Emergence of polarized ideological opinions in multidimensional topic spaces

Opinion polarization is on the rise, causing concerns for the openness of public debates. Additionally, extreme opinions on different topics often show significant correlations. The dynamics leading to these polarized ideological opinions pose a challenge: How can such correlations emerge, without assuming them a priori in the individual preferences or in a preexisting social structure? Here we propose a simple model that reproduces ideological opinion states found in survey data, even between rather unrelated, but sufficiently controversial, topics. Inspired by skew coordinate systems recently proposed in natural language processing models, we solidify these intuitions in a formalism where opinions evolve in a multidimensional space where topics form a non-orthogonal basis. The model features a phase transition between consensus, opinion polarization, and ideological states, which we analytically characterize as a function of the controversialness and overlap of the topics. Our findings shed light upon the mechanisms driving the emergence of ideology in the formation of opinions.


INTRODUCTION
According to classical opinion dynamics models in which social interactions add constructively to opinion formation, the increasing interaction rates of modern societies would eventually lead to a global consensus, even on controversial issues [1,2]. This classical prediction has been recently challenged by the empirical observation of opinion polarization, i.e. the presence of two well-separated peaks in the opinion distribution. Polarization can be found, both offline [3,4], and in online social media [5][6][7], where polarized debates have been observed with respect to several areas and issues, ranging from political orientation [8][9][10], US and French presidential elections [11], to street protests [12]. Interestingly, polarization seems to burst especially in public discussions evolving around politically and ethically controversial issues such as abortion [13] or climate change [14][15][16]. Different modeling approaches have investigated opinion polarization on single topics as the result of repulsive interactions among agents [17], biased assimilation [18], or social re-inforcement mechanisms [19][20][21].
Topics are rarely discussed in isolation. Especially with growing connectedness [22] and increased information flow [23], the processes of opinion formation take place simultaneously. For heterogeneous opinion distributions deviating from a global consensus, another striking feature can often be observed: issue alignment [4,24,25], whose presence implies that individuals are much more likely to have a certain combination of opinions than others, a state that can be defined as an ideological opinion state. For some combinations of topics the alignment is quite intuitive. For example, opinions with respect to rights of transgender people [26] and same-sex couples may be correlated. In this case, the majority of individuals would mainly split into two groups, those who deny certain rights to both, transgender people and same-sex couples, and those who support them, while the mixed positions would be rare. While the two gender-related issues can be considered as quite related, in what follows we will show that also opinions on rather unrelated issues might be strongly correlated. Which underlying mechanism might drive such ideological states to emerge?
While considerable efforts have been recently put into measuring and modeling opinion polarization, the phenomenon of issue alignment got much less attention. This problem has been mainly approached by agent-based modeling within multidimensional opinion spaces, inspired by Axelrod's seminal work on cultural diversity [27]. Models based on the concept of a confidence bound illustrated how opinion alignment can result from a dependence between opinion dimensions combined with assimilation and rejection mechanisms [28], and from assumed correlations between individual and immutable agents' attributes [25,29]. Other attempts include the extension of Heider's cognitive balance theory [30] to multiple dimensions, in a well-mixed population [31].
However, all these works assume an a priori, static social network structure (or a well-mixed population) as a substrate for opinion formation, and/or encode issue alignment directly as correlations between individual attributes. On the contrary, social interactions are known to evolve in time [32,33], and such evolution can have a strong impact on the dynamical processes running on top of such time-varying networks, such as opinion formation (see [34] for an extensive review). This is particularly true for social media platforms, which have been shown to be the major news source for up to 62% of adults in the U.S. [35]. On such platforms the process of opinion formation is continuously shaped by the new information and content shared by users on the platform [36].
In this paper, we propose a simple model featuring the emergence of polarized ideological states from microscopic interactions between individuals, assuming neither a preexisting social structure, nor a confidence bound or correlated individual attributes of the agents. We find that the co-evolution of social interactions and opinions can not only lead to extreme opinions, but can also cause issue alignment. Strikingly, such issue alignment emerges also for rather unrelated topics that are sufficiently controversial, due to the reinforcement mecha-nism mediated by social interactions. Our model is based on a minimal set of assumptions. First, opinions evolve according to the social interactions among the agents, which are ruled by homophily: two agents sharing similar opinions are more likely to interact [37,38]. Second, connected agents sharing similar opinions can mutually reinforce each other's stance. Within the theory of group polarization [39,40] this happens when individuals, through the exchange of arguments, influence each other in an additive way [41]. Third, opinions lay in a multidimensional Euclidean space, spanned by a nonorthogonal basis formed by topics. Topics can be controversial and mutually overlapping, i.e. there may exist an intersection of arguments that is valid for several topics.
