Recently, applying
stochastic geometry, Penrose [93, 94], Bettstetter [95], Li et al [96] studied
how to set the transmission radius to achieve the k-connectivity with certain
probability for a network when wireless nodes are uniformly and randomly distributed
over a two-dimensional region. Levcopoulos et al [97], Lukovszki [34],
and Li et al [96] proposed some methods to construct a spanner that can sustain
k-nodes or links failures. How to find a small transmission range (power)
for each node such that the resulted communication graph is k-connected
is also studied in [98, 99]. Another important issue a?®ecting the throughput
of the network topology is interference. Jia, Rajaraman and Scheideler [42]
studied the interference analysis of sparsified Yao graph (Y Yk(V )) recently.
They established an upper bound on the interference number of Y Yk(V ) for
a random node distribution. Martin Burkhart et al [100] provided a concise
138
Chapter 5 Topology Control for Wireless Sensor Networks
and intuitive definition of interference and proposed connectivity-preserving
and spanner constructions that are interference-minimal.
One problem related to topology control is transmission power assignment.
In the previous sections, we have assumed that the transmission power of every
node is equal and is normalized to one unit. However, in practice, each
node can adjust its transmission power according to its neighbors??™ positions.
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