Age of Gossip on a Grid

We consider a gossip network consisting of a source generating updates and n nodes connected in a two-dimensional square grid. The source keeps updates of a process, that might be generated or observed, and shares them with the grid network. The nodes in the grid 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. We find an upper bound for the average version age for a set of nodes in a general network. Using this, we show that the average version age at a node scales as $O\left( {{n^{\frac{1}{3}}}} \right)$ in a grid network. Prior to our work, it has been known that when n nodes are connected on a ring the version age scales as $O\left( {{n^{\frac{1}{2}}}} \right)$, and when they are connected on a fully-connected graph the version age scales as O(logn). Ours is the first work to show an age scaling result for a connectivity structure other than the ring and fully-connected networks that represent two extremes of network connectivity. Our work shows that higher connectivity on a grid compared to a ring lowers the age experience of each node from $O\left( {{n^{\frac{1}{2}}}} \right)$ to $O\left( {{n^{\frac{1}{3}}}} \right)$.