  Finding group elements from noisy measurements of their pairwise ratios is
also known as the group synchronization problem, first introduced in the
context of the group SO(2) of planar rotations, whose usefulness has been
demonstrated recently in engineering and structural biology. Here, we focus on
synchronization over $\mathbb{Z}_2$, and consider the problem of identifying
communities in a multiplex network when the interaction between the nodes is
described by a signed (possibly weighted) measure of similarity, and when the
multiplex network has a natural partition into two communities. When one has
the additional information that certain subsets of nodes represent the same
unknown group element, we consider and compare several algorithms based on
spectral, semidefinite programming (SDP) and message passing algorithms. In
other words, all nodes within such a subset represent the same unknown group
element, and one has available noisy pairwise measurements between pairs of
nodes that belong to different non-overlapping subsets. Following a recent
analysis of the eigenvector method for synchronization over SO(2), we analyze
the robustness to noise of the eigenvector method for synchronization over
$\mathbb{Z}_2$, when the underlying graph of pairwise measurements is the
Erd\H{o}s-R\'{e}nyi random graph, using results from the random matrix theory
literature. We also propose a message passing synchronization algorithm that
outperforms the existing eigenvector synchronization algorithm only for certain
classes of graphs and noise models, and enjoys the flexibility of incorporating
constraints that may not be easily accommodated by other spectral or SDP-based
methods. We apply our methods both to several synthetic models and a real data
set of roll call voting patterns in the U.S. Congress across time, to identify
the two existing communities, i.e., the Democratic and Republican parties.
