—hence, a coevolving network model. Such a feedback mechanism is embedded into the model via the presence of attractors (a concept fundamental to dynamical systems) at the latent (i.e., unobserved) level. This class of coevolving network models is, to the best of our knowledge, novel.
In order to capture, disentangle, and quantify key components of polarization that may drive social network behavior online in our setting, we define a twogroup CLSNA model with two attractors mimicking attractive and repulsive forces, which is a specific version of the aforementioned CLSNA model class. In this version of the model, each node in the network is assumed to fall into one of two groups with known labels, and their movements are influenced by their neighbors through specially defined attractors. We also account for persistence of links in the dynamic evolution of the network, which is a necessary control informed by social network theory.
This article is organized as follows: In the ‘Materials and Methods’ section we provide an overview of the two network data sets used in this study through an exploratory analysis. We also introduce the statistical network model proposed in this work, including a discussion of model behaviors and parameter interpretation. Adopting a Bayesian perspective, we develop a MetropolisHastings (MH) within Gibbs MCMC framework for posterior inference. In the ‘Results’ section we present the application of this model to the two data sets and quantitatively analyze the evolution of the key factors in affective polarization. Finally, we conclude the study with a discussion of the results and the model proposed.
2 Materials and Methods
In this section, we introduce data used in this study and outline our specific statistical modeling and inference approach. Data and code are available at https://github.com/KolaczykResearch/CLSNA2PartyPolarization.
2.1 Data
We construct and explore two different online longitudinal social interaction networks for evidence of polarization, one of the elite, via Twitter, and one of the public, via Reddit.
2.1.1 Twitter congressional hashtag networks
On Twitter, we collected tweets for every US congressperson with a handle from 2010 to 2020. This yielded 796 accounts, 843,907 tweets (after retweets and tweets with no hashtags are removed) and 1,252,455 instances of hashtag sharing. This data was used to construct a binary network for each year, wherein nodes correspond to sitting members of congress and edges between any two nodes indicates that the number of common hashtags tweeted by both members of congress that year was more than the average. The nodes for the resulting networks vary from year to year since some members of Congress were reelected, while others may have joined Twitter at a later stage, left early or both. In our analysis, we focus on 207 members of Congress who served in office and stayed active on Twitter over the entire course of our study from 2010 to 2020, among whom 131 are Democrats and 76 are Republicans.
2.1.2 Reddit comment networks
In the case of Reddit, we collected the full data on submissions and comments since the site’s inception through pushshift.io. We focus on active Reddit users whose ideologies can be identified from their comments or flairs with declarative patterns, e.g., “I am a Republican.”. We thereby selected 102 Republicans and 267 Democrats who made 1) at least one comment in each month during April 2015—March 2020, and 2) more than 50 comments in a year across political subreddits (e.g., ‘politics’, ‘Libertarian’, ‘PoliticalDiscussion’, ‘Conservative’, etc.). We then constructed longitudinal binary networks on these 369 active users for each oneyear period based on their interactions in comments, wherein an edge between two active users indicates that they commented on the same submission.
2.2 Exploratory Analysis
Figure 1 and Figure 2 show plots of the density of edges within and between Democrats and Republicans as they evolve over time for the Twitter and Reddit networks, respectively. In Figure 1 for Twitter congressional hashtag networks, while initially growing together in density, over the last four years, we notice divergent trends among the subsets of Democrats and Republicans, with Democrats increasing in their social media connections to each other and Republicans decreasing. While connections have increased overall, the presidential election year of 2016 brought about a drop in interparty connections. In Figure 2 for Reddit comment networks, while intraparty connections for the two parties share similar trends, with both of them initially increasing then decreasing, they differ in the extent of drop over the last three years, with more interaction ties dissolved among the subset of Republicans than Democrats. The interparty connections also have decreased following the election year of 2016.
This straightforward analysis is the first hint at a nuanced perspective on the polarization hypotheses from the Twitter congressional network data of the elite, and the Reddit comment network data of the public. Of course, this is only descriptive, and thus we turn to inferential methods with our proposed twogroup CLSNA model with attraction and repulsion.
