  The study of the optimality of low-complexity greedy scheduling techniques in
wireless communications networks is a very complex problem. The Local Pooling
(LoP) factor provides a single-parameter means of expressing the achievable
capacity region (and optimality) of one such scheme, greedy maximal scheduling
(GMS). The exact LoP factor for an arbitrary network graph is generally
difficult to obtain, but may be evaluated or bounded based on the network
graph's particular structure. In this paper, we provide rigorous
characterizations of the LoP factor in large networks modeled as
Erd\H{o}s-R\'enyi (ER) and random geometric (RG) graphs under the primary
interference model. We employ threshold functions to establish critical values
for either the edge probability or communication radius to yield useful bounds
on the range and expectation of the LoP factor as the network grows large. For
sufficiently dense random graphs, we find that the LoP factor is between 1/2
and 2/3, while sufficiently sparse random graphs permit GMS optimality (the LoP
factor is 1) with high probability. We then place LoP within a larger context
of commonly studied random graph properties centered around connectedness. We
observe that edge densities permitting connectivity generally admit cycle
subgraphs which forms the basis for the LoP factor upper bound of 2/3. We
conclude with simulations to explore the regime of small networks, which
suggest the probability that an ER or RG graph satisfies LoP and is connected
decays quickly in network size.
