SGMR co-evolves strategies and multilayer economic links under one dynamic potential, converging faster with higher welfare and stronger noise resilience on synthetic networks

The strategy and link blocks share one exact dynamic potential even with state-dependent multilayer weights, so global maximizers of the potential correspond to equilibria. Adaptive-network evolutionary games usually prescribe rewiring through behavioral heuristics without a common certificate, while distributed potential-game methods optimize over a fixed communication graph; this work unifies both in one time-indexed potential. The paper provides a theorem and proof of the exact-potential identity and equilibrium existence under Assumption 1 (regularity and bounded observations).

The mirror strategy revision recovers the replicator equation component-wise in the small-step limit, so strategies with above-average marginal payoff grow and those below decline. Unlike a standard replicator process on a fixed graph, the payoff landscape here changes endogenously because the second block updates multilayer links. Proposition 2 gives a component-wise derivation, dividing by the step size to obtain the continuous-time replicator equation.

The spectral guard yields monotone potential improvement and a finite-horizon sublinear stationarity guarantee, plus a local input-to-state stability bound under observation disturbances. The guard is not a post hoc diagnostic but restricts both update steps before each update using the current graph spectrum, turning spectral information into an operational stability decision. Theorems 2, 3, and 4 establish one-step potential improvement, sublinear stationarity, and a local ISS bound; experiments directly evaluate the residual decay and ISS envelope.

On synthetic economic networks SGMR achieves the best welfare in every topology, with the largest advantage on the modular graph, and retains about 0.846 of normalized welfare at noise standard deviation 0.18 versus 0.715 for SMA and 0.616 for RD. Ablations show that removing the target link degrades welfare and raises turnover, removing the spectral guard most affects potential variance and final residual, and removing temporal inertia yields the highest link turnover. Experiments use 40–300 agents, 3 behavioral categories, and 3 layers, comparing against SMA, RD, and DFP; Welch tests remain significant after Bonferroni correction with large standardized effects, though all networks are synthetic.

The framework targets decentralized economic and social systems that must remain stable while strategies and relationships change together, for example settings where market transactions, information diffusion, and institutional coordination coexist. It lets researchers update using local payoffs, neighbor strategies, and incident links without a centralized welfare oracle, and it exposes data confidence, relationship inertia, and diffusion pressure as interpretable policy parameters. For practitioners, the paper sketches a concrete mapping: the market layer from transaction, supplier–buyer, credit-exposure, or co-investment records; the information layer from communication, news co-mention, repost, search, or forecast-sharing data; and the institutional layer from shared contracts, regulatory jurisdictions, standards, or eligibility constraints.

The paper states explicitly that the present evidence is synthetic and mechanism-oriented and does not establish causal validity, forecasting accuracy, or policy effectiveness for any particular economy. A real-data study would need to estimate transaction and exposure links, information-sharing links, and institutional compatibility from time-stamped records, infer rewards from measurable outcomes, and evaluate out-of-sample fit and policy counterfactuals. Open questions also include how the multilayer structure should handle missing links and strategic privacy, how to incorporate directed, signed, higher-order, and coalition interactions, and whether mechanism-design variants can learn incentives that improve welfare, fairness, resilience, and sustainability under explicit budget constraints. The loaded text is an incomplete read, with some equations, table values, and figure captions not fully rendered, which limits restating specific numerical and derivation details.