An Analytical Model of the Small World Effect in D2D Wireless Networks

Small-world networks are characterized by large clustering coefficients and small characteristic path lengths. These properties are induced by replacing a small fraction of short-range local scale links of a geometric/Euclidean graph with long-range global scale links. Advances in wireless networks allow for cost-efficient addition of secondary long-range wireless interfaces in devices. We derive analytical mean-field solutions for 1) clustering coefficient and 2) characteristic path length of peer-to-peer D2D wireless networks with topologies mimicking small-world networks. In graph-theoretic terms these topologies correspond to geometric graphs of randomly deployed nodes in two-dimensions with range limited shortcuts. The models show that in spite of the fact that links used to create shortcuts are range limited, the network still retains the essential phase difference between characteristic path length and clustering coefficient that is the hallmark of small-world networks when a small fraction of all nodes, as little as 1%-5%, have range limited shortcut links. We also demonstrate the utility of these models as design tools for determining deployment parameters of small-world wireless sensor networks.