Age of Gossip on Generalized Rings

We consider a gossip network consisting of a source forwarding updates and $n$ nodes placed geometrically in a ring formation. Each node gossips with $f(n)$ nodes on either side, thus communicating with $2f(n)$ nodes in total. $f(n)$ is a sub-linear, non-decreasing and positive function. The source keeps updates of a process, that might be generated or observed, and shares them with the nodes in the ring network. The nodes in the ring network communicate with their neighbors and disseminate these version updates using a push-style gossip strategy. We use the version age metric to quantify the timeliness of information at the nodes. Prior to this work, it was shown that the version age scales as $O(n^{\frac{1}{2}})$ in a ring network, i.e., when $f(n)=1$, and as $O(\log{n})$ in a fully-connected network, i.e., when $2f(n)=n-1$. In this paper, we find an upper bound for the average version age for a set of nodes in such a network in terms of the number of nodes $n$ and the number of gossiped neighbors $2 f(n)$. We show that if $f(n) = \Omega(\frac{n}{\log^2{n}})$, then the version age still scales as $\theta(\log{n})$. We also show that if $f(n)$ is a rational function, then the version age also scales as a rational function. In particular, if $f(n)=n^\alpha$, then version age is $O(n^\frac{1-\alpha}{2})$. Finally, through numerical calculations we verify that, for all practical purposes, if $f(n) = \Omega(n^{0.6})$, the version age scales as $O(\log{n})$.