即时收敛流言
Finite-time Convergent Gossiping
摘要：
Gossip算法广泛应用于现代分布式系统中，其应用范围从传感器网络和对等网络到移动车辆网络和社交网络。大量的研究工作一直致力于分析和改善的渐近收敛速度的八卦算法。在这项工作中，我们研究有限时间收敛的确定性流言。我们表明，存在一个对称的八卦算法，收敛在有限时间内，当且仅当网络节点的数量是2的幂，而总是存在一个非对称的八卦算法与有限时间收敛，独立于节点的数量。对于$n=2^m$节点，我们证明了最快的收敛可以达到$nm=n\log_2 n$节点更新通过对称八卦。另一方面，在具有$0\leq r<2^m$的$n=2^m+r$节点之间的非对称流言下，至少需要$mn+2r$节点更新来实现有限时间收敛。它还表明，有限时间收敛八卦的存在往往对底层的相互作用图的结构要求很强。最后，我们将我们的结果应用于量子网络中的八卦算法，其目标是通过成对相互作用控制量子系统的状态。我们表明，有限时间收敛是永远不可能的，这样的系统。
英文摘要：
Gossip algorithms are widely used in modern distributed systems, with applications ranging from sensor networks and peer-to-peer networks to mobile vehicle networks and social networks. A tremendous research effort has been devoted to analyzing and improving the asymptotic rate of convergence for gossip algorithms. In this work we study finite-time convergence of deterministic gossiping. We show that there exists a symmetric gossip algorithm that converges in finite time if and only if the number of network nodes is a power of two, while there always exists an asymmetric gossip algorithm with finite-time convergence, independent of the number of nodes. For $n=2^m$ nodes, we prove that a fastest convergence can be reached in $nm=n\log_2 n$ node updates via symmetric gossiping. On the other hand, under asymmetric gossip among $n=2^m+r$ nodes with $0\leq r<2^m$, it takes at least $mn+2r$ node updates for achieving finite-time convergence. It is also shown that the existence of finite-time convergent gossiping often imposes strong structural requirements on the underlying interaction graph. Finally, we apply our results to gossip algorithms in quantum networks, where the goal is to control the state of a quantum system via pairwise interactions. We show that finite-time convergence is never possible for such systems.
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