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 networks, 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.

中文翻译：

---

具有社区结构的网络中的八卦时代

我们考虑一个由单个源和$ n $个接收者节点组成的网络，这些节点被分组为$ m $个相等大小的社区，即群集，其中每个群集包括$ k $个节点，并由一个专用的群集头提供服务。源节点保留观察到的进程的版本，并通过关联的簇头更新每个簇。每个群集中的节点根据给定的网络拓扑相互连接。基于此拓扑，每个节点通过$ local $ $ gossiping $将其当前更新中继到其相邻节点。我们使用$ version $ $ age $度量来量化接收方节点的信息及时性。我们考虑每个群集的断开，环形和完全连接的网络拓扑。对于这些网络拓扑中的每一种，我们描述每个节点的平均版本年龄，并根据网络规模$ n $找到版本年龄缩放比例。我们的结果表明，可以实现$ O（\ sqrt {n}）$，$ O（n ^ {\ frac {1} {3}}）$和$ O（\ log n）$的每个节点版本年龄缩放分别处于断开连接，环形和完全连接的网络中。最后，通过数值评估，我们确定了年龄最佳的$（m，k）$对作为源，群集头和节点更新率的函数。