Network structure, nationalization and mass partisan polarization

Partisan polarization in the United States is one of the most studied political phenomenon in recent decades. And it manifests in quite a few different ways. Republicans and Democrats increasingly dislike each other, a process known as affective polarization [10,11]. And they increasingly disagree on important policy topics such as immigration, climate change, abortion, LGBT rights, gun control and the size of welfare state [12,13]. This paper focuses on mass partisan ideological polarization, the increasingly divergence in ideological lining between Democrats and Republicans at the population level [14,15].

Computational modeling of polarization is rooted in opinion dynamics and cultural dissemination. [16] demonstrated that local convergence need not imply global consensus. When agents are more likely to interact with similar neighbors and then become more similar through interaction, local homogeneity can coexist with global diversity. This model is not a direct model of U.S. partisanship, but it establishes a key insight: polarization can emerge from decentralized interaction rather than from top-down coordination.

Bounded-confidence models extend this logic by allowing agents to interact only when their opinions fall within a threshold. In the Hegselmann-Krause model, agents update toward the average opinion of those within their confidence bounds [17]; in the Deffuant-Weisbuch model, randomly paired agents adjust toward one another only when their opinions are sufficiently close [18]. These models can generate consensus, multiple opinion clusters, or fragmentation depending on the confidence threshold and initial opinion distribution. They are especially useful for theorizing how exposure may fail to moderate attitudes if citizens discount sufficiently distant views.

However, bounded confidence alone explains clustering more than it explains polarization. To offer a better account of polarization, later models introduced contrast effect, negative influence, biased assimilation, or affective mechanisms. [19] modeled attitude change through assimilation, contrast, and indifference, showing how uniformity, bi-polarization, and pluralism can emerge from a small set of interaction rules. [20] showed that weak ties and small-world shortcuts do not automatically integrate a society; depending on micro-level influence assumptions, bridges can instead intensify cultural polarization. [21] demonstrated that homophily alone is insufficient to polarize a DeGroot averaging process, but homophily combined with biased assimilation can produce polarization. [22] pushed opinion dynamics toward affective and emotional mechanisms by modeling opinion change as a function of information and emotional arousal.

The network dimension of partisan polarization begins with the homophily effect. Similarity brews friendship, social and partisan networks [23]. In politics, homophily can operate through geography, religion, ethnicity, and partisan identity. Homophily matters because it changes both the information flow and the social cost of taking a particular position on a policy issue. When political disagreement is rare within a local network, out-party views are less likely to be perceived as legitimate disagreement and more likely perceived as an ideological threat. Digital platforms complicate this process. [24] found that Facebook users’ cross-cutting exposure depends on friend networks, ranking algorithms, and individual clicking choices, with social ties and user choices playing a major role. [25] showed that Twitter following networks can be used to infer ideological positions and that online political networks exhibit substantial ideological structure. Using a field experiment on Twitter, [26] found that exposure to opposing partisan elites can sometimes increase polarization rather than reduce it. These findings caution against a simple “more exposure equals more moderation” assumption. [27] further warns that the Internet alone cannot explain the full demographic pattern of rising U.S. partisan polarization.

Nationalization of politics refers to the increasing alignment of local, state, and national political behavior around the same national partisan cleavage and political agenda. In a more localized party system, state and local candidates could cultivate reputations, coalitions, and issue agendas that partially diverge from national party brands. In a nationalized political environment, local elections are increasingly interpreted through national partisan identities and conflict. [2] argues that American political behavior has become increasingly nationalized: voters’ evaluation of politics has become less anchored in local contexts and more organized around national parties, national media, and national political competition. [28] added a geographic dimension by showing that left-lining parties, including the U.S. Democratic Party, have become spatially concentrated in cities, while right-lining parties often win more efficiently across suburban, exurban, and rural constituencies. The result is not simply ideological sorting, but partisan geographical sorting [29].

Nationalization of politics can contribute to polarization through at least four mechanisms. First, uniform national cues reduce local variation. Citizens in different places receive similar elite messages, campaign framing, and partisan narratives. Second, nationalization turns the out-party into a large and morally charged national coalition rather than a set of neighbors with locally familiar interests. Third, national parties bundle policy issues into coherent partisan packages, making disagreement on one issue more likely to imply social distance across many related issues. Fourth, nationalized media and elections increase the salience of national partisan conflict even in local contests, weakening the capacity of local identities to cross-cut national party identity. In modeling terms, nationalization of politics is a parameter governing the relative weight of national versus local cues in opinion and ideological position updating.