With these assumptions, our model generates three different scenarios: i) convergence toward a global consensus, ii) polarization of non-correlated opinions, and iii) polarization with issues alignment, i.e. a polarized ideological state. Interestingly, ideology emerges from uncorrelated polarization simply by relaxing the assumption of an orthogonal basis of the topic space. We analytically and numerically characterize the transitions between these three states, in dependence on the controversialness and overlap of the topics discussed. We compare the model's behavior with empirical opinion polls from the American national election surveys (ANES) [42]. In a pairwise comparison of a broad selection of topics, we can observe several realizations of the scenarios proposed by the model. In particular, we found a number of non-trivial cases where opinions are polarized and aligned, but the opinion correlation cannot be simply traced back to the similarity between topics, validating the model's behavior.
Our framework is built on the generalization of a simple one-dimensional model describing polarization dynamics [19] to multiple dimensions, assuming the non-orthogonal topic basis. This assumption implies that topics, forming the basis of the space where opinions lay, may not be completely independent but rather can show a certain degree of overlap. As suggested by argument exchange theory [43], a non-vanishing overlap between two topics might arise due to a common set of arguments which simultaneously supports or rejects certain stances on both topics. Thus large overlaps are characteristic for pairs of closely related topics such as our example of rights of transgender people and same-sex couples. As we will show, however, also small overlaps critically determine the opinion formation, and hence, ideological opinion states may also emerge for rather unrelated topics.
Interestingly, non-orthogonal bases (equivalently, skew coordinate systems) have been recently proposed to solve some well-known problems of classical vector space models for representing text documents [44]. Within this framework, documents are represented as vectors in an underlying space, whose basis is formed by the terms used in the documents. Crucially, if the terms are assumed as orthogonal, similarity measures (such as cosine similarity) can not precisely describe the relationship between documents, if terms are not independent. When the assumption of orthogonality is relaxed, such as in Latent Semantic Indexing or distance met-ric learning, similarity measures work much better [45]. Our approach follows a similar idea: if the orthogonality of topics is relaxed, i.e. if topics can overlap, the correlation between opinions with respect to different topics can naturally emerge through the proposed reinforcement dynamics from social interactions.

A MODEL OF OPINION DYNAMICS IN A MULTIDIMENSIONAL TOPIC SPACE
Let us consider a system of N agents. Each agent i holds opinions towards T distinct topics, represented by the opinion vector denotes the opinion of agent i towards topic v. For each topic v, the sign of the opinion x v i , sgn(x v i ), describes the qualitative stance of agent i towards the topic (i.e., in favor or against the issue), while the absolute value of x v i , |x v i |, quantifies the strength of his/her opinion, or the conviction, with respect to one of the sides. The opinion vector x i represents the position of an agent i in the T -dimensional topic space T . The opinion vector x i can be written as where {x v i } are the coordinates of agent i and {e v } form a basis of the Euclidean space T , representing the topics under consideration. To form the basis in T , {e v } have to be assumed linearily independent, but are not necessarily orthogonal.
The opinion vectors of agents evolve in time, i.e. x i = x i (t), where we will omit the dependence on t in the following for brevity. We assume that the evolution of opinions follows a radicalization dynamics, a recently proposed mechanism that reproduces polarization and echo chambers found in empirical social networks [19,46]. Within this framework, the opinions of an agent are reinforced by interactions with other agents sharing similar views. The mechanism is inspired by the phenomenon of group polarization [39], by which interactions within a group can drive opinions to become more extreme. The social interactions responsible for the opinion dynamics are not static but evolve in time as well [36,47] , forming a time-varying social network that can be represented by a temporal adjacency matrix A ij (t), with A ij (t) = 1 if agents j and i are connected at time t, A ij (t) = 0 otherwise. The opinion dynamics is solely driven by interactions among the agents and is described by the following set of N × T ordinary differential equations, where K > 0 denotes the social influence strength acting globally among agents -the larger K, the stronger the social influence exerted by the agents on their peers [19]. The interpretation of the sigmoidal non-linearity tanh(. . .) and the matrix Φ will be discussed a couple of lines below. According to Eq. (1), the opinion of agent i towards topic v, x v i , evolves depending on the aggregated inputs from his/her neighbors, determined by the temporal adjacency matrix A ij (t). The social input of each agent j contributing to the change of x v i , [Φx j ] v , is smoothed by the influence function tanh(αΦx), which tunes the mutual influences that the opinions of different agents exert on each other. As suggested by experimental findings [48], the social influence of extreme opinions is capped, and therefore has to be described by a sigmoidal function. As a particular realization of such function we use tanh(x), as it was done in the previous work [19]. The shape of this function is controlled by the parameter α: for small α, the social influence of individuals with moderate opinions on other peers is weak, while for large α, even moderate agents can exert a strong social influence on others.
