Diffusion on assortative networks: from mean-field to agent-based, via Newman rewiring

Abstract

In mathematical models of epidemic diffusion on networks based upon systems of differential equations, it is convenient to use the heterogeneous mean field approximation (HMF) because it allows to write one single equation for all nodes of a certain degree k, each one virtually present with a probability given by the degree distribution P(k). The two-point correlations between nodes are defined by the matrix P(h|k), which can typically be uncorrelated, assortative or disassortative. After a brief review of this approach and of the results obtained within this approximation for the Bass diffusion model, in this work, we look at the transition from the HMF approximation to the description of diffusion through the dynamics of single nodes, first still with differential equations, and then with agent-based models. For this purpose, one needs a method for the explicit construction of ensembles of random networks or scale-free networks having a pre-defined degree distribution (configuration model) and a method for rewiring these networks towards some desired or “target” degree correlations (Newman rewiring). We describe Python-NetworkX codes implemented for the two methods in our recent work and compare some of the results obtained in the HMF approximation with the new results obtained with statistical ensembles of real networks, including the case of signed networks.