More Benefits of Adding Sparse Random Links to Wireless Networks: Yet Another Case for Hybrid Networks

Abstract

We theoretically and experimentally analyze the process of adding sparse random links to random wireless networks modeled as a random geometric graph. While this process has been previously proposed, we are the first to prove theoretical bounds on the improvement to the graph diameter and random walk properties of the resulting graph as a function of the frequency of wires used, where this frequency is diminishingly small. In particular, given a parameter k controlling sparsity, any node has a probability of 1 / k 2 n r 2 for being a wired link station. Amongst the wired link stations, we consider creating a random 3-regular graph superimposed upon the random wireless network to create model G 1 , and alternatively we consider a sparser model G 2 as well, which is a random 1-out graph of the wired links superimposed upon the random wireless network. We prove that the diameter for G 1 is O ( k + log ( n ) ) with high probability and the diameter for G 2 is O ( k log ( n ) ) with high probability, both of which exponentially improve the Θ ( n / log n ) diameter of the random geometric graph around the connectivity threshold, thus also inducing small-world characteristics as the high clustering remains unchanged. Further, we theoretically demonstrate that as long as k is polylogarithmic in the network size, G 1 has rapidly mixing random walks with high probability, which also exponentially improves upon the mixing time of the purely wireless random geometric graph, which yields direct improvement to the performance of distributed gossip algorithms as well as normalized edge connectivity. Finally, we experimentally confirm that the algebraic connectivities of both G 1 and G 2 exhibit significant asymptotic improvement over that of the underlying random geometric graph. These results further motivate future hybrid networks and advances in the use of directional antennas.