This paper proposes and investigates a framework for clique gossip protocols. As complete subnetworks, the existence of cliques is ubiquitous in various social, computer, and engineering networks. By clique gossiping, nodes interact with each other along a sequence of cliques. Clique-gossip protocols are defined as arbitrary linear node interactions where node states are vectors evolving as linear dynamical systems. Such protocols become clique-gossip averaging algorithms when node states are scalars under averaging rules. We generalize the classical notion of line graph to capture the essential node interaction structure induced by both the underlying network and the specific clique sequence. We prove a fundamental eigenvalue invariance principle for periodic clique-gossip protocols, which implies that any permutation of the clique sequence leads to the same spectrum for the overall state transition when the generalized line graph contains no cycle. We also prove that for a network with $n$ nodes, cliques with smaller sizes determined by factors of $n$ can always be constructed leading to finite-time convergent clique-gossip averaging algorithms, provided $n$ is not a prime number. Particularly, such finite-time convergence can be achieved with cliques of equal size $m$ if and only if $n$ is divisible by $m$ and they have exactly the same prime factors. A proven fastest finite-time convergent clique-gossip algorithm is constructed for clique-gossiping using size- $m$ cliques. Additionally, the acceleration effects of clique-gossiping are illustrated via numerical examples.

中文翻译：

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派系闲聊

本文提出并研究了集团八卦协议的框架。作为完整的子网，团体的存在在各种社交，计算机和工程网络中无处不在。通过小组闲聊，节点沿着一系列小组相互交互。Clique-gossip协议定义为任意线性节点交互，其中节点状态是演变为线性动力学系统的向量。当节点状态在平均规则下为标量时，此类协议成为集团闲话平均算法。我们概括了线图的经典概念，以捕获由基础网络和特定集团序列两者诱导的基本节点交互结构。我们证明了周期性集团闲话协议的基本特征值不变性原理，这意味着当广义线图不包含循环时，集团序列的任何置换都会导致整个状态转换的频谱相同。我们还证明了对于具有 $ n $ 节点，由以下因素决定的较小规模的集团 $ n $ 可以始终构造为导致有限时间收敛的集团闲话平均算法 $ n $ 不是素数。特别是，可以使用大小相等的派系实现这种有限时间收敛 $ m $ 当且仅当 $ n $ 被...整除 $ m $ 它们具有完全相同的主要因素。构造了一种经过验证的最快的有限时间收敛式团簇算法，用于使用size- $ m $ 集团。此外，还通过数值示例说明了集团闲聊的加速效果。