In this paper, we propose and study a randomized Boolean gossiping process, where nodes taking value from $\{0,1\}$ pairwise meet over an underlying graph in a random manner at each time step and the two interacting nodes update their states by random logistic rules drawn from the set $\{{\it AND}, {\it OR}\}$. This model is a generalization of the classical gossiping process and serves as a simplified probabilistic Boolean network. First of all, using standard theories from Markov chain we show that almost surely, the network state asymptotically converge to a consensus. We also establish a characterization of the distribution of this limit for large-scale networks with all-to-all communication in light of mean-field approximation methods. Next, we study how the number of communication classes in the network state space relates to the topology of the underlying interaction graph and obtain a full characterization: A line interaction graph with $n$ nodes generates $2n$ communication classes; A cycle graph with $2n$ nodes generates $n+3$ communication classes, and a cycle graph with $2n+1$ nodes generates $n+2$ communication classes; For any connected graph which is not a line or a cycle, the number of communication classes is either five or three, where three is achieved if and only if the graph contains an odd cycle.
['Bo Li', 'Hongsheng Qi', 'Guodong Shi']
Randomized Boolean Gossiping