Some Properties of Stochastic Matrices and Non-Homogeneous Markov Chains Generated by Nonlinearities in the Resource Network Model

Abstract

The resource network is a non-linear threshold model where vertices exchange resource in infinite discrete time. The model is represented by a directed weighted graph. At each time step, all vertices send their resources along all output edges following one of two rules. For each vertex, the threshold value for changing the operation rule is equal to the total weight of its outgoing edges. If all vertices have resources less than their thresholds, the network is completely described by a homogeneous Markov chain. If at least one of the vertices has a resource above the threshold, the network is described by a non-homogeneous Markov chain. The purpose of this article is to describe and investigate non-homogeneous Markov chains generated by the resource network model. It is proven that they are strongly ergodic. In addition, stochastic matrices of a special form were studied. A number of new properties were revealed for them. The results obtained were generalized to arbitrary stochastic matrices.