On the incompatibility of connectivity and local pooling in Erdős-Rényi Graphs

For a wireless communications network, Local Pooling (LoP) is a desirable property due to its sufficiency for the optimality of low-complexity greedy scheduling techniques. However, LoP in network graphs with a primary interference model enforces an edge sparsity that may be prohibitive to other desirable properties in wireless networks, such as connectivity. In this paper, we investigate the impact of the edge density on both LoP and the size of the largest component under the primary interference model, as the number of nodes in the network grows large. For Erdös-Rényi graphs, we employ threshold functions to establish critical values for the edge probability necessary for these properties to hold. These thresholds demonstrate that LoP and connectivity (or even the presence of a giant component) cannot both hold asymptotically for a large class of edge probability functions. A similar incompatibility for random geometric graphs is suggested by our simulation results.