A Bounded-Confidence Model of Opinion Dynamics on Hypergraphs

Abstract

People's opinions evolve with time as they interact with their friends, family, colleagues, and others. In the study of opinion dynamics on networks, one often encodes interactions between people in the form of dyadic relationships, but many social interactions in real life are polyadic (i.e., they involve three or more people). In this paper, we extend an asynchronous bounded-confidence model (BCM) on graphs, in which nodes are connected pairwise by edges, to an asynchronous BCM on hypergraphs, in which arbitrarily many nodes can be connected by a single hyperedge. We show that our hypergraph BCM converges to consensus for a wide range of initial conditions for the opinions of the nodes, including for nonuniform and asymmetric initial opinion distributions. We also show that, under suitable conditions, echo chambers can form on hypergraphs with community structure. We demonstrate that the opinions of nodes can sometimes jump from one opinion cluster to another in a single time step; this phenomenon (which we call ``opinion jumping"") is not possible in standard dyadic BCMs. Additionally, we observe a phase transition in the convergence time of our BCM on a complete hypergraph when the variance \sigma 2 of the initial opinion distribution equals the confidence bound c. We prove that the convergence time grows at least exponentially fast with the number of nodes when \sigma 2 > c and the initial opinions are normally distributed. Therefore, to determine the convergence properties of our hypergraph BCM when the variance and the number of hyperedges are both large, it is necessary to use analytical methods instead of relying only on Monte Carlo simulations.

A density description of a bounded-confidence model of opinion dynamics on hypergraphs

“…This result is consistent with the steady state and convergence results for agent-based models in [33]. We also consider a special type of hypergraph in which each hyperedge has three nodes -such hypergraphs occur, for example, in the study of folksonomies [19] -and numerically examine the steady-state distributions of opinion clusters for different discordance functions, which are analogous to the confidence bounds of dyadic BCMs [23]. We observe numerically for both bounded and unbounded opinion distributions that the cluster locations undergo a periodic sequence of bifurcations as we increase the variance of the initial opinion distribution.…”

“…The edges in a graph connect only two nodes at a time (or a single node to itself, in the case of a self-edge), whereas the hyperedges in a hypergraph connect any number of nodes [9,38] and thereby allow one to study the collective interactions of arbitrarily many nodes. Very recently, two papers have generalized BCMs to hypergraphs [23,44]. Hickok et al [23] showed that polyadic interactions in a BCM can enhance the convergence to opinion consensus and that BCMs on hypergraphs can possess qualitative dynamics, such as opinion jumps, that do not occur for BCMs on regular graphs.…”

“…Very recently, two papers have generalized BCMs to hypergraphs [23,44]. Hickok et al [23] showed that polyadic interactions in a BCM can enhance the convergence to opinion consensus and that BCMs on hypergraphs can possess qualitative dynamics, such as opinion jumps, that do not occur for BCMs on regular graphs.…”

“…We introduce scaling constants α p to reduce the discrepancy between different choices of the parameter p; we discuss the selection of α p in Section 3. Other choices of discordance functions include the L ∞ norm [44] and the sample variance [23].…”

Dynamics on higher-order networks: a review

“…Apart from the hypergraph models like block hypergraphs, analysis has been performed on real-world networks as well. In [ 93 ], the authors formulated a hypergraph bounded confidence model and showed the appearance of a scenario named ‘opinion jumping’, in which individuals’ opinions can jump from one cluster of opinions to another, which one does not observe in dyadic connectivity. Moreover, echo chambers are witnessed to emerge on hypergraphs with community structure.…”