The parameter α can thus be interpreted as the controversialness of the topic, which has been shown to be an important factor driving the emergence of polarization in debates on online social media [49]. For the sake of simplicity, we assume α to denote the overall controversy of the discussion around all considered topics, i.e., the same value of α is set for all topics. The general case of a different controversy for each topic gives rise to additional opinion states that can also be found in the empirical data, as shown in the Supplementary Information (SI). According to Eq. (1), an agent j exerts social influence on a connected agent i with respect to all topics under consideration, and the opinion of an agent towards a specific topic is not only influenced by the opinion of others on the same topic but, in general, also about other topics. This is reflected in the symmetric topic overlap matrix Φ, which encodes the relation between topics. If the element Φ v,z is different from zero, the opinions of agents on topic v can influence the opinions of other agents with respect to topic z, and vice versa.
The matrix Φ has a geometric interpretation in the latent topic space. The element Φ v,z can be interpreted as a scalar product of topics v and z, Φ v,z = e v · e z = cos(δ vz ), where δ vz represents the angle between topics v and z, as shown in Fig. 1 for T = 2. In relation to our introductory example, cos(δ vz ) quantifies the overlap between topic v (rights of transgender people) and z (rights of same-sex couples). The scalar product between two opinion vectors x i and x j in the topic space T spanned by such non-orthogonal topics, is computed as involving the overlap matrix Φ . Note that it always holds Φ vv = 1, so that if all topics are orthogonal, Φ vz = 0, the matrix Φ reduces to a unit matrix, and Eq. (1) decouples with respect to topics. The contact patterns among the agents, which sustains the opinion formation, evolves according to the activity driven (AD) model [50][51][52][53]. This gives rise to a temporal network which changes at discrete time intervals. According to the original AD model, each agent i is characterized by an activity a i ∈ [ε, 1], representing his/her propensity to contact FIG. 1. Illustration of two non-orthogonal topics as basis for the topic space T . For T = 2, the non-orthogonal, normalized basis is uniquely defined by the angle δ. Geometrically, cos(δ) quantifies the overlap between basis vectors, interpreted as a topical overlap, here the rights of same-sex couples (e v ) and transgender people (e z ). The opinion distance between two agents i and j, d(xi, xj), is computed by the scalar product defined in Eq. (2). m distinct other agents chosen at random. Activities are extracted from a power law distribution F (a) ∼ a −γ , as suggested by empirical findings [50,52]. The set of parameters (ε, γ, m) fully encodes the basic AD dynamics. Furthermore, we assume that social interactions are ruled by homophily, a well-known empirical feature in both offline [54,55] and online [56,57] social networks. To this end, the probability p ij that an active agent i will contact a peer j is modeled as a decreasing function of the distance between their opinions, where d(x i , x j ) is the usual Euclidean distance between opinion vectors (cf. Fig. 1) generated by the scalar product defined in Eq. (2), while the exponent β controls the power law decay of the connection probability with opinion distance. As a result of Eq. (3), two agents i and j are more likely to interact if they are close in the topic space T , i.e. the distance d(x i , x j ) is small. Upon such interaction (i.e., if A ij (t) = 1), the opinions of agent j influence all opinions of agent i, following the sigmoidal influence function in Eq. (1). In the case of orthogonal topics (Φ = 1) social influence takes place only between opinions on the same topic. If the stances of two interacting agents i and j on a topic v are equal, i.e. sgn(x v i ) = sgn(x v j ), they will increase their current conviction on topic v, which is given by the absolute values of the opinion coordinates |x v i | and |x v j |. On the contrary, for sgn(x v i ) = sgn(x v j ), they will tend to decrease their conviction on that topic and converge towards a consensus. Crucially, for non-orthogonal topics v and z, cos(δ vz ) = 0, the opinion with respect to topic v of agent j, x v j , will influence the opinion of agent i on topic z, x z i : an argument supporting a topic is logically connected to the other topic.

