Weak convergence of a measure-valued process for social networks
This article formalizes the problem of modeling social networks into an interacting particle system on random geometric graphs. Each vertex of the graph is associated with a geometric position in a latent space describing the unobserved affinity between the network members and characterizes the strength of the interaction between them. We endow the system with two recruitment mechanisms that depend on the positions of the particles in the latent space and one departure mechanism independent of the latent position. We characterize each spatial position by a Dirac measure and the system state by a measure-valued process which is the sum of all Dirac masses. Therefore, we investigate the large-scale behavior of the system. In particular, using a renormalization technique, we study the system's behavior when the initial number of particles goes to infinity. We thus show the weak convergence of the rescaled measure-valued process towards the solution of a deterministic integrodifferential equation. Finally, we present some Monte Carlo simulations with different parameter sets to illustrate the merit of our approach in modeling some phenomena encountered in real-world networks.