2.1 Belief systems and ideologies

– Converse, The nature of belief systems in mass publics (Reference Converse1964).

Political attitudes rarely exist in isolation. For instance, support for increasing government spending on social welfare is usually linked to favoring higher income taxes for wealthy citizens in many Western democracies (Roosma et al., Reference Roosma, Gelissen and andWim Van2013). Public opinion scholars have long relied on the concept of constraint, that is, the functional interdependence between two or more issue positions (Converse, Reference Converse1964), to study the prevalence of connected political beliefs at the individual (Zaller, Reference Zaller1992; Kuklinski and Peyton, Reference Kuklinski and Peyton2007) and the collective level (Boutyline and Vaisey, Reference Boutyline and Vaisey2017; Brandt and Sleegers, Reference Brandt and Sleegers2021; Warncke, Reference Warncke2025). Viewed through this lens, an ideology, then, is a package of constrained beliefs that have been logically, quasi-logically, or socially developed to form an integrated framework (Converse, Reference Converse1964; Federico, Reference Federico2019).

Building upon this conceptualization, our understanding of mass ideologies further rests on two key premises. First, we view ideology as a latent construct at a level of higher abstraction, which stems from, and might exert influence on multiple issue positions and identities (Jost et al., Reference Jost, Nosek and Gosling2008; Federico, Reference Federico2019). Yet, ideologies and issue positions are not always aligned (Free and Cantril, Reference Free and Cantril1967; Ellis and Stimson, Reference Ellis and Stimson2012; Groenendyk et al., Reference Groenendyk, Kimbrough and Pickup2023); ideological misalignment can have broad and meaningful implications for understanding political cognition and behavior (Zaller, Reference Zaller1992; Converse, Reference Converse1964; Ellis and Stimson, Reference Ellis and Stimson2012; Kinder and Kalmoe, Reference Kinder and Kalmoe2017). We therefore refrain from assuming any alignment between ideologies and issue positions a priori, that is, defining ideological orientation based on particular issue positions, and vice versa (e.g., fixating opposition to redistribution as a right-wing attitude or right-wing identity to entail opposing redistribution). Furthermore, we explicitly acknowledge that the way ideologies and issue positions align may vary over time and across different political contexts (Johnston and Ollerenshaw, Reference Johnston and Ollerenshaw2020). We, therefore, believe it particularly beneficial to let ideological structure emerge organically from observed co-endorsement patterns within political attitude data.

Understanding why variation in ideological alignment exists is key to our second premise: we consider ideologies as socially constructed attitude bundles. This assertion rests on the well-documented observation that the co-occurrence of response patterns is often shaped by an underlying social structure, including socialized partisan leanings (Campbell et al., Reference Campbell, Converse, Miller and Stokes1960; Converse, Reference Converse1964; Green et al., Reference Green, Palmquist and Schickler2004; Baldassarri and Gelman, Reference Baldassarri and Gelman2008; Mason, Reference Mason2015). For example, support for social security policies is far from randomly distributed across the population; it is significantly more prevalent among Democrats than among Republicans. The collective identity formed by social and partisan groups often serves as ideological heuristics when individuals try to position themselves on a new issue or update their former beliefs. For instance, when a Democrat positions themself on a new issue, they might first take into account how other Democrats position themselves on this very issue. Consequently, ideological alignment implies more than a sorting process along various ideological dimensions. It should also be understood as a consolidation and polarization process in which political beliefs can become fused with group identities (ibid.).

2.2 Ideological polarization and belief network analysis (BNA)

Despite substantial research effort investigating political polarization, debates among social and political scientists persist regarding its overall trends in the mass public and the underlying mechanisms that drive it (Baldassarri and Bearman, Reference Baldassarri and Bearman2007; Abramowitz and Saunders, Reference Abramowitz and Saunders2008; Abramowitz, Reference Abramowitz2010; Fiorina et al., Reference Fiorina, Abrams and Pope2011; Federico, Reference Federico2019). One reason such debates persist is that ideological polarization has been conceptualized and measured in various different ways. Two influential conceptualizations of polarization are divergence and alignment (Lelkes, Reference Lelkes2016). Divergence describes a process of people moving apart from one another toward the extreme ends of an ideological dimension, such as the left–right spectrum (Fiorina et al., Reference Fiorina, Abrams and Pope2011). Alignment, by contrast, means that people develop ideologically coherent beliefs (i.e., constraint; Converse (Reference Converse1964)) because their partisan identities are increasingly aligned with political issue positions (i.e., partisan sorting Abramowitz and Saunders (Reference Abramowitz and Saunders2005)). Regarding alignment, a system is considered strongly polarized if people’s position on one issue or their partisan identity can be used to reliably predict their position on various other issues. Conversely, a system in which people’s position on one issue or their partisan identity is a weak predictor of issue positions would be considered less polarized.

