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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. The usefulness of synchronization over the group   $\mathbb {Z}_2$   has been demonstrated in recent algorithms for localization of sensor networks and three-dimensional structuring of molecules. In this paper, 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 (and possibly weighted) measure of similarity, and when the multiplex network has a natural partition into two communities, of possibly different sizes. In the setting where one has the additional information that certain subsets of nodes represent the same (unknown) group element, we consider and compare several algorithms for synchronization over   $\mathbb {Z}_2$  , based on spectral and SDP relaxations, 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 analyse the robustness to noise of the eigenvector method for synchronization over   $\mathbb {Z}_2$  , when the underlying graph of pairwise measurements is the Erdős–Rényi random graph, using results from the random matrix theory literature on the largest eigenvalue of rank-1 deformation of large random matrices. We also propose a message passing synchronization algorithm, inspired by the standard belief propagation algorithm, that outperforms the existing eigenvector synchronization algorithm only for certain classes of graphs and noise models, and enjoys the flexibility of incorporating additional constraints that may not be easily accommodated by any of the other spectral or SDP-based methods. We apply the synchronization methods both to several synthetic models and a real data set of roll call voting patterns in the US Congress across time, to identify the two existing communities, i.e., the Democratic and Republican parties. Finally, we discuss a number of related open problems and future research directions.

Synchronization over Z 2 and community detection in signed multiplex networks with constraints