We consider a network consisting of a single source and $n$ receiver nodes that are grouped into equal-sized clusters. We use cluster heads in each cluster to facilitate communication between the source and the nodes within that cluster. Inside clusters, nodes are connected to each other according to a given network topology. Based on the connectivity among the nodes, each node relays its current stored version of the source update to its neighboring nodes by $local$ $gossiping$. We use the $version$ $age$ metric to assess information freshness at the nodes. We consider disconnected, ring, and fully connected network topologies for each cluster. For each network topology, we characterize the average version age at each node and find the average version age scaling as a function of the network size $n$. Our results indicate that per node average version age scalings of $O(\sqrt{n})$, $O(n^{\frac{1}{3}})$, and $O(\log n)$ are achievable in disconnected, ring, and fully connected cluster models, respectively. Next, we increase connectivity in the network and allow gossiping among the cluster heads to improve version age at the nodes. With that, we show that when the cluster heads form a ring network among themselves, we obtain per node average version age scalings of $O(n^{\frac{1}{3}})$, $O(n^{\frac{1}{4}})$, and $O(\log n)$ in disconnected, ring, and fully connected cluster models, respectively. Next, focusing on a ring network topology in each cluster, we introduce hierarchy to the considered clustered gossip network model and show that when we employ two levels of hierarchy, we can achieve the same $O(n^{\frac{1}{4}})$ scaling without using dedicated cluster heads. We generalize this result for $h$ levels of hierarchy and show that per user average version age scaling of $O(n^{\frac{1}{2h}})$ is achievable in the case of a ring network in each cluster across all hierarchy levels.

中文翻译：

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集群八卦网络中信息的版本时代

我们考虑一个由单个源和 $n$ 个接收器节点组成的网络，这些节点被分组到相同大小的集群中。我们在每个集群中使用集群头来促进源和该集群内节点之间的通信。在集群内部，节点根据给定的网络拓扑相互连接。基于节点之间的连通性，每个节点通过 $local$$gossiping$ 将其当前存储的源更新版本中继到其相邻节点。我们使用 $version$ $age$ 指标来评估节点上的信息新鲜度。我们为每个集群考虑断开连接、环形和完全连接的网络拓扑。对于每个网络拓扑，我们表征每个节点的平均版本年龄，并找到作为网络大小 $n$ 的函数的平均版本年龄缩放。我们的结果表明，$O(\sqrt{n})$、$O(n^{\frac{1}{3}})$ 和 $O(\log n)$ 的每个节点平均版本年龄缩放是可分别在断开连接、环形和完全连接的集群模型中实现。接下来，我们增加网络中的连接性，并允许簇头之间的闲聊以提高节点的版本年龄。有了这个，我们证明了当集群头在它们之间形成一个环形网络时，我们获得了每个节点的平均版本年龄缩放 $O(n^{\frac{1}{3}})$, $O(n^{ \frac{1}{4}})$ 和 $O(\log n)$ 分别在断开、环和完全连接的集群模型中。接下来，关注每个集群中的环形网络拓扑，我们将层次结构引入到所考虑的集群八卦网络模型中，并表明当我们采用两个层次的层次结构时，我们可以在不使用专用簇头的情况下实现相同的 $O(n^{\frac{1}{4}})$ 缩放。我们将这个结果推广到 $h$ 层次的层次，并表明在每个集群中的环形网络的情况下，每个用户的平均版本年龄缩放 $O(n^{\frac{1}{2h}})$ 是可以实现的跨越所有层级。