Over the past few decades, researchers have not seen a vast divergence along many political issues in the United States (DiMaggio et al., Reference DiMaggio, Evans and Bryson1996; Hill and Tausanovitch, Reference Hill and Tausanovitch2015; Lelkes, Reference Lelkes2016); however they do observe alignment in terms of the increasingly pronounced partisan identities (DellaPosta et al., Reference DellaPosta, Shi and Macy2015; Baldassarri and Gelman, Reference Baldassarri and Gelman2008) that now have stronger ties with political beliefs (Levendusky, Reference Levendusky2009; Han, Reference Han2011; DellaPosta, Reference DellaPosta2020). Our study uses a belief network perspective to better understand this process.

To assess the degree of ideological alignment, researchers have explored linear methods (e.g., Mason, Reference Mason2015; Davis and Dunaway, Reference Davis and Dunaway2016; Baldassarri and Gelman, Reference Baldassarri and Gelman2008), and more recently belief network analysis (BNA) (e.g., Boutyline and Vaisey, Reference Boutyline and Vaisey2017; Brandt et al., Reference Brandt, Sibley and Osborne2019; Fishman and Davis, Reference Fishman and Davis2022). Following Converse’s idea of belief constraint, BNA operationalizes belief systems by representing issues (e.g., abortion restrictions, government spending on welfare, and tax cuts) as nodes while modeling the associations between them as links, assuming that the constraint between two issues can be measured by the degree to which they are aligned in the mass public (Boutyline and Vaisey, Reference Boutyline and Vaisey2017). Hence, BNA imposes a greater link strength among more highly correlated issue pairs.

Leveraging the network properties of belief system graphs, scholars have attempted to identify the central belief elements (Boutyline and Vaisey, Reference Boutyline and Vaisey2017; Brandt et al., Reference Brandt, Sibley and Osborne2019) to examine how belief centrality may predict the change in the system (Fishman and Davis, Reference Fishman and Davis2022), and more pertinently to our present study, to explain the mechanism of polarization (DellaPosta, Reference DellaPosta2020). According to DellaPosta (Reference DellaPosta2020), polarization can unfold both in the “heightening alignment across pairs of issues,” and in the “broadening alignment across a wider range of issues.” In BNA, these processes manifest through a rise in edge weights (i.e., increased correlation among issues), or an expansion in the size of connected components.

While inter-issue correlations can serve as indicators for the strength of dependency between variables, they cannot fully capture non-monotonic relationships where the change of one variable with regard to the other is not simply increasing or decreasing (Carpentras et al., Reference Carpentras, Lüders and Quayle2024). Moreover, while BNA can capture the issue-wise alignment process, they do not explicitly encode ideologies (i.e., a coherent set of issue positions) as a spatial component of a belief network. Thus, we see an important gap between the theoretical framing of ideologies as the core product of (organized) belief systems (Converse, Reference Converse1964), and the operational obscurity of how ideologies spatially manifest within a belief network. It is this gap that motivates our application of Response Item Networks (ResIN) to the analysis of ideological polarization within belief networks.

3.1 American national election studies (ANES)

The current study utilizes the American National Election Studies (ANES) from 2000 to 2020, a cross-sectional nationally representative survey of the general U.S population. We select six waves conducted in each presidential election year with a four-year interval. Using ResIN–a spatial belief network model described in further detail below–we create one snapshot for each wave to visualize the shifts in political beliefs throughout this period. In each snapshot, we include a set of politically relevant issues that were consistently assessed in all six waves of ANES (i.e., government service spending, government healthcare insurance, guaranteed jobs and income, aid to blacks, legal abortion), which allows for meaningful temporal comparisons. ANES respondents are asked to self-report their positions (i.e., attitudes) on these issues on either a 7 or 4 point scale. The complete set of issues is listed in Table 1, and Appendix A (ANES Included Items) provides the detailed issue descriptions and response options.

