We consider the binary freshness metric for gossip networks that consist of a single source and $n$ end-nodes, where the end-nodes are allowed to share their stored versions of the source information with the other nodes. We develop recursive equations that characterize binary freshness in arbitrarily connected gossip networks using the stochastic hybrid systems (SHS) approach. Next, we study binary freshness in several structured gossip networks, namely disconnected, ring and fully connected networks. We show that for both disconnected and ring network topologies, when the number of nodes gets large, the binary freshness of a node decreases down to 0 as $n^{-1}$, but the freshness is strictly larger for the ring topology. We also show that for the fully connected topology, the rate of decrease to 0 is slower, and it takes the form of $n^{-\rho}$ for a $\rho$ smaller than 1, when the update rates of the source and the end-nodes are sufficiently large. Finally, we study the binary freshness metric for clustered gossip networks, where multiple clusters of structured gossip networks are connected to the source node through designated access nodes, i.e., cluster heads. We characterize the binary freshness in such networks and numerically study how the optimal cluster sizes change with respect to the update rates in the system.

中文翻译：

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八卦二进制新鲜度指标

我们考虑由单个源和 $n$ 个端节点组成的八卦网络的二进制新鲜度度量，其中允许端节点与其他节点共享其存储的源信息版本。我们使用随机混合系统 (SHS) 方法开发了表征任意连接八卦网络中二进制新鲜度的递归方程。接下来，我们研究了几个结构化八卦网络中的二进制新鲜度，即断开连接、环形和完全连接的网络。我们表明，对于断开连接和环形网络拓扑，当节点数量变大时，节点的二进制新鲜度下降到 0 为 $n^{-1}$，但环形拓扑的新鲜度严格更大。我们还表明，对于全连接拓扑，下降到 0 的速度较慢，当源节点和端节点的更新率足够大时，对于小于 1 的 $\rho$，它采用 $n^{-\rho}$ 的形式。最后，我们研究了集群八卦网络的二进制新鲜度度量，其中多个结构化八卦网络集群通过指定的访问节点（即簇头）连接到源节点。我们描述了此类网络中的二进制新鲜度，并通过数值研究了最佳集群大小如何随系统中的更新速率而变化。