Multifractal analysis of microscopic fluctuation and quasi-equilibrium landscape of evolutionary games with environmental feedback on complex networks

Abstract The evolution of societies within civilizations is characterized by the emergence of cooperative traits through continuous adaptation within their unique complex networks. Notably, cooperative behaviours can sustain local persistence for extended durations before succumbing to total defection. Consequently, this behaviour leads the study to examine evolutionary games with environmental feedback across diverse network topologies, integrating replicator dynamics with resource-dependent pay-offs and agent-based simulations. By exploring and quantifying macroscopic (potential landscape) and microscopic (multifractal fluctuation) perspectives, the research elucidates the influence of network structure and population size on the transition from oscillatory to quasi-equilibrium cooperation regimes. Additionally, this study introduces the innovative application of multifractal detrended fluctuation analysis (MF-DFA) to evolutionary game dynamics, revealing that intermittent, scale-invariant bursts of cooperation in metastable states are not random noise. Moreover, the quantified parameters of the quasi-stationary fluctuations around that interior attractor bear the potential to be used in predicting the transition of the system. Furthermore, findings reveal that larger networks exhibit deeper and narrower potential wells, enhancing metastability and prolonging cooperative interactions. Overall, these results deepen understanding of cooperation dynamics in structured populations and propose a novel method to quantify quasi-stationary fluctuations for predicting cooperation collapse or stability.