3.2 Creating resIN snapshots using ANES

While BNA uses individual nodes to represent a single issue (e.g. legalization of abortion), ResIN, instead, treats individual nodes as issue positions (e.g. the response “strongly agree” to legalization of abortion). Furthermore, as a “spatial network,” each node in ResIN is located in an N-dimensional space, which can form the basis for a spatial model of ideology as an underlying left–right continuum. In what follows, we describe how we apply the main components of ResIN to the ANES data.Footnote ¹ Readers should be advised that this procedure follows the exact steps as described in the work by Carpentras et al. (Reference Carpentras, Lüders and Quayle2024), which provides a comprehensive description of the method. In this article, we briefly walk through the main steps involved in ResIN-based analysis and focus on how this methodology can be used to explore the polarization dynamics in the belief system.

For a given issue selected from ANES, we first dummy code each issue position as a response variable. For instance, for the issue “guaranteed jobs and income,” respondents indicate their attitudes via the response options ranging from 1 (i.e., “strongly agree; government should guarantee job and good standard of living”) to 7 (i.e., “strongly disagree; government should let each person get ahead on their own”). For each response $x$ from 1 to 7, we construct a set of binary variables guar_jobs:x indicating whether or not a given respondent has chosen response $x$ (1 for yes, 0 for no). These binary variables are represented as nodes in ResIN. A positiveFootnote ² association between a given pair of binary variables from two different issues produces a weighted link between the corresponding two nodes. The link weight equals the strength of association, in this case, the phi correlation, which is an equivalent measure of the Pearson’s correlation index for binary variablesFootnote ³ (Carpentras et al., Reference Carpentras, Lüders and Quayle2024). Intuitively, the correlation between two responses A and B is a measurement of how often people who select A also select B (and vice versa).

After setting up nodes and weighted links, we implement the force-directed layout algorithm (Fruchterman and Reingold, Reference Fruchterman and Reingold1991), where node positions are determined by an equilibrium state that balances the attractive and repulsive forces among them (Carpentras et al., Reference Carpentras, Lüders and Quayle2024). The greater the link weights between any two response node pairs, the more powerful the attractive force between them. This allows ResIN to visualize the network structure of political beliefs organically, placing nodes connected by stronger links closer to each other. We also use principal component analysis (PCA) to identify the main dimension among the obtained spatial coordinates among all nodes, which we align with the X-axis by rotating the initial solution. Thus, ResIN enables a spatial organization of attitudes (i.e., nodes) within a 2-dimensional plane, where the distribution of nodes along the (major) X-axis reflects their relative position along the most relevant latent variable. As we will later show in Section 4.1, the X-axis happens to align well with partisan identities, a node-level attribute that averages respondents’ partisan identities. Carpentras et al. (Reference Carpentras, Lüders and Quayle2024) have shown that node positioning in ResIN is practically equivalent to latent variable modeling using Item Response Theory, in that distances between nodes represent corresponding distances in a latent space. Thanks to these unique features, ResIN is capable of identifying asymmetries in attitudinal patterns across the latent space of interest, as is shown in a recent application on vaccination attitudes (Carpentras et al. Reference Carpentras, Lüders and Quayle2022).

To further assist in visualizing how node positions correlate with other covariates—such as party identity—we can use node color to indicate the average covariate value of all respondents who endorse the corresponding attitude in a given year. Here, we choose to color nodes based on partisan leaning, which, at the individual level, is measured by a Likert scale ranging from 1 (i.e., strong Democrat) to 7 (i.e., strong Republican) (see Appendix A Party Identification of Respondent). For instance, the color of the node guar_jobs:1 in 2020 is determined by the average partisan leaning of all respondents who strongly believe that government should guarantee jobs and a good standard of living, according to their responses in 2020 ANES.