EMERGENCE OF CONSENSUS, POLARIZATION AND IDEOLOGICAL PHASES
The model in a one-dimensional space, corresponding to a single topic (T = 1), has been shown to reproduce empirical data for polarized debates on Twitter, with respect to polarization of opinions and segregation of social interactions [19]. A phase transition between a global consensus and polarized state emerged as social influence (tuned by parameter K) and the controversialness of the topic discussed (represented by α) increased. In the following, we explore the impact of multiple topics and their potential overlap within this framework for T > 1. Following empirical observations, we set the parameters of the basic AD model to ( , γ, m) = (0.01, 2.1, 10) [50][51][52][53], and consider a regime of strong social influence and strong homophily, by setting K = 3 and β = 3.
We investigate the emergence of different opinion states for long times in dependence of α and of the topics overlaps. Due to the fluctuations induced by the stochastic interaction dynamics, the states other than consensus are not stable for t → ∞. However, for sufficiently high values of β (i.e. homophily), they been shown to be meta-stable [19], numerically indistinguishable from stable states. Therefore, we will refer to them as steady states in the following. Furthermore, we focus on a regime of fast-switching interactions, i.e. opinions evolve at a slower rate than social interactions. This choice is motivated by the assumption that multiple social inputs are necessary to change an agents opinion substantially while attitude change has been shown to be slow, especially in the case of important issues [58]. We therefore choose an integration time step of dt = 0.01, which corresponds to an effective time-scale separation by a factor of 100 between the network and the opinion dynamics.
For the sake of simplicity (and convenient illustrations), in the following we will show the behavior of the model for a system of N = 1000 agents interacting with respect to two topics (T = 2, v = 1, 2). In this case, Eqs. (1) readṡ where Φ is fully defined by a single angle δ, with cos(δ) giving the overlap between the two topics considered. Fig. 2 shows the three dynamical regimes of the model, which strongly depend on the controversialness of topics α and the topic overlap cos(δ). The opinion trajectories of single agents are depicted as grey lines, while their steady state positions are shown as colored dots. To clarify the visualization, we use polar coordinates (r, ϕ), with r corresponding to the overall conviction of an agent, who is colored according to its opinion, in the polar coordinate ϕ.
If topics are not controversial (i.e. for α small), agents reach a global consensus, as shown in Fig. 2(a). Starting from normally distributed opinions in the two-dimensional topic space, opinions converge towards the state of vanishing convictions, i.e. ||x i (t → ∞)|| = 0 ∀i. In this regime, the dynamics is dominated by the decay terms (−x 1 i , −x 2 i ) in Eq. (4), which mimic the agents' finite opinion memory. The fast relaxation toward the global consensus is due to the lack of sufficient social influence from interacting peers. This situation is also depicted in the final opinion distributions P 1 (x) and P 2 (x), plotted on the marginals of Fig. 2(a): For both topics, the opinion distribution is peaked around x = 0.
If topics are controversial -for larger values of α -the situation is drastically different, cf. Fig. 2(b)-(c). The social influence among the agents dominates the opinion evolution, destabilizing the global consensus. The opinions of agents do not converge but are widely spread and potentially reach convictions much stronger than in the initial configuration. Note that for polarization to emerge, the presence of homophily is a necessary condition [19]. In this regime, the overlap between topics, encoded by cos(δ), crucially determines the dynamics and the possible emergence of ideological states in the system.
If topics do not overlap, i.e. cos(δ) = 0, the opinions with respect to each topic evolve independently. That is, the opinion dynamics with respect to each topic decouple, and can be effectively captured by the one-dimensional model of [19]. In this regime of strong social influence, homophily and controversial topics, a polarized state emerges, as shown in Fig. 2(b). In polarized states, the opinion distributions are bimodal for each topic, as shown on the marginals plots of Fig. 2 Fig. 2(b), represent individuals taking all different stances as expected when the two topics are orthogonal. Note that the opinion correlation in both polarized and consensus states is low, as reported in Fig. 2(a) and Fig. 2 This situation radically changes if topics overlap (cos(δ) > 0), i.e. they are non-orthogonal in the underlying space. In this case, according to Eq. (4), the opinions with respect to one topic can influence the opinions with respect to the others, and vice versa.  This state of the system, characterized by opinions which are both polarized, σ 2 1 (x), σ 2 2 (x) 0, and correlated, ρ(x 1 , x 2 ) 0, is characterized as a polarized ideological state. In the underlying topic space, this situation translates into a symmetry breaking and consequent dimensionality reduction: The opinion of an agent towards one topic is able to predict his/her opinion towards ones. For example, an individual who strongly opposes the idea of same-sex marriage, will also mostly likely argue against transgender people being allowed to use the toilets corresponding to their identified genders.
