Network Evolution Induced by the Dynamical Rules of Two Populations

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 ($κ_a$ and $κ_b\llκ_a$). 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 $θ_0=\moyenne{k_{ab}}/\moyenne{k_{bb}}$ appears to be an integer in the asymptotic limits of the three time regimes. For early times (defined by $t<t_1=κ_b$) the total number of links presents a linear evolution, where the two populations are indistinguishable and where $θ_0=1$. Interestingly, in the intermediate time regime (defined for $t_1<t<t_2\proptoκ_a$ and for which $θ_0=5$), 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 $θ_0=3$.