On point-to-point communication networks

A point-to-point communication network is represented by (G,C), where G=(V,/spl epsiv/) is a directed graph with vertex set V and edge set /spl epsiv/, and C=[C/sub ij/,(i,j)/spl isin//spl epsiv/] is a nonnegative-valued vector. A vertex in V 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/sub 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 /spl delta/ and recovered at sink node t with arbitrarily small probability of error. We show that the value of a max-flow from node /spl delta/ 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.