Equilibrium resource distribution in a network model

A network is modeled by a weighted undirected graph $G$. Some certain time invariable resource is assigned to each node and is distributed among the incident edges at each time (time is assumed to be discrete). A state of the network corresponds to a distribution of resources of all nodes among the edges of $G$. At each time a vertex $i$ evaluates its relationship with an adjacent vertex $j$ according to a given function $c_{ij}(x_{ij},x_{ji})$ of the resources $x_{ij}$ and $x_{ji}$ provided by the nodes $i$ and $j$ to the edge $(i,j)$. Since resources of the nodes are redistributed at every time, the state of the system varies in time. Some sufficient conditions are found for the existence of the limit and equilibrium states of the model; and precise formulas are given to compute these states in the case of a special function $c_{ij}$ for an arbitrary graph $G$.