Existing account of the link between nationalization of politics and partisan polarization is mostly descriptive and generic in nature. It lacks detailed investigation of the connections between polarization and nationalization and how nationalization of politics contribute to partisan polarization through different network structures. If individuals form and sever partisan ties based on ideological proximity or shared neighborhood, then the same national cue can have different effects on the distribution of individuals’ ideological positions depending on whether it transmits through a sparse cross-cutting network or a clustered small-world network. This paper addresses these unanswered questions from the angle of social network structure. That is, we look at how the association between nationalized politics and polarization manifests differently through different network structures. This analytical approach offers a meso-level view of how partisans’ micro-level behavior results in macro-level changes and enriches the existing literature on nationalized politics and partisan polarization.

Toward integrated modeling of nationalization, network structure, and polarization

More specifically, we examine the following network types: ring network, Erdos-Renyi random network, small-world network, and scale-free network. The ring network is a graph where nodes are arranged in a circle and connected to a fixed number of nearest neighbors. Ring network is vertex-transitive, meaning every node is structurally identical to each other, has the same number of edges, and any node can be mapped to another by a symmetry of the network. Given a ring network with fixed size, the complexity of the network can be increased by adding more edges to the network. And the network diameter scales linearly with the number of nodes. A water supply system where households are connected strictly linearly to their neighbors is a ring network in action. The Erdos-Renyi random network [30] is a network with N nodes where every possible edge exists with a fixed probability p. The network is fully defined by these two parameters: N and p. The total expected number of edges is equal to . The degree distribution follows a binomial distribution with parameters . The clustering coefficient is equal to p. Increasing p will increase the density and connectivity of the network. A real world example is random chance-based encounter of strangers at a massive festival.

The small-world network (Watts-Strogatz model) is characterized by both high local clustering and short average path length, essentially bridging a regular lattice and a random network. It is commonly seen in social network, biological network and infrastructure network. Modeled with the Watts and Strogatz model [31], it is defined by the total number of nodes N or size of the original lattice, the neighborhood within which nodes of the network will be connected, and the rewiring probability p, the probability that an edge connecting two nodes in the regular lattice is disconnected from one end and reconnected to another randomly selected node. The scale-free network (used interchangeably with preference attachment network) is one where edges of node follow a power-law distribution. That is the network’s node degree distribution follows , where many small-degree nodes co-exist with a small number of high-degree nodes. As the scaling exponential of degree-distribution, governs the “rich-get-richer” rule and determines how massive the power hubs become. Modeled with the Barabasi-Albert model [32], the complexity of the network increases with the number of edges initiated for each new node added to the network. Social media is a prime example of scale-free network where a handful of celebrity accounts or massive content creators hold the vast majority of all connections while most users share a tiny fraction of the network. Fig 1 visualizes the four network types.

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Fig 1. Four Network Types.

https://doi.org/10.1371/journal.pcsy.0000121.g001

Polarization process

For mass partisan polarization, we model it as an iterative process where in each time step, individuals of different partisanship interact with each other, and through these interactions, their ideological positions change towards either left or right of their current positions. Building upon existing models on mass partisan polarization [33–35], here we model the partisan ideological polarization process across different types of social network and different nationalization level of political interactions.

Initial state

Individuals start with ideological positions that are normally distributed with mean 0 and standard deviation 0.2, and range between -1 and 1 (plugging in liberal conservative scale values from surveys such as the American National Electoral Studies as the initial ideological positions will give us substantively identical results in terms of model convergence and influence of model parameters). A value above 0 indicates a liberal Democratic position while a value below 0 indicates a conservative Republican position. Individuals are spread across networks of different types representing the social spaces they are involved in. For simplicity, we consider a community of 100 individuals with each one occupying one node in a network. Each individual’s neighborhood is the individuals that have an edge directly connecting her.

Mass partisan polarization depicts a process where individuals interact with both co-partisans and out-partisans. And as a result of these interactions, their ideological positions move either left or right of their current ideological positions. The systematic patterns in individual ideological position change over time, for example, individuals moving closer to the interlocutor’s ideological position when interacting with people within her ideological tolerance threshold and moving farther away from the interlocutor’s ideological position when interacting with people outside her ideological tolerance threshold, result in partisan ideological polarization at the mass level.

