Aug 2026· Entropy· Vol 28, pp. 857· 0 citations· 54 references
Medicine
Abstract
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.
A Minimally-Nonlinear Opinion Dynamics framework is proposed that analytically characterizes dynamic polarization as a truly emergent collective behavior arising solely from local, agent-level opinion-update rules without externally imposed mechanisms.
In a mechanistic model of the dawn chorus, Kaye showed that heterogeneous activation thresholds and a shared feedback signal determined by the population's active fraction can produce abrupt collective activation and hysteresis. We extend this mechanism to a network of interacting agents. Each node has a continuous activation level, and a nonnegative row-stochastic matrix determines how node activities contribute to individual feedback. We prove that sufficiently weak feedback yields a unique globally attracting equilibrium. For all feedback strengths, the homogeneous dynamics exactly reproduce Kaye's scalar equation; consequently, network topology does not alter the folds or cusp of the homogeneous branch, and no heterogeneous mode becomes unstable before the homogeneous mode. For equitable partitions, the network admits an exact quotient system in which nodes within a block receive the same aggregate input from every block. When blocks are uncoupled, the quotient reduces to independent copies of Kaye's scalar equation. We show that every assignment of stable scalar equilibria to blocks persists under sufficiently weak interblock coupling, producing a combinatorial family of stable quotient equilibria that lift to stable full-network equilibria and remain under small perturbations that break exact equitability. For two symmetrically coupled blocks, branches with unequal block activities terminate at a pair of symmetry-related cusp bifurcations. Near the onset of bistability, we derive scaling laws for the interblock coupling at which these bifurcations occur, the activity difference between the blocks at bifurcation, and the corresponding shift of the external stimulus from the scalar cusp. Numerical continuation confirms the scaling laws for gamma, logistic, and normal threshold distributions.
Stochastic binary networks are widely used to describe collective dynamics in complex systems and to perform neuromorphic computation, yet realistic networks often contain both asymmetric interactions and finite signal propagation times that fall outside conventional theories. Here we study stochastic binary networks with asymmetric and time-delayed interactions motivated by experimental observations in coupled superparamagnetic tunnel junctions. We find that time delay fundamentally reshapes the dynamics induced by anti-symmetric couplings, producing strong oscillatory temporal correlations consistent with experiment. At the same time, sufficiently long delays drive the steady-state probabilities toward equal state occupations even in strongly coupled systems. These apparently featureless probability distributions coexist with pronounced temporal correlations, distinguishing them from equilibrium high-temperature behavior. We further show analytically that delay-induced uniform distributions emerge in a broad class of stochastic networks, while symmetry-breaking bias fields restore interaction-dependent steady states with qualitatively modified behavior. Simulations of networks with five coupled spins demonstrate that these effects persist beyond minimal systems with only two spins. Our results establish a unified framework for stochastic binary networks in the intermediate regime between symmetric instantaneous interactions and asymmetric or time-delayed interactions, and suggest that asymmetry and delay can be exploited as functional resources in neuromorphic hardware and complex network dynamics.
Hantao Zhang, Sidra Gibeault, M. Daniels et al.· arXiv.org· 0 citations
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
Explosive death - a discontinuous, first-order transition from oscillatory dynamics to a steady state - is investigated in a framework of interacting networks under mixed coupling. Unlike previous studies focused on monolayer topologies, this work explores the interplay between local diffusive coupling among peripheral nodes and their respective hubs, and conjugate (dissimilar variable) interactions between the hubs of different layers. We demonstrate that by tuning the intralayer and interlayer coupling strengths, the system exhibits a sudden collapse of oscillations into a steady state. The generality of this explosive transition is verified across a diverse range of dynamical systems, including the Stuart-Landau limit cycle oscillator, and the Hindmarsh-Rose bursting neuron model with star and scale-free networks. Our results suggest that the combination of star-type or scale-free network connectivity and mismatch in coupling variables provides a robust mechanism for inducing abrupt transitions to quenching in multilayered nonlinear systems within the explored parameter space.
Ranjib Banerjee, Dibakar Ghosh, Amit Sharma· Europhysics letters· 0 citations
We study nonzero-frequency phase-locked motions in asymmetrically coupled Kuramoto networks. Such motions are relative equilibria with fixed phase differences and a nonzero common angular velocity, and we call them dancing equilibria. Their existence requires all coupling sums to have the same nonzero value. We show that neither symmetric coupling nor an acyclic associated digraph can support a dancing equilibrium. We introduce structurally equitable and $q$-twisted state equitable partitions and prove a partition-based criterion for the resulting class-constant profiles, with standard labeled $q$-twisted profiles recovered from singleton partitions. For the forward $m$-neighbor model, we characterize existence by an exact indivisibility criterion. Stability is studied modulo the common phase-shift direction. For general directed networks, strong connectivity and edgewise phase differences in $\left(-\pi/2,\pi/2\right)$ imply local orbital exponential stability and yield an explicit positively invariant set contained in the local basin of attraction. For arbitrary twisted indices, this contraction argument gives a low-winding stability regime with explicit positively invariant neighborhoods. For each existing $q$-twisted branch of the forward model, a discrete Fourier transform criterion yields local orbital exponential stability when all nonzero Fourier-mode factors are positive and nonlinear instability when at least one is negative. In the unstable case, the proof constructs explicit escaping real Fourier perturbations. We further derive additional explicit stability and instability ranges for arbitrary twisted indices in terms of constants $N$, $m$, and $q$. For the first two twisted branches, sharper arguments yield a first-mode transition criterion for $q=1$ and a complete finite-size classification for $q=2$, with the degenerate case in each branch handled separately.
Wen Sun, Yu-Qing Wang, Jiu‐Gang Dong· 0 citations
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