Equilibrium Resource Distribution in a Model of Group Interaction

We consider a distributed system represented by weighted bipartite graph $G=(I\cup J, \mathcal{E})$. Each vertex $i\in I$ (agent $i$) possesses a certain amount of resource and distributes it among adjacent vertices $j\in J$ (fields of interaction). Agent $i$ evaluates the efficiency of allocation of its resource in the field $j$ according to value of given function $c_{ij}(x_{ij},\hat{X}_{j})$. Here $x_{ij}$ is the quantity of resource assigned to $j$ by $i$ and $\hat{X}_j$ is the total amount of resources allocated in $j$ by all the adjacent agents. A feasible distribution of resources is called equilibrium distribution, if the following condition is satisfied: $c_{ij}(x_{ij}, \hat{X}_j)=c_i$ for each $(i,j)\in\mathcal{E}$.
In this paper we consider the problem of existence of equilibrium resource distributions in systems with linear functions $c_{ij}$ and represented by different kinds of graphs. We formulate sufficient conditions for the existence of equilibriums and obtain explicit expressions to compute these distributions.