  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. These results answer
the following questions which arise naturally from previous work: (a) What is
the significance of $1/\sqrt{n}$ in the scaling laws for the Protocol
interference model obtained by Gupta and Kumar (2000)? (b) Can we obtain a
tight approximation to the "maximum supportable flow" for node distributions
more general than the geometric random distribution, traffic models other than
randomly chosen source-destination pairs, and under very general assumptions on
the channel fading model?
  We first establish that the random source-destination model is essentially a
one-dimensional approximation to the capacity region, and a special case of
product multi-commodity flow. Building on previous results, for a combinatorial
interference model given by a network and a conflict graph, we relate the
product multicommodity flow to the spectral properties of the underlying graphs
resulting in computational upper and lower bounds. 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. As an implication, we obtain
computationally efficient upper and lower bounds on the PMF for any wireless
network with a guaranteed approximation factor.
