  This paper considers the consensus problem for a network of nodes with random
interactions and sampled-data control actions. We first show that consensus in
expectation, in mean square, and almost surely are equivalent for a general
random network model when the inter-sampling interval and network size satisfy
a simple relation. The three types of consensus are shown to be simultaneously
achieved over an independent or a Markovian random network defined on an
underlying graph with a directed spanning tree. For both independent and
Markovian random network models, necessary and sufficient conditions for
mean-square consensus are derived in terms of the spectral radius of the
corresponding state transition matrix. These conditions are then interpreted as
the existence of critical value on the inter-sampling interval, below which
global mean-square consensus is achieved and above which the system diverges in
mean-square sense for some initial states. Finally, we establish an upper bound
on the inter-sampling interval below which almost sure consensus is reached,
and a lower bound on the inter-sampling interval above which almost sure
divergence is reached. Some numerical simulations are given to validate the
theoretical results and some discussions on the critical value of the
inter-sampling intervals for the mean-square consensus are provided.

Systems and Control