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.

翻译：我们考虑由单一源和美元终端节点组成的八卦网络的二元新鲜度衡量标准,允许终端节点与其他节点共享源信息存储版本的零美元,我们开发循环方程式,利用随机连接的混合系统(SHS)方法,在任意连接的八卦网络中以二元新鲜度为特征。接下来,我们研究若干结构化八卦网络的二元新鲜度,即断开、环环和完全连接的网络。我们显示,对于断开和环环网络的顶点,当节点数量大时,结点的二元新鲜度会降低到零美元,但对于环节点表层而言,新点将严格地更大。我们还显示,对于完全连接的八卦网络而言,减到零的速率会放缓,而当源和终端网络的更新速度足够大时,则采取小于1美元的双元新鲜度。最后,我们研究一个节点的节点将节点降低为0美元,对于圆点网络的二元新鲜度度度度指标,对于环节点网络而言,对于环比重的顶端点网络,对于环比重多少组群点网络,我们不会连接。