From Resistance Flow to Payoff Distribution: The Emergent Dynamics of Weakly Rational Network Exchange

Our paper proposes a weakly rational network exchange model in which individuals remember only their best historical payoff and engage in bilateral bargaining governed by the resistance equation under random activation. The model unifies the power-dependence and resistance paradigms within Network Exchange Theory (NET), showing that core concepts——strong power, exclusion, and equalization——emerge as natural consequences of network structure. Specifically, we rigorously prove finite-step termination (Theorem 1), a positive degree–payoff correlation (Theorem 3), and that the Gini coefficient of the payoff distribution is strictly smaller than that of the degree distribution (Theorem 4). Through heterogeneous mean-field equations and extensive numerical simulations, we uncover several counterintuitive phenomena: higher connection density reduces payoff inequality; the rewiring probability of small-world networks is positively correlated with the Gini coefficient; symmetric community structure produces no between-block payoff differences; and core–periphery structure yields a payoff advantage of approximately 11%. Most strikingly, we reveal a "phase transition without a phase transition"——the criticality of exchange activity (e.g., when rewiring probability or initial payoff crosses a threshold) does not induce any jump in the payoff distribution; payoff inequality always evolves continuously. Moreover, the model semi-quantitatively reproduces classic network exchange experimental results without any free parameters. Finally, by means of scaling laws, we derive the above simulation findings analytically and provide deeper regularities governing the relationships among exchange frequency, complex network topology, average payoff, Gini coefficient, and other emergent quantities: these scaling laws also identify a co-movement trend of equity and efficiency (a negative linear relationship between the Gini coefficient and average payoff), and find that this co-movement arises from an increase in the average degree of the network. Taken together, these findings furnish NET with a general foundation based on resistance and stochastic network interaction, and offer a theoretical foundation for designing equalizing exchange mechanisms.