  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.
