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(k,m)-connectivity in Mobile Clustered Wireless Networks

This paper has been withdrawn by the author due to a crucial error in the calculation of Equation (28). We propose a novel concept of $(k,m)$-connectivity in mobile clustered wireless networks, in which there are $n$ mobile cluster members and $n^d$ static cluster heads, where $k,m,d$ are all positive constants and $k\leq m$. $(k,m)$-connectivity signifies that in a time period consisting of $m$ time slots, there exist at least $k$ time slots for each cluster member and in any one of these $k$ time slots the cluster member can directly communicate with at least one cluster head. We investigate the critical transmission range of asymptotic $(k,m)$-connectivity when cluster members move according to random walk or i.i.d. mobility model. Under random walk mobility, we propose two general heterogeneous velocity models in which cluster members may move with different velocities. Under both mobility models, we also define weak and strong parameters conditions, resulting in different accuracies of evaluations on the probability that the network is asymptotically $(k,m)$-connected, denoted as $P(\mathcal {C})$ below for simplicity. For both mobilities, under weak parameters condition, we provide bounds on $P(\mathcal {C})$ and derive the critical transmission range for $(k,m)$-connectivity. For random walk mobility with one kind of velocity model and i.i.d. mobility, under strong parameters condition, we present a precise asymptotic probability distribution of $P(\mathcal {C})$ in terms of the transmission radius. Our results offer fundamental insights and theoretical guidelines on design of large-scale wireless networks.