It is found that opinion-independent Brownian mobility produces no detectable geographic opinion structure in the explored regime, whereas opinion-dependent drift produces spatial domains through a Peclet-controlled crossover near $\chi\ell/D \sim 1$.
Abstract
We study how physical mobility, algorithmic exposure, and repulsive social influence interact in a stochastic multiplex model of opinion dynamics. Agents diffuse in physical space while a directed digital network rewires under a conserved attention budget, so digital exposure displaces rather than supplements local interaction. With purely assimilative bounded-confidence influence, opinion-blind long-range exposure reduces locality-induced fragmentation whereas homophilic recommendation preserves echo chambers. When a contested repulsive response to sufficiently distant opinions is activated, this ordering reverses at the reference parameters: a neutral platform reaches the maximal polarization permitted by the bounded opinion space, controversy-seeking curation drives faster initial separation but slows sharply near the boundary, and homophilic curation delays radicalization by suppressing cross-bloc exposure. In a late-stage symmetric two-bloc reduction, any curation kernel maps to a state-dependent cross-bloc exposure profile $p(y)$ and an exact quadrature for the radicalization time. Pointwise-ordered profiles inherit a global kinetic ordering; crossing profiles yield target- and horizon-dependent rankings. For similarity-driven curation the quadrature has a closed form involving the exponential integral. Simulations, finite-size scans to $N=1600$, structural controls, and a well-mixed particle comparison support the mechanism. Heavy-tailed influence strengths are not required for the inversion; in the well-mixed heavy-tail regime they additionally produce a non-self-averaging stable-weighted asymptotic description. Finally, opinion-independent Brownian mobility produces no detectable geographic opinion structure in the explored regime, whereas opinion-dependent drift produces spatial domains through a P\'eclet-controlled crossover near $\chi\ell/D \sim 1$.
Collective opinions in social networks evolve through local interaction rules, yet how such local updates give rise to dynamic polarization--persistent oscillatory disagreement between opposing opinion clusters at the network level--remains unexplained. This paper proposes a Minimally-Nonlinear Opinion Dynamics (or M-NOD) framework that analytically characterizes dynamic polarization as a truly emergent collective behavior arising solely from local, agent-level opinion-update rules without externally imposed mechanisms. By introducing a minimal cubic nonlinearity, we rigorously prove that, as the reactivity rate exceeds a critical threshold, the network's consensus equilibrium loses stability via a supercritical flip bifurcation. In the post-bifurcation regime, this instability gives rise to a unique, locally asymptotically stable periodic orbit, thereby characterizing symmetric dynamic polarization with balanced bipartite opinion clusters. We further establish the structural robustness of this behavior by proving the existence and local asymptotic stability of asymmetric dynamic polarization under directed graphs with nonuniform influence weights. Finally, to resolve this undesirable cyclic deadlock, we develop local agent-level control strategies. We prove that anchoring the opinion of only a single agent is sufficient to eliminate network-wide oscillatory disagreement and restore asymptotically stable consensus. Numerical simulations substantiate the theoretical analysis, including the emergence of symmetric and asymmetric dynamic polarization, and demonstrate the efficacy of the proposed control interventions.
This work shows that a dual-threshold bootstrap percolation model on random hypergraphs separates a connected active backbone from large-scale endogenous activation, providing a basis for predicting cascade risk and designing targeted node- and group-level interventions in complex systems.
We introduce a majority-rule model in which collective reversal can be activated in highly aligned groups even when a limited number of members dissent. The dissent tolerance $d$ extends the strict-unanimity dynamics by making near-unanimous group compositions eligible for reversal. Mean-field analysis and simulations reveal that this change qualitatively alters the phase structure. Under strict unanimity, a physically accessible transition exists only for $n=3$ and $n=4$. Allowing dissent restores transitions at larger interaction sizes, replacing the fixed interaction-size threshold with an accessibility boundary in the $(n,d)$ plane. When the activation window is sufficiently broad, directional asymmetry can eliminate one of the two ordered attractors through a saddle-node bifurcation, producing a single stable collective state. In the one-sided case, increasing the dissent tolerance can shorten the transient approach to consensus but leaves its leading logarithmic dependence on population size unchanged. Activation selectivity acts as an independent control parameter for collective ordering, bistability, and consensus dynamics.
Roni Muslim, Rinto Anugraha Nqz, Qonuni Gusthaf Haq et al.· 0 citations
Modern societies comprise overlapping communities whose opinions evolve on strongly interacting networks that are often mutually antagonistic. We introduce a minimal antagonistic multiplex consensus model in which each layer follows intra-layer majority-rule dynamics, while inter-layer interactions are inhibitory. A mean-field analysis shows that antagonistic coupling destabilizes the balanced state through an antisymmetric mode and favors two polarized absorbing states with opposite magnetization in the two layers. Network-averaged simulations confirm that small fluctuations near equal initial support determine which polarized state is ultimately reached: trajectories exhibit metastable delay, long convergence times, and a localized peak in the Shannon entropy of outcomes. A finite-size analysis with independent network realizations and bootstrap uncertainty estimates shows that the high-entropy interval narrows as Δr ∼N−γeff, with γeff=0.513 and a 95% bootstrap confidence interval [0.489,0.526], consistent with finite-size sharpening controlled by fluctuations in the initial imbalance. We also perform network topology checks and find that the qualitatively antagonistic mechanism persists beyond random-regular graphs. As an illustrative empirical application, we analyze county-level results from the 2024 U.S. presidential election. The vote-share and entropy landscapes separate low-entropy partisan strongholds from higher-entropy competitive counties. Our results suggest that antagonistic multiplex coupling provides a simple mechanism by which polarized attractors and localized outcome uncertainty can arise together.
J. C. Hughes, A. Kusmartseva, G. Muschert et al.· Entropy· 0 citations
We propose a novel framework for modeling binary opinions (0 or 1) of individuals connected through a weighted directed network, where edge weights quantify interpersonal influence. Unlike classical models that assume complete access to previously expressed opinions, our framework allows individuals to update their biases using structured memory sets that capture limited and delayed information exchange. To analyze these opinion differences, we introduce a mathematically tractable notion of relative bias between pairs of individuals. The relative biases evolve according to a linear update rule involving past expressed opinions specified by the memory sets. We define the belief of an individual as the probability of expressing opinion 1 and derive a time-delayed dynamical system governing the evolution of network beliefs. We establish its asymptotic behavior and characterize its properties. The framework is further extended to networks containing bots, which maintain fixed biases while influencing neighboring individuals. We quantify the effect of bots by comparing the fixed points of the dynamics in their presence and absence. Finally, simulations illustrate the influence of memory, network structure, and bot interactions on the resulting opinion dynamics.
