We propose mathematical tools for social network simulation to obtain sufficient conditions for network ergodicity, defined as the existence of a steady state as time approaches infinity. The proposed model is linear; the network elements form a two-dimensional array (matrix), where each entry represents the state of an element at a specific time. An impact operator, structured as a four-dimensional array, defines the interactions between elements. This operator is also presented as a directed graph where vertices correspond to network elements, and arcs represent the impact of one element on another. The model incorporates boundary elements that influence the internal states of the network. Sufficient conditions for network ergodicity are derived from the connectivity properties of the impact graph, which must contain paths between all pairs of vertices and loops for all vertices. These conditions ensure that the operator's spectrum (with the possible exception of the value 1) is located inside the open unit disk. We prove that 1 is an eigenvalue if and only if the boundary is isolated. These spectral properties guarantee that a steady state exists and can be found using an iterative procedure with linear (geometric) convergence.
Спекторський et al. (Mon,) studied this question.