Network Evolution Induced by the Dynamical Rules of Two Populations

TL;DR

This paper analyzes the evolution of a two-population network with competing link creation and destruction rules, revealing distinct dynamic regimes and a surprising integer ratio of cross to within-group links in the steady state.

Contribution

It introduces a mean field approach to model the network dynamics of two interacting populations with different preferred degrees and validates it with Monte Carlo simulations.

Findings

Three distinct time regimes with different degree behaviors.

The ratio of cross to within-group links becomes an integer in steady state.

Transient states exhibit linear growth and constant contact phases.

Abstract

We study the dynamical properties of a finite dynamical network composed of two interacting populations, namely; extrovert ($a$) and introvert ($b$). In our model, each group is characterized by its size ($N_a$ and $N_b$) and preferred degree ($\kappa_a$ and $\kappa_b\ll\kappa_a$). The network dynamics is governed by the competing microscopic rules of each population that consist of the creation and destruction of links. Starting from an unconnected network, we give a detailed analysis of the mean field approach which is compared to Monte Carlo simulation data. The time evolution of the restricted degrees $\moyenne{k_{bb}}$ and $\moyenne{k_{ab}}$ presents three time regimes and a non monotonic behavior well captured by our theory. Surprisingly, when the population size are equal $N_a=N_b$, the ratio of the restricted degree $\theta_0=\moyenne{k_{ab}}/\moyenne{k_{bb}}$ appears to be an integer in the asymptotic limits of the three time regimes. For early times (defined by $t<t_1=\kappa_b$) the total number of links presents a linear evolution, where the two populations are indistinguishable and where $\theta_0=1$. Interestingly, in the intermediate time regime (defined for $t_1<t<t_2\propto\kappa_a$ and for which $\theta_0=5$), the system reaches a transient stationary state, where the number of contacts among introverts remains constant while the number of connections is increasing linearly in the extrovert population. Finally, due to the competing dynamics, the network presents a frustrated stationary state characterized by a ratio $\theta_0=3$.