The dynamics of the model given by Eq. (1) can, in the thermodynamic limit (N → ∞) and for strong homophily (β 1), be qualitatively captured within a mean-field approximation, as shown in the Methods section. Figures 2(d), (e), and (f) show the attractors of the deterministic, meanfield dynamics for the same values of the parameters α and cos(δ) as in Figures 2(a), (b), and (c), respectively. The resulting dynamics look remarkably similar to the behavior of the full stochastic model. For low α, there is only one stable fixed point, corresponding to the global consensus at x i (t → ∞) = 0 ∀i, as shown in Fig. 2(d). As α increases, the consensus is destabilized. If topics are orthogonal, this results in four stable fixed points corresponding to an uncorrelated polarized state (Fig. 2(e)). If topics overlap the symmetry is broken and only two stable fixed points emerge, corresponding to the ideological state, depicted in Fig. 2(f).
Within the mean-field approximation, the transition between a global consensus and polarization can be described analytically. For T = 2 the stability limits of the consensus phase are determined by the critical controversialness, α c , as which is depicted in Fig. 3 as black dashed line. It depends inversely on the product of social influence strength K, the number of agents contacted by an active agent m, the average activity a , and a factor [1 + cos(δ)] accounting for the overlap of the two topics. The different regimes of polarization, i.e. polarization of non-correlated opinions and the ideological phase can be distinguished numerically, see Methods section for details. Figure 3 shows the stability regions in the α-cos(δ) plane, colored according to the corresponding phases, consensus (green), polarization of uncorrelated opinions (blue), and ideology (red). Note that the phase diagram is symmetric with respect to the line of vanishing overlaps cos(δ) = 0 (orthogonal topics). For this case, no ideological states emerge. By contrast, for finite overlaps, cos(δ) > 0, i.e. non-orthogonal topics, ideological states emerge and their region of stability (red region) widens as the topics' overlap, cos(δ), increases. If topics are sufficiently controversial, i.e. for α > α c , as given by Equation (5) (plotted as a dashed line in Fig 3), consensus is de-stabilized and polarization emerges. The larger the overlap between topics (the larger the value of cos(δ)), the smaller is the critical controversialness α c necessary to de-stabilize consensus and promote polarization.

SOCIAL NETWORK'S TOPOLOGY REFLECTS OPINION SEGREGATION
On social media, opinion polarization can be reflected in the topology of the corresponding social networks: The users interact more likely with peers sharing similar opinions, a situation known as echo-chambers [54,59]. Our model assumes that the opinion evolution is coupled to the dynamics of the underlying social network via Eqs. (1) and (3). This mechanism yields a social network structure which is shaped by the process of opinion formation. Figures 4(a), (b), and (c) show the social networks generated by the model for the same parameters employed in Fig. 2(a), (b), and (c), corresponding to global consensus, uncorrelated polarization, and ideological state, respectively. The networks result from the timeintegration of the last 70 time steps of the temporal adjacency matrix A ij (t), once the system reaches a steady state. Each node corresponds to an agent i, size of the node is proportional to his conviction (given by r i ), while the color represents the opinion in the polar coordinate ϕ i . Fig. 4(a) shows the system approaching global consensus. While nodes with similar opinions are more likely to be connected -an effect caused by homophily, also in the case of low α -no clear groups emerge in the network structure. Fig. 4 (b) shows that in the uncorrelated polarized case, on the contrary, four groups are clearly visible, each one characterized by a different opinion (color coded as in Fig. 2). A similar situation is visible in Fig. 4(c), depicting the ideological state, where the social network is mainly segregated into two groups, holding different opinions.
These observations can be quantified by a community detection analysis. Figs. 4 (d), (e), (f) show the community structure of the corresponding networks, plotted as polar bar plots, as obtained by the Louvain algorithm [60]. Each community is represented as a different angle sector, which is orientated (polar angle) according to the average opinion ϕ within that community. The size of the community is represented by the radius of each bar, while the width and color of each sector represent the average cosine similarity between nodes in that community, the mean scalar product of opinion directions calculated according to Eq. (2) and averaged over all pairs of agents within the community.