2.3 A Twogroup Coevolving Latent Space Network with Attractors (CLSNA) Model with Attraction and Repulsion for Affective Polarization
Let be a network evolving in (discrete) time , with vertex set and edge set . For simplicity, assume that is fixed over time, of order . Let be the (random) adjacency matrix at time corresponding to
. Throughout the paper, we use capitals to denote random variables and lowercases to denote the realizations of them. We assume data come in the form of time series of adjacency matrices
, where if there is an edge between node and node at time and otherwise.To model the dynamic evolution of networks in connection with affective polarization, we use the latent space approach and add attractors in the latent level to capture the notion of attractive and repulsive forces specifically for the mechanism of affective polarization. Let be a timeindexed latent (i.e., unobserved) position for node in dimensional Euclidean space, and . Assume that each of the nodes of the network falls into one of two groups, i.e., Democratic and Republican, with known node label for node , where is the set of group labels. Formally, we define our model as follows:
(1)  
(2)  
(5)  
with initial distribution at time ,
(6)  
(7) 
Here is a similarity function, and and are the two attractor functions for node in . Specifically, we set , and define the two attractors for node as follows,
(8) 
(9) 
which are the discrepancies of from two local averages at time . These latter are the average of latent values of nodes in the following two sets, informed by a combination of group membership, and network connectivity:

, neighbors of node in the same group

, neighbors of node in a different group.
In this proposed model, we assume that each node lies in a dimensional Euclidean latent space, and the smaller the distance between two nodes in the latent space, the greater their probability of being connected, as in (1), (2). The expressions in Eqs. (8) and (9) capture the discrepancy between the current latent position of node and the average of that of its current neighbors in groups and , respectively. The corresponding parameters and represent attractive/repulsive forces, as we discuss below.
In contrast to the existing dynamic latent space network models [23, 25, 26, 27, 24] where the latent process is assumed to evolve over time in a Markov fashion with transition distribution, e.g., , and thus to drive evolution of the networks, as illustrated in Figure 3 (left), our CLSNA model allows the network connectivity to enter the temporal evolution of latent positions in the form of attractors, as illustrated by the blue arrow in Figure 3 (right). Specifically, in our model the evolution of latent positions for each node from to is modeled by the normal transition distribution in Eq. (55
), the mean vector of which depends not only on the latent position of itself at time
, but also on the two local averages, one from its neighbors in the same group, the other from its neighbors in a different group, as captured in (8) and (9). This is an important aspect of our model since it quantifies propensity for attraction/repulsion within/between two groups, and can help us understand how polarization/flocking and interaction coevolve. Strength of attraction/repulsion toward local averages is therefore summarized by the attractor functions and the associated parameters, the details of which are discussed in the later sections.We also include an effect for edge persistence, as illustrated by the red arrow in Figure 3 (right), which is a necessary control informed by social network theory. captures the impact of having an edge at time on whether or not there is an edge at time . For , the probability of an edge at time will be increased when one exists already at time , and hence the model explicitly captures a notion of edge persistence.
2.4 Model Behavior and Parameter Interpretation
Our model incorporates a level of baseline connectivity (), edge persistence (), two separate withingroup node attraction for the two groups ( and , respectively), betweengroup node attraction (), and a measure of volatility ( initially, and for ). A rich set of behaviors can be generated by varying these parameters. The three attraction parameters , and are of particular interest, in that by varying the sign they allow for the possibility of different combinations of attraction and/or repulsion in the evolution of the latent positions. The sign of these parameters encodes the direction of these forces – a positive sign indicates latent positions being pulled toward the direction of local averages, aka attraction, while a negative sign indicates being pushed toward the opposite direction, aka repulsion. For example, when , we can interpret this as two groups flocking together, while for but , the two groups are flocking separately— that is, we have a notion of affective polarization.
In Figure 4, we illustrate the behavior of latent positions and network connectivity in simulated models for the two scenarios, one reflecting two group flocking, and the other, polarization among the same two groups. For convenience of visualization, the latent space is taken to be onedimensional. We can see that initialized with different latent positions, the time courses for positions of the nodes in this network cluster together under flocking. But initialized together, they diverge into two clusters under polarization. At the same time, while the network becomes ever more densely connected over time under flocking, it evolves towards two fully connected subgraphs under polarization.
