Cooperation dynamics in networked geometric Brownian motion

Recent works suggest that pooling and sharing may constitute a fundamental mechanism for the evolution of cooperation in well-mixed fluctuating environments. The rationale is that, by reducing the amplitude of fluctuations, pooling and sharing increases the steady-state growth rate at which individuals self-reproduce. However, in reality interactions are seldom realized in a well-mixed structure, and the underlying topology is in general described by a complex network. Motivated by this observation, we investigate the role of the network structure on the cooperative dynamics in fluctuating environments, by developing a model for networked pooling and sharing of resources undergoing a geometric Brownian motion. The study reveals that, while in general cooperation increases the individual steady state growth rates (i.e., is evolutionary advantageous), the interplay with the network structure may yield large discrepancies in the observed individual resource endowments. We comment possible biological and social implications and discuss relations to econophysics.

Figure 1

Networked GBM with pooling and sharing of resources. The resources of three individuals grow according to GBM and after that they are pooled in $n$ and $m$. Finally, the pools distribute the pooled resources equally among its participants. For visualization purposes we set $dt=1$.

Figure 2

Graph representation. (a) Example of a random graph with 5 individuals. (b) The bipartite representation modeling interactions in a pooling-sharing game, as used in the numerical experiments. (c) The replacement graph capturing effective reallocation of resources between the individuals. The edges are the nonzero elements of the transition matrix $\mathbf{A}$, as in Eq. (8). (d) The transition matrix $\mathbf{A}$.

Figure 3

Transient regime dynamics. Individual growth rate dynamics for sample RR, ER, BA and WS graphs, for (a) small and (b) large networks. Filled lines represent simulated values while the dashed lines are the analytical solutions of the individual growth rate. The GBM parameters are set to $\mu =0.3$ and ${\sigma }^2=0.4$. The results are averaged across 100 realizations of pooling and sharing processes with each graph each having an average degree $\langle d\rangle =5$.

Figure 4

Steady state distribution of resources. Estimated PDF for the rescaled resources for four different types of random graphs—RR, ER, WS, and BA, each having an average degree $\langle d\rangle =5$. (a) Results for small graphs ($N=10$). (b) Results for large graphs ($N=100$). The inset plots depict the ratio of the estimated growth rate and the drift parameter for the same graphs. Filled lines represent the simulated values while the dashed lines are the analytical solutions of the corresponding variables. In the simulation $\mu =0.3$ and ${\sigma }^2=0.4$. For each graph type, the results are averaged across 100 realizations.

Figure 5

Network sparsity and steady-state resource distribution. The variance of rescaled resources as a function of the average degree $\langle d\rangle$ for four different types of random graphs – RR, ER, WS, and BA. (a) Results for small graphs ($N=10$). (b) Results for large graphs ($N=100$). The corresponding $g_i/\mu$ ratios are depicted in the insets. Filled lines represent the simulated values while the dashed lines are the analytical solutions of the corresponding variables. In the simulation $\mu =0.3$ and ${\sigma }^2=0.4$. For each graph type, the results are averaged across 100 realizations.