Age of Gossip in Random and Bipartite Networks

In this paper we study gossip networks where a source observing a process sends updates to an underlying graph. Nodes in the graph communicate to their neighbors by randomly sending updates. Our interest is studying the version age of information (vAoI) metric over various classes of networks. It is known that the version age of $K_n$ is logarithmic, and the version age of $\overline{K_n}$ is linear. We study the question `how does the vAoI evolve as we interpolate between $K_n$ and $\overline{K_n}$' by studying Erd\H{o}s-Reyni random graphs, random $d$-regular graphs, and bipartite networks. Our main results are proving the existence of a threshold in $G(n,p)$ from rational to logarithmic average version age, and showing $G(n,d)$ almost surely has logarithmic version age for constant $d$. We also characterize the version age of complete bipartite graphs $K_{L,R}$, when we let $L$ vary from $O(1)$ to $O(n)$.