2.5 Quantifying the Extent of Inter/Intraparty Attraction/Repulsion and Edge Persistence
While the sign of each attraction parameters encodes attraction (positive sign) or repulsion (negative sign) within group 1, within group 2 and between the two groups, respectively, the absolute value of these parameters can be used to quantify the extent of inter/intraparty attraction/repulsion. Similarly, the value of can be used to quantify the relative importance of edge persistence. Accordingly, we take the effect size of interparty repulsion as a measurement for negative partisanship, and that of intraparty attractions for positive partisanship.
We can answer an array of questions regarding the proposed mechanism for affective polarization by investigating the values of these parameters. For example, does the phenomenon of affective polarization occur in our Twitter and Reddit data? If so, are negative feelings about the other party or positive feelings about one’s own the dominant factor that drives the interaction of affective partisan polarization among elites and the public online? These questions can be answered respectively by assessing whether ; and by comparing the values of , with . Essentially, positive (negative) indicates positive (negative) affect towards the outparty, and measures the extent of outparty favorability or disdain. Similarly for , , the signs encode inparty positive/negative affect, and the magnitude encode the extent of inparty favorability or disdain.
2.6 Bayesian Inference
The parameters in our model are natural and interpretable candidates for statistical inference. Given the hierarchical nature of our model, Bayesian inference based on appropriate interrogation of the posterior distribution makes sense. That is, given an observed network time series or, more specifically, a time series of the corresponding adjacency matrices , we can make inference of the latent positions and model parameters based on the posteriors
. A closedform expression for this distribution is not available, but we can use Markov chain Monte Carlo. We have implemented an adaptive MetropolisHastings (MH) within Gibbs MCMC scheme for posterior sampling. The implementation in nontrivial, as certain issues of scaling (regarding the volatility parameters
and latent positions) and rotational invariance (in the latent space) must be resolved. Details of the MCMC algorithm are given in the SI Appendix.3 Results
In this section, we fit our model to both Twitter data and Reddit data, with a latent space of dimension , and present the estimates for model parameters and latent positions. The choice of two dimensions is consistent with DWNOMINATE, one of the most popular established ideal point models of congressional ideology, for which 2 dimensions explain up to 90% of variation in roll call voting [21]. To evaluate how well the model explains the data used to fit the model, we obtain the insample edge predictions by plugging the estimates into the linkage probability function and compute the AUC (area under the ROC curve) [11].
3.1 Twitter Data Analysis
We first fit our model with timeinvariant parameters to the whole sequence of longitudinal networks in Twitter from year 2010 to 2020. The AUC values computed at each year are all above , and the overall AUC value computed across all times is , providing evidence that our model fits the data very well.
The summary statistics for the posterior distribution of model parameters are provided in Table 2
. The edge persistence coefficient indicates that the logodds that an edge appears increase by
if the same edge appeared in the previous time frame. The betweengroup coefficient is , demonstrating polarization across the sets of Republican and Democratic members of Congress. Additionally, the within group coefficient is for Democrats, and for Republicans, which means that while they have moved away from one another, they generally flocked to their own. Moreover, the Democratic members have a higher extent of intraparty attraction on Twitter, meaning that for Democratic members the positive feelings toward their own party were stronger than that for Republican members. Comparing the magnitude of betweengroup coefficients to the two withingroup coefficients, we can see that for Democrats the positive feelings toward their own party were stronger than the negative feelings toward the other party in the movement of affective polarization, while for Republicans negative partisanship was dominant.Figure 5 shows the posterior means of latent positions for each member of Congress in the twitter hashtag networks. The dynamics of the clustering of latent positions exhibits a clear consistency with the evolution of within/between party edge densities seen in Figure 1, with Democratic members of Congress (blue dots) tending to converge over time, while Republican members of Congress (red dots) initially converge then disperse following the presidential election year of 2016. The inflection around 2016 seen in the bottom panel of Figure 5 suggests that dynamics driving partisanship have changed.
Mean  2.809  1.500  0.493  0.105  0.155 

SD  0.022  0.018  0.026  0.025  0.014 
2.5% Quantile 
2.766  1.464  0.444  0.055  0.183 
97.5% Quantile  2.850  1.537  0.543  0.153  0.127 
So far, we have seen that the CLSNA model is quite powerful in terms of revealing polarization in social network interactions, as well as disentangling and quantifying the two sides of polarization: positive and negative partisanship. These results motivate questions about the dynamics of the relationships uncovered above. In particular, given a host of major political, social and economic events over the past decade, can our model help us to pinpoint changes in polarization and edge persistence over this period?
