  Distributed consensus computation over random graph processes is considered.
The random graph process is defined as a sequence of random variables which
take values from the set of all possible digraphs over the node set. At each
time step, every node updates its state based on a Bernoulli trial, independent
in time and among different nodes: either averaging among the neighbor set
generated by the random graph, or sticking with its current state.
Connectivity-independence and arc-independence are introduced to capture the
fundamental influence of the random graphs on the consensus convergence.
Necessary and/or sufficient conditions are presented on the success
probabilities of the Bernoulli trials for the network to reach a global almost
sure consensus, with some sharp threshold established revealing a consensus
zero-one law. Convergence rates are established by lower and upper bounds of
the $\epsilon$-computation time. We also generalize the concepts of
connectivity/arc independence to their analogues from the $*$-mixing point of
view, so that our results apply to a very wide class of graphical models,
including the majority of random graph models in the literature, e.g.,
Erd\H{o}s-R\'{e}nyi, gossiping, and Markovian random graphs. We show that under
$*$-mixing, our convergence analysis continues to hold and the corresponding
almost sure consensus conditions are established. Finally, we further
investigate almost sure finite-time convergence of random gossiping algorithms,
and prove that the Bernoulli trials play a key role in ensuring finite-time
convergence. These results add to the understanding of the interplay between
random graphs, random computations, and convergence probability for distributed
information processing.