In the global consensus case (Fig. 4 (d)), many communities are present and rather randomly oriented. Each community is characterized by a heterogeneous spectrum of opinions, (low values of the average cosine similarity). On the contrary, when consensus is broken, the average opinion of the agents within each community is aligned with the dynamical attractors shown in Fig. 2(e) and (f). In the uncorrelated polarized case, Fig. 4 (e), the communities are characterized by four typical average opinions, corresponding to the four colors shown in Fig. 4 (b). Within each community, opinions are very similar, with large values of the average cosine similarity. In the ideological phase - Fig. 4 (f), communities are characterized by only two typical averages opinions and a strong homogeneity of opinions (very high average cosine similarity).

COMPARISON WITH EMPIRICAL DATA
The presence of three different scenarios suggested by our model can be compared with empirical data. In what follows, we investigate the degree of polarization and correlation between opinions with respect to different topics using data collected by the American National Election Study (ANES). The ANES study is a continuation of a series of surveys run since 1948, with the main objective of analyzing public opinion and voting behavior in the U.S. presidential elections by interviewing a representative sample of U.S. citizens. The ANES data have been proven to be suitable for a variety of research purposes, ranging from examining the drivers for public attitudes towards specific topics like immigration [61], observing  Fig. 2(a)-(c), i.e. α = 0.05, δ = π/2 (a), α = 3, δ = π/2 (b), α = 3, δ = π/4 (c). In the network illustrations each node is colored according its opinion angle ϕ, size is proportional to its conviction r. Communities are represented in the polar bar plot below each network. Each community is represented by a bar: the radius represents the size, color and width correspond to the average cosine similarity between all pairs of agents within the community. The orientation represents the average opinion angle ϕ of all agents within the community. Communities containing less than 5% of the total number of nodes are not shown.
longitudinal developments of trust in the American government [62], or characterizing long-term trends of polarization [4,63].
For our analysis, we select a total of 67 questions with overall 253984 valid responses from the 2016 ANES. See Methods for details on the selection criteria and the SI for a complete list of analyzed questions. Respondents are assigned an individual ID, such that their answers to different questions can be related to each other. In the following, we will focus on two key features of the ANES data: i) the distribution of responses with respect to each question, quantifying the degree of polarization or consensus toward a certain topic, and ii) the correlation between responses with respect to different pairs of questions, revealing which issues are aligned and thus contribute to an ideological state.
A schematic illustration of the subset of considered issues is given in Fig. 5. On top of Fig. 5(a), we plot the variance σ 2 v (x) of the response distribution to question v. Questions are sorted according to σ 2 v (x) in descending order, from questions with most polarized responses to less polarizing ones. While for the majority of questions (on the right side of the marginal plot) a consensus looks achievable, few questions (on the left side of the plot) are strongly polarized, such as the question of whether "voting is a duty". Panel (a) shows the correlation matrix of the responses, sorted according to their variance.
The cell (v, z) is color coded according to the absolute value of the Pearson correlation between the opinion distributions P v (x) and P z (x), |ρ vz |. The full distribution of correlation values for all investigated pairs of questions is reported in the SI. The average correlation value is 0.2, but the distribution is broad: some pairs of questions are weakly correlated, while others are strongly so. Note that although there is a small dependence of the strength of correlation on the variance (slight decay of correlation towards the bottom right), both large and small correlation values can be observed in all parts of the matrix.
Panels (b)-(d) of Fig. 5 show three prototypical cases corresponding to the three steady states found in our model: consensus (d), polarization (b) and ideological state (c). The first case corresponds to questions whose responses are both peaked around a neutral opinion, with a low variance of the opinion distribution. This case is shown in Fig. 5  lated is shown in Fig. 5 (c), with the questions "Should transgender people have to use the bathrooms of the gender they were born as, or should they be allowed to use the bathrooms of their identified gender?" vs. "Do you favor, oppose building a wall on the U.S. border with Mexico?".
One may expect strong opinion correlations only for a pair of questions dealing with very similar topics, such as the one stated in our initial example, about transgender bathrooms and same-sex marriage, which seem intimately related to each other. In the SI we show that the responses to these questions are indeed strongly correlated. The question about building the wall to Mexico, however, seems to be rather unrelated to the issue of transgender bathrooms, so that the high correlation in Fig. 5(c) comes as a surprise. This is not a rare example, and three more are shown in Fig. S3(c)-(f) of the SI. Our model proposes a mechanism which explains the emergence of correlations between opinions with respect to topics with small overlap: If topics are sufficiently controversial, social interactions can reinforce the stance of individuals and trigger the formation of ideological states, as suggested by Fig. 3.

