Emergence of Polarized Ideological Opinions in Multidimensional Topic Spaces

Open Access

Emergence of Polarized Ideological Opinions in Multidimensional Topic Spaces

Fabian Baumann, Philipp Lorenz-Spreen, Igor M. Sokolov, and Michele Starnini

Phys. Rev. X 11, 011012 – Published 20 January 2021

Abstract

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 individual preferences or in a preexisting social structure? Here, we propose a simple model that qualitatively 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 of opinions unfolding in a multidimensional space where topics form a nonorthogonal basis. 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. The model features phase transitions between a global consensus, opinion polarization, and ideological states. Interestingly, the ideological phase emerges by relaxing the assumption of an orthogonal basis of the topic space, i.e., if topics thematically overlap. Furthermore, we analytically and numerically show that these transitions are driven by the controversialness of the topics discussed; the more controversial the topics, the more likely are opinions to be correlated. Our findings shed light upon the mechanisms driving the emergence of ideology in the formation of opinions.

Received 22 July 2020
Revised 10 November 2020
Accepted 2 December 2020

DOI:https://doi.org/10.1103/PhysRevX.11.011012

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Collective behavior in networks Network phase transitions Social systems

Social dynamics Social networks

Networks

Authors & Affiliations

Fabian Baumann^1,*, Philipp Lorenz-Spreen², Igor M. Sokolov ^1,3, and Michele Starnini^4,†

¹Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
²Center for Adaptive Rationality, Max Planck Institute for Human Development, Lentzeallee 94, 14195 Berlin, Germany
³IRIS Adlershof, Humboldt-Universität zu Berlin, Zum Großen Windkanal 6, 12489, Berlin, Germany
⁴ISI Foundation, via Chisola 5, 10126 Torino, Italy

^*fabian.olit@gmail.com
^†michele.starnini@gmail.com

Popular Summary

Public opinion polarization is on the rise, as suggested by data collected across a variety of political and ethical issues. In the U.S., for example, society recently split with respect to so-called bathroom bills, which restrict public restroom access by gender assigned at birth. Opinions on different topics can also be correlated: Individuals advocating rights for transgender people, for example, are likely to also support same-sex couples. Strikingly, data show correlations between unrelated topics, such as rights of transgender people and the debate about whether the U.S. should build a wall at the border with Mexico. Here, we propose a simple model that reproduces such polarized ideological states, in which opinions are extreme and correlated with respect to different topics.

We develop a mathematical formalism in which opinions evolve in a multidimensional space, representing the relation between different topics. We show that a simple reinforcement mechanism coupled with homophilic social interactions leads to the emergence of polarized, ideological opinions, even between unrelated topics that are sufficiently controversial.

People usually influence each other while discussing a certain topic. But when topics are controversial, the reinforcement mechanism proposed here spills over into a plethora of different topics. Two users advocating for transgender rights in a Twitter debate, for example, will reinforce their stances with respect to other controversial topics, such as the U.S. building a wall at the border with Mexico.

Our findings shed light on the mechanisms—both offline and online—driving the process of opinion formation.

