A Stochastic Model for Layered Self-organizing Complex Systems

In this paper we study a problem common to complex systems that dynamically self-organize to an optimal configuration. Assuming the network nodes are of two types, and that one type is subjected to a an upward pressure according to a preferential stochastic model , we wish to determine the distribution of the active nodes over the levels of the network. We generalize the problem to the case of layered graphs as follows. Let

G

be a connected graph with

M

vertices which are divided into

d

levels where the vertices of each edge of

G

belong to consecutive levels. Initially each vertex has a value of 0 or 1 assigned at random. At each step of the stochastic process an edge is chosen at random. Then, the labels of the vertices of this edge are exchanged with probability 1 if the vertex on the higher level has the label 0 and the lower vertex has the label 1. The labels are switched with probability

λ

, if the lower vertex has value of 0 and the higher vertex has the value of 1. This stochastic process has the Markov chain property and is related to random walks on graphs. We derive formulas for the steady state distribution of the number of vertices with label 1 on the levels of the graph.