CONCLUSIONS
To sum up, we proposed a simple model able to reproduce crucial features of opinion dynamics as measured in survey data, such as consensus, opinion polarization, and correlation of opinions on different issues, i.e. ideological states. Our model is based on three main ingredients, inspired by empirical evidence: i) The opinion formation is driven by timevarying, homophilic social interactions among the agents, ii) agents sharing similar opinions can mutually reinforce each other's stance, and iii) opinions lay in a multidimensional space, where topics form a non-orthogonal basis (i.e. they can overlap) and can be controversial. Opinion correlations emerge as soon as the assumption of an orthogonal basis is relaxed and topics are allowed to partly overlap. Ideological states appear as a purely collective phenomenon without explicit assumptions of individual attributes of agents favoring one partisanship over another. We analytically and numerically characterize the transitions between the three states, consensus, polarization, and ideology, in dependence on the controversialness and overlap of the topics discussed. The model describes the possibility of strong correlations between opinions with respect to rather unrelated topics provided they are controversial enough, which prediction is corroborated by empirical data of questionnaire surveys.
Of course, our work comes with limitations. With respect to the modelling perspective, it is important to note that our model is based on a minimal number of assumptions. It disregards some empirical features of social interactions such as individual preferences of the agents. This is, however, a necessary trade-off between including realistic features of human behavior and the need to keep the model as simple as possible and the number of parameters small. With respect to the empirical validation, the direct tests about the role of social interactions and the impact of the temporal dimension (evolution of opinions) are not possible. However, a data set which is comprehensive of a large set of topics, such as the ANES, and includes the aforementioned temporal and network information is absent, to the best of our knowledge, and would be quite difficult to collect, also for privacy constraints. The ideal venue to build such data sets could be online social media, where users can take advantage of anonymity in expressing their opinions and social interactions could be reconstructed. We left the design of such as study as important future work. The proposed framework also suggests another interesting direction for future work: to investigate the relation between opinion polarization and issue alignment, whose empirical evidence remains unclear [4]. Finally, it would be extremely interesting to directly quantify topic overlaps in surveys, such as the ANES. This challenge could be addressed by topic modelling of large data sets related to the topics under consideration, such as news articles, and then projecting the trained model (i.e., the topics forming the basis of the space) to the survey data under consideration.

Numerical simulations
For the numerical simulations of Eqs. (2) we set the basic simulation parameters to the following values: N = 1000, T = 2, β = 3, K = 3. The parameters of the basic AD model are set to (m = 10, = 0.01, γ = 2.1), the activity of agents is drawn from the distribution F (a) = 1−γ 1− 1−γ a −γ . The results depicted in Figs. 2-4 differ with respect to the values of α and δ, as reported in the captions and the main text. The initial opinions are sampled from a two dimensional Gaussian distribution with zero mean and unit variance (µ = 0, σ 2 = 1).
The temporal network A ij (t) and the opinion vectors x i are updated at each time step t as follows.
• The temporal network A ij (t) is initially empty. Each agent i is activated with probability a i .
• Each active agent i contacts m distinct agents. Each agent j is chosen according to Eq. (3), where the opinion distance d(x i , x j ), between agents i and j, is computed involving Eq. (2). The elements of the temporal adjacency matrix A ij (t) are set to A ij (t) = A ji (t) = 1 if agent i contacts agent j, or vice-versa.
• After the temporal adjacency matrix A ij (t) is generated, for each agent i the aggregated social input coming from its neighbors is computed and the opinion vector x i (t + 1) is updated by numerically integrating Eq. (1) using an explicit Runge-Kutta 4th order method [65] with dt = 0.01.

Mean-field approximation
For an arbitrary number of topics T , in case of a large number of agents (N 1) and strong homophily (β 1), an agent's opinions will be close to the opinions of its interaction partners, i.e. we have (2). In this approximation, the dynamics of a single agent is then effectively described solely by interactions with neighbors holding the same opinion, i.e., a self-interacting agent. For fast switching interactions, the average number of interactions received by an agent at each time step can approximated by m a . Hence, Eqs. (1) reduce tȯ which describes the opinion dynamics of agents, depending on the topic overlap matrix Φ.