Key Image

Article Text

Click to Expand

Supplemental Material

Click to Expand

References

Click to Expand

Issue

Vol. 11, Iss. 1 — January - March 2021

Subject Areas

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Illustration of two nonorthogonal topics as a basis for the topic space $T$ . For $T = 2$ , the nonorthogonal, 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 ( $e^{(u)}$ ) and transgender people ( $e^{(v)}$ ). The opinion distance between two agents $i$ and $j$ , $d (x_{i}, x_{j})$ , is computed by the scalar product defined in Eq. (2).
Reuse & Permissions
Figure 2
Temporal evolution of the agents’ opinions in a $T = 2$ topic space. Evolution of opinions from numerical simulations (a)–(c) and corresponding deterministic dynamics (d)–(f) from mean-field approximation, with identical values of $α$ and $δ$ ; cf. Appendix pp2 for details. The trajectories of the agents’ opinions are depicted as gray lines, and final opinions are colored according to $φ$ . The system reaches a global consensus if topics are not controversial, for small $α = 0.05$ (a), while polarization emerges for controversial topics, for larger $α = 3$ (b),(c). This result is indicated by the marginal distributions $P_{u} (x)$ and $P_{v} (x)$ : The values of the variances are $σ_{u}^{2} (x) = 0.04$ and $σ_{v}^{2} (x) = 0.035$ in (a), $σ_{u}^{2} (x) = 7.27$ and $σ_{v}^{2} (x) = 7.17$ in (b), and $σ_{u}^{2} (x) = 11.22$ and $σ_{v}^{2} (x) = 11.2$ in (c). If topics do not overlap ( $δ = π / 2$ ), all combinations of opinion stances appear in consensus (a) and uncorrelated polarized states (b) with low correlation values of $ρ (x^{(u)}, x^{(v)}) = 0.01$ (a) and $ρ (x^{(u)}, x^{(v)}) = 0.024$ (b), respectively. If topics overlap [ $δ = π / 4$ matching the angle between $x$ and $y$ axis in (c)], opinions become correlated and ideological states emerge.
Reuse & Permissions
Figure 3
Stability regions of the mean-field approximation as a function of the topic overlap $\cos (δ)$ and controversialness $α$ for $2 K m ⟨ a ⟩ = 1$ . The different regions in the phase space are colored according to the corresponding states: consensus (green), uncorrelated opinion polarization (blue), and ideological state (orange). The black dashed line depicts the critical controversialness $α_{c}$ separating the regions of consensus and opinion polarization, as given by Eq. (5). The phase diagram and $α_{c}$ are symmetric with respect to $\cos (δ) = 0$ , i.e., $δ = π / 2$ ; see Appendix pp2 for details. The symbols (square, cross, and rhombus) depict the parameter combinations of $α$ and $\cos (δ)$ used in Figs. 2 and 5.
Reuse & Permissions
Figure 4
Temporal evolution of the agents’ opinions in a $T = 3$ topic space. Evolution of opinions from numerical simulations for strong social influence ( $K = 3$ ), controversial topics ( $α = 3$ ), and high homophily ( $β = 3$ ). The grey lines and black dots depict the time evolution of agents’ opinions and the steady states, respectively. In both cases (a) and (b) the topics $u$ and $v$ are orthogonal. In panel (a) topic $z$ is also orthogonal to both topics $u$ and $v$ . In this case an uncorrelated polarized state emerges, with weak correlations among the three opinions: $ρ (x^{(u)}, x^{(v)}) = 0.15$ , $ρ (x^{(u)}, x^{(z)}) = 0.17$ and $ρ (x^{(v)}, x^{(z)}) = 0.11$ . In panel (b) topic $z$ has a finite overlap with both topics $u$ and $v$ , i.e. $δ_{u z} = δ_{v z} = π / 4$ . In this case, an ideology state emerges: opinions with respect to the three topics are correlated, $ρ (x^{(u)}, x^{(v)}) ≃ ρ (x^{(u)}, x^{(z)}) ≃ ρ (x^{(v)}, x^{(z)}) ≃ 1$ . Note that for simplicity of illustration the opinion space in panel (b) is depicted using orthogonal axes, although $δ_{u z} = δ_{v z} < π / 2$ .
Reuse & Permissions
Figure 5
Community structure of the social networks. Visualization of the social networks aggregated over the last 70 time steps (top) and corresponding community detection (bottom) for three different dynamical regimes: (approaching) consensus (a), uncorrelated polarized state (b), and ideological state (c). The model parameters are set as in Figs. 2, i.e., $α = 0.05$ , $δ = π / 2$ (a), $α = 3$ , $δ = π / 2$ (b), and $α = 3$ , $δ = π / 4$ (c). Top: In the network illustrations, each node is colored according to its opinion angle $φ$ , and its size is proportional to its conviction $r$ . Bottom: Communities are represented in the polar bar plot below each network. Each community is represented by a bar: The radius represents the size, and 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.
Reuse & Permissions
Figure 6
Responses to questions from the ANES. (a) Variance of all responses and absolute value of pairwise Pearson correlation. (b)–(d) Scatter plots of selected pairs of questions $v$ and $z$ , where each dot represents one respondent by his or her responses to both questions. The marginal plots represent the response distributions $P_{v} (x)$ and $P_{z} (x)$ . To improve the visualization, data are jittered [74], and uniformly distributed noise is applied, to up to 80% of the bin size, to avoid overplotting of the categorical data and at the same time assure discrete separation. The examples are selected to represent different combinations of response variance (opinion polarization) and response correlation: (d) low variance ( $σ_{v}^{2} = 0.08$ , $σ_{z}^{2} = 0.25$ ) and low correlation [ $| ρ (v, z) | = 0.02$ ] for questions V162176x vs V161154, denoting ANES IDs (see Supplemental Material [56] for a complete list of IDs); (b) high variance ( $σ_{v}^{2} = 0.58$ , $σ_{z}^{2} = 0.64$ ) and low correlation [ $| ρ (v, z) | = 0.03$ ] for V161151x vs V161114x; and (c) high variance ( $σ_{v}^{2} = 0.62$ , $σ_{z}^{2} = 0.49$ ) and high correlation [ $| ρ (v, z) | = 0.44$ ] for V161228x vs V161196x.
Reuse & Permissions

Physical Review X