Product Multicommodity Flow in Wireless Networks

We provide a tight approximate characterization of the
n
-dimensional product multicommodity flow (PMF) region for a wireless network of
n
nodes. Separate characterizations in terms of the spectral properties of appropriate network graphs are obtained in both an information-theoretic sense and for a combinatorial interference model (e.g., protocol model). These provide an inner approximation to the
n

²
-dimensional capacity region. Our results hold for general node distributions, traffic models, and channel fading models. We first establish that the random source-destination model assumed in many previous results on capacity scaling laws, is essentially a one-dimensional approximation to the capacity region and a special case of PMF. We then build on the results for a wireline network (graph) that relate PMF to its spectral (or cut) properties. Specifically, for a combinatorial interference model given by a network graph and a conflict graph, we relate the PMF to the spectral properties of the underlying graphs resulting in simple computational upper and lower bounds. These results show that the 1/radic
n
scaling law obtained by Gupta and Kumar for a geometric random network can be explained in terms of the scaling law of the conductance of a geometric random graph. For the more interesting random fading model with additive white Gaussian noise (AWGN), we show that the scaling laws for PMF can again be tightly characterized by the spectral properties of appropriately defined graphs-such a characterization for general wireless networks has not been available before. As an implication, we obtain computationally efficient upper and lower bounds on the PMF for any wireless network with a guaranteed approximation factor.