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

The evolution of societies within civilizations is characterized by the
emergence of cooperative traits through continuous adaptation within their
unique complex networks. Cooperative behaviors can sustain local
persistence for extended durations before succumbing to total defection,
motivating this study to investigate evolutionary games with environmental
feedback across diverse network topologies, integrating replicator
dynamics with resource-dependent payoffs 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. This
study introduces the innovative application of multifractal detrended
fluctuation analysis (MF-DFA) to evolutionary game dynamics, revealing
that intermittent, scale-invariant bursts in cooperation are not random
noise but carry predictive information about metastable states. Employing
MF-DFA quantifies these bursts of cooperative behavior, demonstrating how
network size affects cooperation’s stability and potential landscape.
Findings reveal that larger networks exhibit deeper and narrower potential
wells, enhancing metastability and prolonging the endurance of cooperative
interactions. These results advance understanding of cooperative evolution
within finite, structured populations, providing valuable insights into
mechanisms sustaining cooperation in real-world adaptive systems and
suggesting a novel method to detect early-warning signals of cooperation
collapse or stability.