We consider three important parameters when modeling the polarization process: nationalization of partisan interactions , the extent to which communications among partisans are national in nature and increasingly involve individuals outside their social network neighborhood; tolerance threshold for ideological distance, ; and responsiveness, . Tolerance threshold for ideological distance refers to the distance in ideological position above which a randomly selected individual moves away from the interlocutor’s ideological position when interacting and below which she moves closer to the interlocutor’s ideological position when interacting. The existence of a threshold that determines the direction of change in interpersonal interactions has been confirmed by both the bounded confidence models discussed above [17,18] and the social judgment theory on latitudes of acceptance, non-commitment and rejection [36]. A smaller tolerance threshold will intensify the polarization process as partisans are less tolerant of ideological difference. Responsiveness or step size refers to the proportion of the ideological distance a randomly selected individual moves closer to or further away from the interlocutor’s ideological position when interacting. A larger responsiveness will accelerate the polarization process as partisans make a larger ideological change in each interaction with others.

Interaction rules for polarization process

We model a polarization process where ideologically like-minded individuals become closer in their ideological positions when interacting with each other while ideologically dissimilar individuals become further apart when interacting [33]. As shown in previous studies [15,34], the more partisan interactions are online and not bounded by neighborhood proximity and geographical distance, the less face-to-face interactions and a higher level of nationalized partisan interactions and a higher exposure to salient partisan media and news contents online, the more intense the partisan polarization process becomes.

The evolution of ideological positions happen within a finite population of N agents over discrete time steps t. The population is embedded in a static network G = (V, E), where V represents the set of agents and E represents fixed structural connections (neighbors). At each step t, a randomly selected agent i interacts with its connections consisting of its neighbors and a random sample of non-neighbors. Let be the set of neighbors of i with degree , and let be the set of non-neighbors, then the set of non-neighbors interacting with i is , where is the nationalization parameter. Then, the combined pool of interlocutors for i is: .

Agent i has ideological position at time t and the ideological distance between i and interlocutor j is . If the interlocutor’s ideological position is within the tolerance threshold , the agent is attracted; otherwise, the agent is repulsed:

(1)

[33], where is the step size or responsiveness, the fraction of the ideological distance that is reduced or increased after an interaction. In addition, if the interlocutor j is a neighbor of i () and is of the opposite partisanship, then, their ideological distance will be reduced a fraction based on the intergroup contact theory [37,38]. That is:

(2)

Finally, the updated ideological position is clipped to the defined domain of to determine the state at t + 1:

(3)

As an illustration, if an individual’s ideology is 0.4 and the tolerance threshold is 0.25, the actor will move closer to anyone with ideological position between 0.15 and 0.65 when interacting. If responsiveness is 0.06 and the person interacts with another person who has an ideological position of 0.6, the person will move from its current position at 0.4 to the other person’s position at 0.6, and has a new ideological position of . If the other person has ideology 0.1, which is outside the ideological tolerance threshold, the person will move away from the other person and has a new ideological position of .

The nationalization parameter determines the size of interlocutors an individual interacts with that are not neighbors relative to her neighborhood size. The bigger is, the more an individual interacts with non-neighbors online. When , an individual interacts with the same number of non-neighbors as the number of neighbors she has; and when , an individual only interacts with her neighbors. , therefore, governs the relative proportion of global versus local interlocutors for each person. It does not influence the overall connection probability or interaction density while it does influence the geographical scope of information dissemination with higher corresponding to more global rather than local information dissemination. Table 1 lists the default parameter values for the polarization process. The default values of closely match values used in the existing literature [33]. A tolerance threshold of is one eighth of the maximum possible ideological distance while a step size of means 8 percent of the ideological distance between two interlocutors. We show results for N = 100, 200, 400, 800 below.

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Table 1. Default Parameter Values of Polarization Process.

https://doi.org/10.1371/journal.pcsy.0000121.t001

We measure polarization as the variance of ideological positions , where is the population mean [33]. The initial polarization level is 0.04 as individuals’ starting ideological positions follow a normal distribution with standard deviation 0.2. The lowest possible polarization level is 0 when all individuals have the same ideological position and the highest possible value is 1 when half of the individuals are at the liberal extreme position of 1 and the other half are at the conservative extreme position of -1. As an example, Fig 2 shows the small world network and its distribution of ideological positions at iterations 0, 5000 and 10000 with rewiring probability 0.5 and nationalization of partisan interactions 0.5. We define convergence to polarization not as the cessation of movement in ideological position, but as the stabilization of the macroscopic variance. That is the population is said to have converged to a polarization state at time if, for a defined time window and a stability threshold :. In simple terms: Convergence occurs when the rate of change (slope) of the polarization curve effectively flattens to zero, indicating that the forces of attraction and repulsion have reached a dynamic equilibrium.

