Generating Dependent Random Variables Over Networks

In this paper we study the problem of generation of dependent random variables, known as the "coordination capacity" [4,5], in multiterminal networks. In this model $m$ nodes of the network are observing i.i.d. repetitions of $X^{(1)}$, $X^{(2)}$,..., $X^{(m)}$ distributed according to $q(x^{(1)},...,x^{(m)})$. Given a joint distribution $q(x^{(1)},...,x^{(m)},y^{(1)},...,y^{(m)})$, the final goal of the $i^{th}$ node is to construct the i.i.d. copies of $Y^{(i)}$ after the communication over the network where $X^{(1)}$, $X^{(2)}$,..., $X^{(m)}, Y^{(1)}$, $Y^{(2)}$,..., $Y^{(m)}$ are jointly distributed according to $q(x^{(1)},...,x^{(m)},y^{(1)},...,y^{(m)})$. To do this, the nodes can exchange messages over the network at rates not exceeding the capacity constraints of the links. This problem is difficult to solve even for the special case of two nodes. In this paper we prove new inner and outer bounds on the achievable rates for networks with two nodes.