Conversion of Dynamic Social Network Stochastic Differential Equation Model to Fokker-Planck Model

Stochastic differential equation (SDE) models offer one formulation for introducing uncertainty in human interactions in a dynamic social network model based on static and/or deterministic ordinary differential equation (ODE) models. A coupled SDE system for agent characteristics and connectivities was developed and investigated in [4]. This SDE model (which tacitly assumed instantaneous influence between agents with connectivity) may be improved by including delays in an SDE model or in an equivalent Fokker-Planck (FP) model if such exists. The coupled model of [4] involved discontinuities and did not yield a Markov diffusion process (for which an equivalent Fokker-Planck formulation is possible). In this project we formulate a new smooth vector SDE system and demonstrate that it generates a Markov diffusion process and provides computational results equivalent to those of the earlier model of [4]. We derive an equivalent Fokker-Planck formulation to this new SDE system. Numerical methods to implement the FP model are formulated. This illustrates the disadvantages of including delays in this already complex framework, suggesting the SDE model may, after all, be more tractable for the consideration of delays.