On the critical phase transition time of wireless multi-hop networks with random failures

In this paper, we study the critical phase transition time of large-scale wireless multi-hop networks when the network topology experiences a partition due to increasing random node failures. We first define two new metrics, namely the last connection time and first partition time. The former is the last time that the network keeps a majority of surviving nodes connected in a single giant component; while the latter is the first time that the remaining surviving nodes are partitioned into multiple small components. Then we analyze the devolution process in a geometric random graph of n nodes based on percolation-theory connectivity and obtain the sufficient condition under which the graph is percolated. Based on the percolation condition, the last connection time and first partition time are found to be on the same order. Particularly, when the survival function of node lifetime is exponential, they are on the order of log(log n); while if the survival function is Pareto, the order is (log n)1/ρ, where ρ is the shape parameter of Pareto distribution. Finally, simulation results confirm that the last connection time and first partition time serve as the lower and upper bounds of the critical phase transition time, respectively. Further, an interesting result is that the network with heavy-tailed survival functions is no more resilient to random failures than the network with light-tailed ones, in terms of critical phase transition time, if the expected node lifetimes are identical.