  This paper has been withdrawn by the author due to a crucial error in the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.
