Representing and Analyzing the Dynamics of an Agent-Based Adaptive Social Network Model with Partial Integro-Differential Equations

We formulated and analyzed a set of partial integro-differential equations that capture the dynamics of our adaptive network model of social fragmentation involving behavioral diversity of agents. Previous results showed that, if the agents’ cultural tolerance levels were diversified, the social network could remain connected while maintaining cultural diversity. Here we converted the original agent-based model into a continuous equation-based one so we can gain more theoretical insight into the model dynamics. We restricted the node states to 1-D continuous values and assumed the network size was very large. As a result, we represented the whole system as a set of partial integro-differential equations about two continuous functions: population density and connection density. These functions are defined over both the state and the cultural tolerance of nodes. We conducted numerical integration of the developed equations using a custom-made integrator implemented in Julia. The results obtained were consistent with the simulations of the original agent-based adaptive social network model we previously reported, confirming the robustness of the original finding. Specifically, when the variance of cultural tolerance d is large enough, the population with low d maintains the original clusters of cultures/opinions, while the one with high d tends to come to the center and connect culturally distant groups. Parameter dependence of the model behavior was also revealed through systematic numerical experiments.