Combinatorial Characterization of Interference Coupling in Wireless Systems

We provide a combinatorial characterization of interference coupling in wireless systems, with the intent of obtaining a better insight into interference coordination and management. We introduce two bipartite graphs, namely the power graph and interference graph. We utilize these graphs and global dependency matrix containing only binary (0 and 1) entries to capture the effects of interference coupling in communication systems. We show that the irreducibility of the global dependency matrix G is related to the connectivity of the power graph and the irreducibility of the matrix GG^T is related to the connectivity of the interference graph. We prove that for strictly positive and strictly log-convex interference functions, the irreducibility of the matrices G and GG^T are necessary and sufficient conditions for the considered utility sets to be strictly convex. In this case there exists a unique optimizer for the problem of maximizing the product of utilities. We show that an interference balancing function is strictly log-convex, if and only if matrices G and GG^T are irreducible. We provide a simple yet comprehensive combinatorial characterization of interference coupled systems which abstracts away certain complexities of the physical layer.