Age of Gossip in Networks with Community Structure

We consider a network consisting of a single source and $n$ receiver nodes that are grouped into $m$ equal size communities, i.e., clusters, where each cluster includes $k$ nodes and is served by a dedicated cluster head. The source node keeps versions of an observed process and updates each cluster through the associated cluster head. Nodes within each cluster are connected to each other according to a given network topology. Based on this topology, each node relays its current update to its neighboring nodes by $local$ $gossiping$. We use the $version$ $age$ metric to quantify information timeliness at the receiver nodes. We consider disconnected, ring, and fully connected network topologies for each cluster. For each of these network topologies, we characterize the average version age at each node and find the version age scaling as a function of the network size $n$. Our results indicate that per node version age scalings of $O(\sqrt{n})$, $O(n^{\frac{1}{3}})$, and $O(\log n)$ are achievable in disconnected, ring, and fully connected cluster models, respectively. Finally, through numerical evaluations, we determine the version age-optimum $(m,k)$ pairs as a function of the source, cluster head, and node update rates.