Oscillatory and Excitable Dynamics in an Opinion Model with Group Opinions

TL;DR

This work extends opinion-dynamics modeling to include group opinions by introducing a hypergraph framework with binary node and group (triangle) states. A mean-field reduction yields a three-parameter map for the order parameters $V^t$, $U^t$, and $Y^t$, capturing both dyadic and triadic influences via sigmoidal update rules. The analysis reveals rich behavior, including group–node discordance, excitability, and self-sustained oscillations that arise when dyadic and triadic degrees are not perfectly correlated, with bifurcations such as SNIC and Hopf governing transitions. The findings highlight the significant impact of higher-order interactions on collective opinion dynamics and suggest multiple avenues for extending the framework and validating against data.

Abstract

In traditional models of opinion dynamics, each agent in a network has an opinion and changes in opinions arise from pairwise (i.e., dyadic) interactions between agents. However, in many situations, groups of individuals possess a collective opinion that can differ from the opinions of its constituent individuals. In this paper, we study the effects of group opinions on opinion dynamics. We formulate a hypergraph model in which both individual agents and groups of 3 agents have opinions, and we examine how opinions evolve through both dyadic interactions and group memberships. In some parameter regimes, we find that the presence of group opinions can lead to oscillatory and excitable opinion dynamics. In the oscillatory regime, the mean opinion of the agents in a network has self-sustained oscillations. In the excitable regime, finite-size effects create large but short-lived opinion swings (as in social fads). We develop a mean-field approximation of our model and obtain good agreement with direct numerical simulations. We also show -- both numerically and via our mean-field description -- that oscillatory dynamics occur only when the number of dyadic and polyadic interactions per agent are not completely correlated. Our results illustrate how polyadic structures, such as groups of agents, can have important effects on collective opinion dynamics.

Figures (10)

• Figure 1: A schematic illustration of how the opinions of nodes and groups are influenced by the opinions of other nodes and groups in our model. (a) A hypergraph with 9 nodes and 3 groups. Nodes 1, 3, 4, 6, and 8 have opinion 1 (in blue), and nodes 2, 5, 7, and 9 have opinion 0 (in light red). Groups I and III have opinion 1 (in blue), and group II has opinion 0 (in light red). (b) Node 5's opinion is influenced by the opinion of its neighboring nodes 1, 2, 6, 8, and 9 (thin arrows) and by the opinions of groups I and II (thick arrows). (c) Group I's opinion is influenced by the opinions of its constituent nodes 1, 2, and 5 (thick arrows).
• Figure 2: An example of a bifurcation of the steady-state solutions when the dyadic and triadic degrees are equal (i.e., $r = 1$) in simulations of our stochastic opinion model (\ref{['node_model']})--(\ref{['edge_model']}) and solutions of the mean-field equations (\ref{['reducedmeandyn']}) for the parameter values $a = b = c = d = \mu = 0.5$, power-law exponent $\gamma = 4$, and mean degrees $\langle k \rangle = \langle q \rangle = 20$. The bifurcation parameter is the inverse-width parameter $m$ of the sigmoidal influence function \ref{['sigf']}. We show the values of (a) $V^*$ and (b) $Y^*$ that we obtain from the mean-field equations (solid and dashed curves) and from means of 100 simulations of our stochastic opinion model (dots).
• Figure 3: The group--node discordance $D(V^*,Y^*)$ versus the node-opinion influence parameter $a$, with $a = b$, for a single numerical solution $(V^*,Y^*) = (F(V^*),G(V^*))$ of Eqs. (\ref{['FG']}) (solid curves) and the mean of $16$ independent simulations of the stochastic opinion model (\ref{['node_model']})--(\ref{['edge_model']}) for a single configuration-model hypergraph with $N = 2000$ nodes, inverse-width parameter $m = 4$, power-law exponent $\gamma = 4$, mean degrees $\langle k \rangle = \langle q \rangle = 20$, and several values of the group-influence parameters $c$ and $d$. We consider (a) $\mu = 0.5$ and (b) $\mu = 0.25$. The initial conditions of the 16 simulations are evenly spaced in the unit square (see Appendix \ref{['ICS']}).
• Figure 4: The group--node discordance $D(V^*,Y^*)$ versus the sigmoid inverse-width parameter $m$ for a single numerical solution $(V^*,Y^*) = (F(V^*),G(V^*))$ of Eqs. (\ref{['FG']}) (solid curves) and the mean of $16$ independent simulations of the stochastic opinion model (\ref{['node_model']})--(\ref{['edge_model']}) for a single realization of a configuration-model hypergraph with $N = 2000$ nodes, power-law exponent $\gamma = 4$, mean degrees $\langle k \rangle = \langle q \rangle = 20$, and several values of $a$, $b$, $c$, and $d$. We consider (a) $\mu = 0.5$ and (b) $\mu = 0.25$. The initial conditions of the 16 simulations are evenly spaced in the unit square (see Appendix \ref{['ICS']}).
• Figure 5: An example of opinion pulses in a single simulation of our stochastic opinion model (\ref{['node_model']})--(\ref{['edge_model']}) with parameter values $a = 1$, $b = -0.5$, $c = d = 0.25$, $\mu = 0.25$, and $m = 8$ for a configuration-model hypergraph with equal dyadic and triadic degrees (i.e., $r = 1$) that we draw from an approximate power-law distribution with exponent $\gamma = 4$ and mean-degrees $\langle k \rangle = \langle q \rangle = 20$. We plot the expected node fraction $V^t$ in red and the expected triangle fraction $Y^t$ in blue. The dashed lines show the fixed points that we obtain by solving Eqs. (\ref{['reduceV*']})--(\ref{['reduceY*']}).