Here, we provide a simple simulation to showcase ResIN’s ability to spatially organize nodes in a 2-dimensional space, which is a key feature absent from classic BNA models. Figure 1 displays BNA and ResIN using the same synthetic data, where 1000 respondents self-report their positions on 12 issues via 3 possible response levels (i.e., agree, neutral, disagree). When all issues are correlated at similar levels, BNA displays a single connected component with no clear modular structure. In contrast, ResIN can reveal that responses corresponding to the same position are strongly connected (e.g., people who agree on one issue tend to agree on all other issues). Furthermore, we see that the neutral position (position 2) is closer to agree (position 1) than to disagree (position 3), which implies that people who remain neutral on some issues are more likely to agree rather than disagree on other issues. This simulated example shows how ResIN can reveal an asymmetric shape of a belief system that remains hidden in BNA.

Figure 1.

Comparison between BNA and ResIN on the same synthetic dataset. While BNA represents issues as nodes and association between issues as links, ResIN represents issue positions (i.e., attitudes) as nodes and association between issue positions as links.

3.3 Multi-level constraint and polarization measures afforded by resIN

Moving beyond revealing simple structural asymmetries within belief networks, we next discuss how ResIN can be used to study attitude structuration and polarization dynamics by offering quantitative, network-based polarization measures. In total, we propose five measures which operate at different levels: (i) two system-level measures that summarize the overall structuration and thus indicate the degree of constraint and polarization for an entire ResIN-network, and (ii) three attitude-level measures that articulate the specific role a given attitude plays in a (polarized) belief system.

3.3.1 System-level: How constrained and polarized is a given attitude space?

Based on Converse’s definition of belief system constraint, we expect a highly constrained belief system to produce more interlocked issue positions, that is, more attitude pairs held together via stronger links. Therefore, an intuitive network metric to assess the system-level constraint is link density $D$ , that is, the proportion of manifest links among all possible links within in a given network. Hence, let $G = (V, E, W)$ denote a ResIN with a set of nodes $V$ , edges $E$ and the corresponding edge weights $W$ , produced by responses to a set of issues $K$ , each with $L_{k}$ possible response options. Thus, the numerator is the sum of link weights; and the denominator, the number of all possible links, equals the number of links that can connect nodes sourced from different issues. The link density of a given ResIN is therefore:

\begin{equation*} D = \frac {\sum _{e \in E}W_{e}}{\sum _{m \in K} L_{m}(\sum _{n \in K, n!=m}L_{n})} \end{equation*}

Apart from computing $D$ for the whole graph, we can also compute $D_{G'}$ for a partisan subgraph $G'$ in a single ResIN. For instance, we can compute $D_{G_{rep}}$ for a subgraph $G_{rep}$ consisting of only Republican-leaning nodes, that is, attitudes that are selected mainly by respondents who lean closer to Republicans.

Next, we introduce linearization as a measure of polarization, that is, the degree to which a network gets squeezed into a dominant latent ideological dimension (e.g., Stimson, Reference Stimson1975). In combination with force-directed layout algorithm, node positions in ResIN are determined by a varimax rotation that aligns the main orientation of the network with the X axis (i.e., the dominant latent ideological dimension). In this way, the spatial information of each node meaningfully indicates their position in a latent ideological space—a feature that we will demonstrate empirically in Section 4.1. In this space, a highly polarized belief system would flatten-out, or distribute linearly, such that knowing one’s position along a single ideological dimension can well predict their positions on a variety of issues (we elaborate on this through simulation in Section 3.3.2). We leverage ResIN’s spatial property and quantify the degree to which attitudes toward the five issues collapse into a single, dominant dimension, which can be visually grasped by the degree to which the nodes are being “squeezed” into a flat shape linear line. The linearization score $E$ equals the ratio of X coordinates spread to Y coordinates spread for all nodes in a given ResIN network.

\begin{align*} E &= \frac {X_{max} - X_{min}}{Y_{max} - Y_{min}} \end{align*}

To assess the robustness of these system-level metrics, we conducted a subsampling-based sensitivity analysis by randomly re-sampling 80% of the yearly ANES data (without replacement). In each of 200 iterations, we regenerate the ResIN network, yielding a distribution of polarization scores for each year. Here, we report the interquartile ranges (IQRs) to show the degree of variability across subsampled results. We choose subsampling rather than classical bootstrapping, as duplicating responses from individual respondents–necessary in bootstrapping–may distort the network structure in ways that do not reflect realistic data variation. Furthermore, subsampling provides a more conservative estimate of robustness by testing how sensitive the polarization score is to partial data omissions.