The relation between the controversialness α and the topic overlap cos(δ), marking the transition between a global consensus and the emergence of opinion polarization, can be derived using the Jacobian of Eq. (6). To capture the transition analytically, we additionally assume that all pairwise topic overlaps are equal, i.e. the angles between topic are δ vz = δ ∀v, z. The Jacobian of Eqs.
where we have defined Λ = Km a for brevity. The largest eigenvalue of J(0), λ max , is given as If λ max < 0 the full consensus is stable. Finally, setting Eq. (8) to zero and solving for α yields which relates the critical controversialness α c to the topic overlap cos(δ) for an arbitrary number of topics T . For the sake of simplicity, in the paper we consider the case of two topics. Setting T = 2 in Eq. (9) yields Eq. (5). In this case, Eqs. (6) is reduced to the following non-linear system of equationṡ which give rise, for Km a = 1, to the attractor dynamics depicted in subpanels (d)-(f) of Fig. 2. The stability regions in the α − cos(δ) space, depicted in Fig. 3, are computed based on the Jacobian of Eqs. (10). While the critical controversialness (black dashed line in Fig. 3) is analytically given by Eq. (5), the regions of stability for correlated and uncorrelated polarization must be determined numerically. In the mean-field approximation, we define as uncorrelated polarized states all situations in which the system has two stable fixed points x * with [sgn(x 1 * ), sgn(x 2 * )] = (−, +) and [sgn(x 1 * ), sgn(x 2 * )] = (+, −), respectively. The stability of these fixed points is determined numerically in a two-step procedure. Upon discretizing the α − cos(δ) plane, we first compute, for each {α, cos(δ)} parameter combination, the values of the two fixed points by using the Newton-Raphson method [65]. In a second step, we numerically determine the stability of these fixed points x * by computing the largest eigenvalue of J(x * ). If negative, the corresponding fixed points are stable, and the system is in an uncorrelated polarized state. Otherwise, they are unstable and the system will fall to a polarized ideological state.
Note that for cos(δ) < 0 (δ ∈ ]π/2, π[) the stability of the system is reversed giving rise to negatively correlated opinions, as shown in the SI. This does, however, not lead to qualitatively new dynamical features. With respect to our empirical data analysis, this merely corresponds to re-formulating one of the two questions with a reversed scale. Therefore, we omit this range of negative topic overlap and focus on δ ∈]0, π/2], i.e. positive overlaps.

Empirical Data
The data set analyzed for this work is the 2016 American National Election Survey (ANES) [42]. It includes a total set of 1842 questions. Each of the 4270 respondents is assigned an individual ID, which allows us to correlate responses given by a respondent to different questions. In order to quantify the degree of polarization and issue alignment we compute the variances of responses to single questions and the Pearson correlation coefficients ρ between the responses to pairs of questions. In the caption of Fig. 5 we report these values for the three examples discussed in the main text, other values can be found in the SI.
This procedure requires a numerical scale for the responses. Therefore, we first exclude all questions with free-text answers, such as "What kind of work did you do on your last regular job?". The remaining questions are multiple-choice questions, not all well suited for our purpose. We only select those questions which allow us to extract the extent of approval or disapproval of the respondent with respect to a certain issue. In particular, we choose questions whose response scale allows us to quantify both the qualitative stance (favor or oppose) and the conviction (e.g., favor a great deal, . . . , neutral, . . . , strongly oppose) of the respondent towards the issue, with at least a 4-point scale. Questions whose response scale do not ensure this or questions which do not ask about a specific opinion, such as "Which of the following radio programs do you listen to regularly?" are excluded. In the last step, we exclude questions regarding political parties or presidential candidates. These selection criteria reduce the 2016 ANES data set to a total of 67 questions, depicted in Fig. 5. We report the complete list of selected questions in the SI, together with the question IDs to locate them in the data set provided by [42].
where α 1 and α 2 denote the controversialness values of topics 1 and 2, respectively. While in the main paper, we focused on situations of consensus or polarization in both considered topic dimensions (v = 1, 2), here we shortly discuss a case of different α-values. For one small and one large value of α, as shown in Fig. S1(a), the dynamics of each topic dimensions strongly depends on the respective controversialness. Due to the small value of α 1 (= 0.05), agents approach consensus with respect to topic v = 1, while polarization emerges in the other topic dimension v = 2 as α 2 = 3 is large. A similar behavior arises for the mean-field approximation, were two stable fixed points can be observed at x * (0, ±1), cf. Fig. S1(c). Interestingly, similar states are can also be found in the ANES data set, e.g. see Fig. S3(a), where the responses with respect to one issue ("attitude towards Muslims") show a (neutral) consensus-like situation, while answers with respect to the second question ("service to same-sex couples") are strongly polarized.