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 $O(n^{\frac{1}{3}})$ 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 $O(n^{\frac{1}{2}})$, and when they are connected on a fully-connected graph the version age scales as $O(\log n)$. 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 $O(n^{\frac{1}{2}})$ to $O(n^{\frac{1}{3}})$.

翻译：我们考虑一个由源节点生成更新并与二维方格网络中的$n$个节点相连的八卦网络。源节点维护着一个可能被生成或观测的过程更新，并将其共享给网格网络。网格网络中的节点与邻居节点通信，并采用推送式八卦策略传播这些版本更新。我们使用版本年龄度量来量化节点信息的新鲜度。我们找到了通用网络中一组节点平均版本年龄的上界。基于此，我们证明网格网络中节点的平均版本年龄按$O(n^{\frac{1}{3}})$规模增长。此前的研究表明，当$n$个节点连接成环状网络时，版本年龄按$O(n^{\frac{1}{2}})$规模增长；当节点连接成全连接图时，版本年龄按$O(\log n)$规模增长。我们的工作是首个针对环状网络和全连接网络（代表网络连通性的两个极端）之外的其他连接结构展示年龄缩放规律的研究。研究表明，与环状网络相比，网格网络更高的连通性将每个节点的年龄体验从$O(n^{\frac{1}{2}})$降低至$O(n^{\frac{1}{3}})$。