Network information flow with correlated sources

Consider the following network communication setup, originating in a sensor networking application we refer to as the "sensor reachback" problem. We have a directed graph G=(V,E), where V={v₀v₁...v_n} and E⊆V×V. If (v_i,v_j)∈E, then node i can send messages to node j over a discrete memoryless channel (DMC) (X_ij,p_ij(y|x),Y_ij), of capacity C_ij. The channels are independent. Each node v_i gets to observe a source of information U_i(i=0...M), with joint distribution p(U₀U₁...U_M). Our goal is to solve an incast problem in G: nodes exchange messages with their neighbors, and after a finite number of communication rounds, one of the M+1 nodes (v₀ by convention) must have received enough information to reproduce the entire field of observations (U₀U₁...U_M), with arbitrarily small probability of error. In this paper, we prove that such perfect reconstruction is possible if and only if H(U_s | U_S(c)) < Σ_{i∈S,j∈S(c)} for all S⊆{0...M},S≠O,0∈S(c). Our main finding is that in this setup, a general source/channel separation theorem holds, and that Shannon information behaves as a classical network flow, identical in nature to the flow of water in pipes. At first glance, it might seem surprising that separation holds in a fairly general network situation like the one we study. A closer look, however, reveals that the reason for this is that our model allows only for independent point-to-point channels between pairs of nodes, and not multiple-access and/or broadcast channels, for which separation is well known not to hold. This "information as flow" view provides an algorithmic interpretation for our results, among which perhaps the most important one is the optimality of implementing codes using a layered protocol stack.