In order to confirm and quantify change in attraction, repulsion and edge persistence, we fit a series of models that allow a single changepoint to vary from 2012 to 2019. Specifically, for each choice of changepoint, we parameterized our model separately within the two corresponding subperiods of time, thus obtaining a set of parameter values up to the given changepoint, and similarly another set of parameter values after the changepoint. The resulting eight fitted models with different changepoints were compared through deviance information criteria (DIC) [29], and the one with the lowest DIC value was selected (the DIC values for all competing models are provided in SI Appendix). Our modeling identified 2015 as the year in which the network relationships changed the most. The AUC values computed at each year for this model are all above , and the overall AUC value computed across all times is
. From this model, we obtain the posterior means and 95% credible intervals for the parameters
, , and , for each of the two time periods 20102014 and 20152020.Edge persistence appears to be fairly stable in the two time periods (shown in Figure 1 in the SI Appendix). Figure 6 illustrates the evolution of withingroup attraction/repulsion for Democrats, for Republicans, and betweengroup attraction/repulsion . The betweengroup coefficient (yellow bars) is negative in both time periods, although its magnitude increases a bit (mean increase = .027, SE= .051, P(increase0)=0.701) in the second time period from 2015 to 2020. This suggests polarization across the sets of Republican and Democratic members of Congress appeared throughout the past decade, with some indication that it started to rise in 2015.
The withingroup attraction coefficients for Democrats (blue bars) remain fairly large for the two time periods, albeit with a slight drop (mean decrease = .170, SE = .053) in the second period, while those for Republicans (red bars) exhibit a steeper downward trend falling from positive to negative. That is, for Democratic members of Congress the positive feelings toward their own party have remained fairly strong, even after 2015 and during unified government under the Trump administration. For Republicans, however, the positive feelings toward their own party are weaker (mean difference = .174, SE = .087) than they are for Democrats prior to 2015. Most intriguing, perhaps, Republican inparty feelings became negative in 2015. That is, during the Trump administration, Republican members of Congress not only remained opposed to Democrats, but also grew in opposition to their own, i.e., a decrease in strength of ingroup affect.
By comparing the magnitude of withingroup coefficients (blue or red bars) with betweengroup coefficients (yellow bars), we find that for Democratic members of Congress positive partisanship dominated the entire time period of study from 2010 to 2020. Democrats favorability of their own was a binding feature of their polarization. In contrast, for Republican members of Congress negative feelings about the other party started to dominate positive feelings about their own in 2015 (mean difference = .075, SE = .028). Indeed, Republicans’ feelings towards their own party became negative as well. Although not explicitly tested, the evidence here suggests that Trump’s extensive appearance in social media and candidacy declaration either caused or occurred in conjunction with the takeover of negative partisanship for Republican members of Congress. Whether due to long evolving attitudes within members of Congress or concurrent political trends, or simply reactions to the new presidential candidate, from 2015 to 2020 Republican members of Congress were defining their online partisanship more in terms of their opposition to Democrats than in support of their own.
We have so far restricted to only one changepoint. We could of course continue this analysis with more than one changepoint. For example, we have run a series of models with two changepoints chosen between 2012 and 2019, and selected that model with the lowest DIC value, which places changepoints at year 2014 and 2019. However, the relative improvement of this model over that with a single changepoint is quite modest. See SI Appendix for details.
3.2 Reddit Data Analysis
In this section we carry out the same line of analysis on the Reddit comment networks for the public. Recall that the Reddit data collected are from April 2015 to March 2020. Each network constructed represents the interaction during a oneyear period from April of a given year to March the year after, and hence there are in total five networks constructed. We fit a model with a single set of parameters for the entire 20152020 period. In addition, we fit a series of models with a single changepoint and selected the one with lowest DIC value, which places that changepoint at the year 2018, three years after the changepoint for elites. Again, the models appear to fit the data quite well, although arguably slightly worse than in the case of the Twitter networks (e.g., with AUC values for the best fitting changepoint model computed at each oneyear period above and overall AUC values above ).
