从图信号中盲提取均匀划分
Blind Extraction of Equitable Partitions from Graph Signals
摘要：
找到公平的分区是密切相关的提取图的对称性和兴趣在各种应用程序的上下文中，如节点的角色检测，集群同步，共识动态和网络控制问题。在这项工作中，我们研究了一个盲识别问题，我们的目标是恢复一个公平的网络分区，而不知道网络的边缘，但仅仅基于一个未知的图形过滤器的输出的观察。具体来说，我们考虑两个设置。首先，我们考虑一种情况，在这种情况下，我们可以控制输入到图过滤器，并提出了一种方法来提取分区的启发，著名的Weisfeiler-Lehman（颜色细化）算法。其次，我们将这个想法推广到一个设置，只观察输出的随机，低秩激励的图过滤器，并提出了一个简单的谱算法来提取相关的公平分区。最后，我们建立了理论上的误差范围，这种光谱检测方案所招致的，并进行数值实验，说明我们的理论结果，并比较两种算法。
英文摘要：
Finding equitable partitions is closely related to the extraction of graph symmetries and of interest in a variety of applications context such as node role detection, cluster synchronization, consensus dynamics, and network control problems. In this work we study a blind identification problem in which we aim to recover an equitable partition of a network without the knowledge of the network's edges but based solely on the observations of the outputs of an unknown graph filter. Specifically, we consider two settings. First, we consider a scenario in which we can control the input to the graph filter and present a method to extract the partition inspired by the well known Weisfeiler-Lehman (color refinement) algorithm. Second, we generalize this idea to a setting where only observe the outputs to random, low-rank excitations of the graph filter, and present a simple spectral algorithm to extract the relevant equitable partitions. Finally, we establish theoretical bounds on the error that this spectral detection scheme incurs and perform numerical experiments that illustrate our theoretical results and compare both algorithms.
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