On point-to-point communication networks

A point-to-point communication network is represented by (G,C), where G = (V,E) is a directed graph with vertex set V and edge set E, and C = [C-ij, (i,j) epsilon E] is a nonnegative-valued vector. A vertex in upsilon represents a node in the communication network, and an edge (i, j) represents a point-to-point discrete memoryless channel (DMC) from node i to node j whose capacity is C-ij. We assume that the channels in the network are independent of each other. An information source with entropy rate h is generated at source node B and recovered at sink node t with arbitrarily small probability of error, We show that the value of a max-flow from node a to node t in (G, C) must be greater than or equal to h. This results implies a separation theorem for network coding and channel coding in such a communication network.