Table 4 and Figure 7 show the results from fitting without a changepoint, analogous to Table 2 and Figure 5 for the Twitter data. Figure 8 displays the evolution of the within group coefficients for the two groups and the betweengroup coefficient, in analogy to Figure 6 for theTwitter data. (Similarly, Figure 2 in the SI Appendix displays the evolution of edge persistence, in analogy to Figure 1 in the SI Appendix.)
Mean  3.079  0.937  0.748  0.401  0.128 

SD  0.011  0.010  0.032  0.041  0.027 
2.5% Quantile  3.056  0.917  0.686  0.321  0.182 
97.5% Quantile  3.101  0.957  0.811  0.482  0.075 
Some conclusions regarding evolution across the two time periods: 1) edge persistence increased (mean increase = .165, SE = .020); 2) betweengroup repulsion was present, demonstrating polarization across the sets of Democratic and Republican users of Reddit, though with some evidence that such polarization was mitigated (mean decrease = .023, SE = .054, P(decrease0)=0.668) in the second time period starting in 2018; 3) while the two groups have moved away from one another, both experienced positive partisanship (withingroup attraction) and became less concentrated over time, as both groups experienced a decline (Democrats: mean decrease = .291, SE = .066; Republicans: mean decrease = .213, SE = .082) in withingroup attraction in the second time period; and 4) positive partisanship dominated the entire time period for both Democrat and Republican users of Reddit. The latter finding is particularly notable, since it suggests different polarization trends among the public than what we found above among members of Congress. Though they are different social media platforms and we have a shorter timespan on Reddit, the consistent dominance of positive partisanship for both Republicans and Democrats among the public and the dominance of negative partisanship among Republican elites over the same period suggests a disconnect between elites and the public, an early focus of debate in the polarization literature [9, 1].
4 Discussion
We develop a twogroup coevolving latent space network with attractors (CLSNA) model for characterizing the mechanism of affective polarization using dynamic social networks. This model incorporates the effects of both attraction and repulsion by specifying appropriate attractor functions to explain the factors driving interactions of polarization. This model may be viewed as a type of causal modeling framework, specifically designed to combine dynamical systems from mathematical modeling with principles of hierarchical statistical modeling. The former allows us to incorporate precise notions of attraction/repulsion relevant to polarization, while the latter permits principled and computationally tractable inferences in the form of statistical estimation, testing and prediction.
While we focus on the context of polarization with the twogroup version of CLSNA model, our proposed class of CLSNA models is a flexible framework which may incorporate a variety of attractor functions, making it general and quite broadly applicable to other coevolutionary social dynamics where behaviors and beliefs impact social interactions, and vice versa. One limitation of our model, as implemented here, is that we assume the node set is fixed over time, which restricts our focus on individuals who are active for the entire period of study. Those who come and go and stay active for only a certain period of time, which is common in practice, is not currently accounted for in our model. It is an interesting subject for future research to design dynamic network models allowing for varying set of nodes.
In this article, we focus the application of our model on two online longitudinal social networks, one of the elite via Twitter for Congress, and one of the public, via Reddit. Our model has captured, disentangled and quantified the two key aspects of affective polarization, positive and negative partisanship, as well as a concept in social network theory, edge persistence. Our results show that for members of Congress active on Twitter polarization across the two parties appeared throughout the past decade. For Republican members of Congress, negative feelings about the other party began to dominate feelings about their own in 2015, while feelings about their own also became more negative at this time. Thus, among Republican members of Congress we find that increasing disdain for the opposing party is not necessarily accompanied by strong inparty attachments. In fact, withinparty forces for Republican members of Congress became negative after 2015. In contrast, for Democratic members of Congress positive partisanship was strongest throughout the entire period of study. We also find evidence of affective polarization among the public on Reddit. However, here positive partisanship dominated the full length of study for both Democrats and Republicans. Thus, the results provide only select support for the increasing role of negative partisanship in polarization. In all, through the modeling and analyses of social media data of both the public and the political elite in the US, this work provides new insights into the nature and presence of affective polarization, as well as how positive and negative partisanship play roles and evolve in driving polarization.
Acknowledgement
We are very grateful to Luis Carvalho and Herbert Weisberg for their valuable insights and discussions. This research was supported by ARO award W911NF1810237, NSF DMS2107856, and NSF SES2120115.
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