  In this paper we propose a distributed algorithm for the estimation and
control of the connectivity of ad-hoc networks in the presence of a random
topology. First, given a generic random graph, we introduce a novel stochastic
power iteration method that allows each node to estimate and track the
algebraic connectivity of the underlying expected graph. Using results from
stochastic approximation theory, we prove that the proposed method converges
almost surely (a.s.) to the desired value of connectivity even in the presence
of imperfect communication scenarios. The estimation strategy is then used as a
basic tool to adapt the power transmitted by each node of a wireless network,
in order to maximize the network connectivity in the presence of realistic
Medium Access Control (MAC) protocols or simply to drive the connectivity
toward a desired target value. Numerical results corroborate our theoretical
findings, thus illustrating the main features of the algorithm and its
robustness to fluctuations of the network graph due to the presence of random
link failures.