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Fig 2. Small world network of partisans (Red for Republicans and Blue for Democrats) with rewiring probability 0.1 when nationalization of partisan interactions is 0.5.

https://doi.org/10.1371/journal.pcsy.0000121.g002

Calibrating time in the polarization process is through the unit of agent activations called step, and we present results using the unit of iteration. In one iteration, an individual interacts once with each of her neighbors and same number of non-neighbors. Suppose an individual has one interaction with another person either online or offline in one day, and she has 8 neighbors, an iteration would roughly equal 16 days in the polarization process. And the mass partisan polarization process starts to converge after 9 years with default model parameter values (the time taken to convergence in years is for a population of 100 individuals).

Simulation experiments design

To disentangle the complex interplay between network topology and information exposure mechanisms, we implemented a comprehensive set of computational experiments using a parallelized computing framework. The technical implementation was designed to execute a full factorial sweep across three primary dimensions: network topology, structural density, and the degree of nationalized partisan interactions (). For each of the four canonical network models, Erdős–Rényi (Random), Watts–Strogatz (Small-World), Barabási–Albert (Preferential Attachment), and Ring network, we vary both the network size (N, 100, 200, 400, 800) and average degree (). And for the small world and preference attachment networks, we also vary the rewiring probability (0.01, 0.05, 0.2) and the scaling exponential of degree distribution () respectively. And, the nationalization parameter (with an interval of 0.25) was varied to control the probability of agents interacting with non-neighbors outside their local neighborhood, thereby, simulating the transition from strictly local social interactions to nationalized information consumption and exchange. To ensure valid comparison, all analytical scenarios for each network type have identical initial network distribution of partisanship and ideological positions. And to isolate the relationship between network topology and polarization, we only make comparisons between different networks with identical network size and average degree. To ensure statistical robustness against stochastic initialization effects and randomness, for each unique parameter combination, 20 independent runs were executed and presented results below are means over these 20 runs.

Comparison with a real world network

Analyses above focus purely on theoretical networks, it is worthwhile to investigate how real world network can function differently from them. Here, we make use of a Facebook network [39] that exemplifies a social media network. The network has 1034 nodes and average degree of 103.5, while the average path length is 2.95 and the diameter is 9. Fig 14 shows the degree distribution of the network, which indicates a power law distribution of connections. Then we run the partisan polarization process on this network and compare its polarization trajectory with those of four theoretical networks that have the same network size and average degree. The initial distribution of both partisanship and ideology is identical across the Facebook network and the theoretical ones. The only difference is how the distributions are spread across the different network structures. Fig 15 shows the partisan polarization process under different nationalization levels for these networks. We can observe that the Facebook network converges slower to polarization than the ring lattice and small world network with rewiring probability 0.05 and random graph in most scenarios. The exception is when nationalization level is 0.25, there is a drop in polarization for the random graph. The preference attachment network with degree distribution exponential 1 always shows a rise-then-fall phenomenon and mostly significantly at low nationalization level.

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Fig 14. Facebook network degree distribution.

The shape resembles a power law distribution.

https://doi.org/10.1371/journal.pcsy.0000121.g014

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Fig 15. Polarization of Facebook and four theoretical networks with same network size and average degree and same initial distributions of ideological positions and partisanship.

The Facebook network converges slower to polarization than the ring lattice, the small world network with rewiring probability 0.05 and the random graph in most scenarios. When nationalization level is 0.25, there is a drop in polarization level for the random graph. The preference attachment network with degree distribution exponential 1 always shows a rise-then-fall phenomenon.

https://doi.org/10.1371/journal.pcsy.0000121.g015

Table 2 shows the modularity, algebraic connectivity and retention rate of the five networks. Modularity describes the degree to which a network can be partitioned into a set of cohesive and tightly knit communities with sparse connections between them. Algebraic connectivity is the second smallest eigenvalue of the normalized Laplacian matrix , where A is the adjacency matrix and D is the degree matrix of the network. It indicates the speed at which a network converges to a consensus. Retention rate is the ratio between the final polarization level and the peak level, indicating how much polarization is maintained after convergence. We can observe that the networks that sustain polarization most are those that have low algebraic connectivity and high modularity - Facebook ( 0.02, modularity 0.54, only 6 percent edges crossing communities) and the ring lattice ( 0.004, modularity 0.77) — whereas the random graph and preferential-attachment network ( 0.73, modularity 0.11, about 76 percent of edges crossing communities) have the most drop in polarization after rising to the peak. Preferential attachment’s rise-then-fall pattern and relatively low retention rate is explained by its combination of high algebraic connectivity and low modularity.

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Table 2. Network Diagnostics Statistics. The combination of low algebraic connectivity and high modularity make the Facebook and ring lattice networks to have sustained polarization while the opposite happens for the random graph and preferential attachment network.

https://doi.org/10.1371/journal.pcsy.0000121.t002