3.3.2 Simulation of a “broken egg”: the linearization process

Why is linearization a meaningful indicator of belief polarization? We answer this by observing how ResIN changes in shape as we move from a perfectly polarized system to a fully random one. To control the degree of polarization, we interpolate between a highly polarized model derived from Item Response Theory (IRT) (Samejima, Reference Samejima2011) and a random model that encodes no polarization. After more detailed model descriptions, we will explain why a highly polarized belief system yields a flat-shaped network and confirm a strong, monotonic association between polarization and linearization through simulation.

We use IRT to set up a polarized attitude system. Under the IRT paradigm, every respondent has (i) a value $\theta$ for a certain latent variable, and (ii) a so-called item characteristic curve $f(\theta )$ for every issue position. $f(\theta )$ determines the probability of this respondent selecting a certain issue position (Samejima, Reference Samejima2011). For instance, let us consider a case where the latent variable is the left–right spectrum and a small (large) value of $\theta$ indicates the left-wing (right-wing) position. For an issue position “support for gun control,” because left-leaning respondents are more likely to support gun control in the U.S., we expect $f(\theta )$ to be greater for smaller $\theta$ (i.e. higher probability for left-leaning respondents to select this issue position), and smaller for larger $\theta$ . Usually, IRT-based probability curves have a bell-shape resembling normal distributions. For a given issue, if the peak of probability functions for issue positions are well separated (i.e., they overlap only minimally), people with similar values of $\theta$ would concentrate their responses on the same issue position, while people with different values of $\theta$ would have distinct issue positions. The random model, in contrast, supposes no latent variable nor any association between issue positions. For each respondent, it assumes that all possible positions are equally probable. Following the previous example, within the random model, a right-leaning respondent is equally likely to select any position for a given issue, regardless of the ideological characteristics of the position. One simple implementation of this model is given by setting all item characteristic curves to be constant and identical one another.

To control the level of polarization, we interpolate between a well-polarized IRT model and the random model using a “randomness” parameter $r$ . Therefore, the final item characteristic curve for a given $r$ is:

\begin{equation*} f_r(\theta )=(1-r)f(\theta )+r/L \end{equation*}

where $f(\theta )$ is the “classic” item characteristic curve, as obtained from Samejima (Reference Samejima2011), and $L$ is the number of issue positions for a certain issue (e.g., corresponding to the number of response options in a survey). In each simulation round, we fix the number of respondents ( $N = 1000$ ), the number of issues ( $K = 8$ ) and the number of positions per issue ( $L = 7$ ), and vary the randomness parameter $r$ . Each respondents has a random value for the latent variable $\theta$ . Based on the issue position probability computed via $f_r(\theta )$ , each respondent selects one position from each of the 8 issues. In this manner, we are able to generate synthetic datasets resembling survey responses. From here, we repeat the same procedure generating ResINs and measure linearization for each simulated network.

We display the simulation results in Figure 2. Panel (a) displays three ResIN snapshots obtained from $r$ = 0, 0.5, and 1 and panel (b) shows the monotonically negative relationship between $r$ and linearization. When the system is fully polarized (i.e., $r=0$ ), the corresponding network is stretched flat and fully elongated, exhibiting a high level of linearization. As we increase the noise by tuning up the randomness parameter, the elongated structure folds on itself and starts to take a more oval shape, and the linearization level drops rapidly. Finally, in the case of pure randomness when $r=1$ and all issue positions are equally probable,Footnote ⁴ the system displays a round shape with a linearization level approximating 1.

Figure 2.

“Broken egg” simulations with varying randomness parameters $r$ . Subfigure (a) shows the ResIN output generated by $r = 0, 0.5, 1$ ; subfigure (b) shows the negative relationship between randomness and linearization level.

