A coding theorem for network information flow with correlated sources

Consider a directed graph, in which vertices represent sensor nodes, and edges represent a discrete memoryless channel of a given capacity over which two neighbors nodes can communicate. The channels are independent. Each node gets to observe one element of a set of discrete sources of information drawn i.i.d., according to some joint probability distribution. Our goal is to solve an incast problem in the given graph: nodes exchange messages with their neighbors, and after a finite number of communication rounds, one of the nodes (the data collector) must have received enough information to reproduce the entire set of observations, with arbitrarily small probability of error. In this paper, we give a complete characterization of the conditions on the sources and the channels under which such perfect reconstruction is possible. Close examination of our proof reveals that in this setup, Shannon information behaves as a classical flow, identical in nature to the flow of water in pipes. This "information as flow" view provides an algorithmic interpretation for our results, among which perhaps the most important one is the optimality of using a layered protocol stack