Age of gossip from connective properties via first passage percolation

In gossip networks, a source node forwards time-stamped updates to a network of observers according to a Poisson process. The observers then update each other on this information according to Poisson processes as well. The Age of Information (AoI) of a given node is the difference between the current time and the most recent time-stamp of source information that the node has received. We provide a method for evaluating the AoI of a node in terms of first passage percolation. We then use this distributional identity to prove matching upper and lower bounds on the AoI in terms of connectivity properties of the underlying network. In particular, if one sets $X_v$ to be the AoI of node $v$ on a finite graph $G$ with $n$ nodes, then we define $m_\ast = \min\{m : m \cdot |B_m(v)| \geq n\}$ where $B_m(v)$ is the ball of radius $m$ in $G$. In the case when the maximum degree of $G$ is bounded by $\Delta$ we prove $\mathbb{E} X_v = \Theta_\Delta(m_\ast)$. As corollaries, we solve multiple open problems in the literature such as showing the age of information on a subset of $\mathbb{Z}^d$ is $\Theta(n^{1/(d+1)})$. We also demonstrate examples of graphs with AoI scaling like $n^{\alpha}$ for each $\alpha \in (0,1/2)$. These graphs are not vertex-transitive and in fact we show that if one considers the AoI on a graph coming from a vertex-transitive infinite graph then either $\mathbb{E} X_v = \Theta(n^{1/k})$ for some integer $k \geq 2$ or $\mathbb{E} X_v = n^{o(1)}$.