Dynamic behavior for social networks with state-dependent susceptibility and antagonistic interactions

In this paper, we investigate a general nonlinear model of opinion dynamics in which both state-dependent susceptibility to persuasion and antagonistic interactions are considered. According to the existing literature and socio-psychological theories, we examine three specializations of state-dependent susceptibility, that is, stubborn positives scenario, stubborn neutrals scenario, and stubborn extremists scenario. Interactions among agents form a signed graph, in which positive and negative edges represent friendly and antagonistic interactions, respectively. We conduct a comprehensive theoretical analysis of the generalized nonlinear opinion dynamics. For stubborn positives and stubborn neutrals scenarios, the general model is well-posed if and only if the system matrix is diagonally dominant. Based on the property of limit set and the existing results, we obtain some sufficient conditions such that the states of all agents converge to specific equilibrium point or bipartite consensus. For stubborn extremists scenario, by using the Perron-Frobenius property of eventually positive matrices, we establish some sufficient conditions such that the states of all agents converge into the subspace spanned by the right positive eigenvector of an eventually positive matrix. When there exists at least one entry of the equilibrium point or right positive eigenvector which is not equal to one, the derived results can be used to describe different levels of an opinion. In addition, for stubborn extremists scenario, we also consider the cases that the system matrix forms the opposing Laplacian and repelling Laplacian matrix, respectively. Finally, we present two examples to demonstrate the effectiveness of the theoretical findings.