Diffusion-induced instabilities promote cooperation in eco-evolutionary networks.

Evolution of cooperation on graphs with degree-dependent inertia

The Impact of Neutral Subpopulations on Cooperation in Two-Layer Coupled Networks

Sustaining cooperation under severe social dilemmas is a fundamental challenge in complex systems. This paper proposes a two-layer coupled network model integrating three neutral subpopulations, combining an upper human layer (Fermi rule) and a lower agent layer (Bush–Mosteller reinforcement learning). The core scientific contribution is revealing that the three-subpopulation structure induces closed invasion cycles. This cross-subpopulation reciprocal suppression effectively halts the global expansion of defectors. Monte Carlo simulations demonstrate that under a severe dilemma (b=1.8), optimizing the coupling strength boosts the cooperation persistence probability (PCC) by 91% and reduces defection persistence (PDD) by 55%, stabilizing the global cooperation rate at approximately 50%. Furthermore, for b>1.26, this model consistently outperforms the canonical BM model. Practically, these findings provide a theoretical foundation and a quantitative reference for designing cooperative mechanisms in human–machine collaboration and public governance.

Effects of strategy-relationship coevolution on the emergence of cooperation in multilayer networks

Since social interactions are inherently embedded in multiple relational contexts, single-network models often fall short in explaining the evolution of cooperation. This study develop a two-layer coevolutionary model where behavioral strategies in the upper interaction layer are coupled with the lower signed emotional layer, representing friendly or hostile ties. The framework of this study allows behavioral strategies, emotional attitudes, and the network structure to coevolve dynamically. It found that the dynamics of the emotional layer influence the evolutionary outcomes. Counterintuitively, a relatively low cross-layer coupling strength proves more favorable for sustaining cooperation. It also showed that stochasticity is crucial for breaking the monostability of the defection-dominated state. It provides the necessary conditions for the system to enter another stable cooperative state or mixed strategy, effectively leading to the existence of system bistability. In the end, although the evolution of emotions changes the distribution of the final probability of cooperation, it is more like a regulator of cooperation frequency and cannot significantly improve the overall level of cooperation. This highlights how the constantly evolving relationship environment affects the trajectory of social cooperation.

Backward bifurcations in spatial replicator models:when invasion criteria fail to predict coexistence

The replicator equation is a central framework for studying frequency--dependent selection in ecology and evolutionary game theory. In well-mixed populations, the long-term outcome of a two-species system is determined by the signs of the pairwise invasion fitnesses, leading to dominance, coexistence, or bistability. However, many ecological systems are spatially structured, with environmental heterogeneity and dispersal shaping local interactions. How these spatial effects modify the classical replicator regimes remains incompletely understood. In this work, we study a spatially heterogeneous extension of the two-species replicator equation in which pairwise invasion fitnesses vary across space, and frequencies evolve under diffusion and advection. In this setting, the classical invasion fitnesses are replaced by spatial invasion rates given by the principal eigenvalues of the associated linearized operators. Using bifurcation theory, we show that these spatial invasion rates do not fully determine the qualitative dynamics of the system. In particular, spatial heterogeneity can induce a backward bifurcation, generating stable coexistence states even in parameter regimes where the well-mixed replicator predicts competitive exclusion. We derive an explicit local condition for this mechanism, characterizing when such coexistence states arise. These results show that spatial structure can fundamentally alter classical replicator dynamics and provide a concrete mechanism through which coexistence may emerge.

Turing instability and bistable epidemic spread on complex networks: Effects of topology and initial seeding.

Understanding how diseases propagate through structured populations is essential for predicting and controlling epidemics. This study develops a reaction-diffusion framework on complex networks to investigate how local bistability, dispersal, and network topology jointly determine infection dynamics. Analytical conditions for Turing instability are derived and examined in Erdős-Rényi and scale-free networks using a bistable susceptible-infected model under mean-field approximation. The analysis shows that high-degree nodes become monostable, whereas low and intermediate-degree nodes exhibit bistability. Simulations confirm that infection outcomes depend strongly on degree distribution, transmission rate, and dispersal intensity. Epidemics seeded at highly connected nodes spread faster and more extensively, while structural differences between network types yield distinct thresholds and outbreak patterns. Together, these results reveal how local nonlinearities and network heterogeneity interact to shape epidemic transitions, offering a theoretical basis for understanding spatial disease persistence and designing targeted control strategies.

Metastability and Entropy Peaks in Antagonistic Multiplex Consensus Dynamics

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.