Consensus Computations over Random Graph Processes

Distributed consensus processing over random graphs and with randomized node dynamics is considered. At each time step $k$, every node independently updates its state with a weighted average of its neighbors' states or stick with its current state. The choice is a Bernoulli trial with success probability $P_k$. The random graph processes, defining the time-varying neighbor sets, are specified over the set of all possible graphs with the given node set. Connectivity-independent and arc-independent graph processes are introduced to capture the fundamental influence of random graphs on the consensus convergence. Necessary and sufficient conditions are presented on the success probability sequence $\{P_k\}$ for the network to reach a global almost sure consensus. For connectivity-independent graphs, we show that $\sum_k P_k^{n-1} =\infty$ is a sufficient condition for almost sure consensus, where $n$ is the number of nodes. For arc-independent graphs, we show that $\sum_k P_k =\infty$ is a sharp threshold, i.e., the consensus probability is zero for almost all initial conditions when the sum converges, while it is one for all initial conditions when the sum diverges. Convergence rates are established by lower and upper bounds of the $\epsilon$-computation time. The results add to the understanding of the interplay between random graphs, random computations, and convergence probability for distributed information processing.