Influence Maximization Dynamics and Topological Order on Erdös-Rényi Networks

Abstract

This paper concerns the study of the linear threshold model in random networks, specifically in Erdös-Rényi networks. In our approach, we consider an activation threshold defined by the expected value for the node degree and the associated influence activation mapping. According to these assumptions, we present a theoretical procedure for the linear threshold model, under fairly general conditions, regarding the topological structure of the networks and the activation threshold. Aiming at the dynamics of the influence maximization process, we analyze and discuss different choices for the seed set based on several centrality measures along with the state conditions for the procedure to trigger. The topological entropy established for Erdös-Rényi networks defines a topological order for this type of random networks. Sufficient conditions are presented for this topological entropy to be characterized by the spectral radius of the associated adjacency matrices. Consequently, a number of properties are proved. The threshold dynamics are analyzed through the relationship between the activation threshold and the topological entropy. Numerical studies are included to illustrate the theoretical results.

Probabilistic Procedures for SIR and SIS Epidemic Dynamics on Erdös-Rényi Contact Networks

“…Usually, the network entropy is the entropy of a stochastic matrix associated with the adjacency matrix A, as set out in [49,50] and references therein. Following this framework, we use the following definition of network entropy for the random networks G (n,p) , already used in [51] for complete networks.…”

“…Some properties for this topological invariant have been presented and proved in [50,51]. In particular, Figure 1b suggests that the topological entropy h top G (n,p) of the contact Erdös-Rényi networks is an increasing function of p, maintaining the network order n constant and varying the value of the connection probability p. Consequently, considering Definitions 1 and 2, we can establish a relationship between the epidemic threshold τ and the topological entropy h top G (n,p) .…”