Why, one may wonder, does ResIN change its shape in response to variations in belief system polarization? And more precisely, why would an increase in polarization bring forward a “broken-egg” process in ResIN? Remember that proximity between nodes (i.e., issue positions) in ResIN indicates a high degree of attitude co-endorsement. Given the structure of IRT, issue positions with similar mean value in their item characteristic curves are likely to be co-selected by the same people, and thus tend to form clusters in ResIN. Within a structured IRT model (i.e., a polarized belief system), the item characteristic curves are well sorted along the natural order of issue positions, such that respondents with the same $\theta$ value are most likely to select the same position, fairly likely to select adjacent positions, but less likely to select more distant positions. For instance, consider three positions (i.e., support, neutral, and against) toward gun control. A structured IRT would predict left-leaning respondents to most likely to choose “support,” moderately likely to choose “neutral,” and least likely to choose the “against” option. The inverse would be true for right-leaning respondents. Meanwhile, centrist respondents would most likely be neutral toward gun control, and less likely to support or be against gun control. A belief system with multiple issues resembling this very item response structure would naturally lead to a ResIN solution in which the neutral positions are connected to the support and against positions, yet the support and against positions are only weakly connected. A force-directed layout balancing the attractive and repulsive patterns would thus result in a linearized, left–right attitude structure indicative of a highly polarized system.

To sum up, the present simulations help us understand how and why the shape of ResIN-networks is indicative of belief polarization along a latent ideological dimension. A weakly polarized belief systems produces an unstructured cloud-like ResIN, while a strongly polarized belief system produces a linear-shape ResIN that map issue positions closely along a single dimension.

3.3.3 Attitude-level: How (de-)polarizing are certain attitudes?

Next we analyze ResIN at the node level. Our main aim here is to reveal heterogeneity in the structural function of different attitude nodes within ResINs. More particular, we are interested in identifying (a) ideological centroids, that is, nodes at the center of divided ideological communities and (b) attitudinal bridge(s), that is, nodes that could help bridge distinct ideological camps.

To accomplish the former, we focus on the node strength centrality as a measure of local importance. Previous research relying on classic BNA models claims that nodes higher in strength are more central to people’s belief systems (Boutyline and Vaisey, Reference Boutyline and Vaisey2017; Brandt et al., Reference Brandt, Sibley and Osborne2019; Warncke, Reference Warncke2025). Leveraging ResIN, we can extend this approach to within-cluster analysis, investigating which issues are relatively more important to Democrats rather than Republicans. Purportedly, these issue positions should also be the most polarizing to the opposite ideological camp.

Now, we turn toward attitudinal bridges. To illustrate how we find issue positions with the most de-polarizing potential, let us consider a scenario featuring two completely sorted issues (i.e., $A$ , $B$ ; Figure 3a), where knowing one’s position on issue $A$ can give full information on their position on issue $B$ (e.g., people who choose position 1 for $A$ , i.e., $A_1$ , would definitely choose position 1 for $B$ , i.e., $B_1$ ). Here, two respondents either agree completely on both issues or share no common position at all. However, if a comparable number of respondents who select $A_1$ and $B_1$ , and those who select $A_3$ and $B_3$ , can agree on another issue $C$ (e.g., if they have all selected $C_3$ ), it would be possible to bridge the debate through finding a common ground on $C$ . In this case, response $C_3$ would thus function as an attitudinal bridge. In Figure 3b, we highlight six such attitude bridges through which respondents can establish partial agreement on one issue while disagreeing on the other(s). Using ResIN, we can measure such bridging power by computing the node-level betweenness centrality, a measure counting how many times a node lies on the shortest path between any pair of nodesFootnote ⁵ (Freeman, Reference Freeman1977). Nodes that are frequently on the shortest paths connecting two or more ideological communities are more likely to serve as the common ground, thus likely showing the highest de-polarization potential in cross-community engagement. Alternatively, we also compute closeness centrality, which is the weighted inverse of the total shortest-path distance from a node to every other reachable node and hence represents a quantitative measure of global connectedness of a given attitude. Conceptually, a high betweenness and closeness centrality should be jointly indicative of node-level bridging power in a belief system.

Figure 3.

Illustration of a toy ResIN model with two issues (a) and three issues (b). Each issue has three positions. The left panel (a) demonstrates the scenario where the population perfectly sorted along issue $A$ and $B$ , and there is no bridge connecting mismatched positions such as $A_1$ and $B_3$ . The right panel (b) shows a scenario where some positions along issue $C$ serve as bridges connecting previously isolated attitudes; similarly positions along $A$ and $B$ can also serve as bridges for $B$ - $C$ and $A$ - $C$ . The green color highlight those bridge attitudes.