Synchronization of Group-Labelled Multi-Graphs

Synchronization refers to the problem of inferring the unknown values attached to vertices of a graph where edges are labelled with the ratio of the incident vertices, and labels belong to a group. This paper addresses the synchronization problem on multi-graphs, that are graphs with more than one edge connecting the same pair of nodes. The problem naturally arises when multiple measures are available to model the relationship between two vertices. This happens when different sensors measure the same quantity, or when the original graph is partitioned into sub-graphs that are solved independently. In this case, the relationships among sub-graphs give rise to multi-edges and the problem can be traced back to a multi-graph synchronization. The baseline solution reduces multi-graphs to simple ones by averaging their multi-edges, however this approach falls short because: i) averaging is well defined only for some groups and ii) the resulting estimator is less precise and accurate, as we prove empirically. Specifically, we present MultiSynch, a synchronization algorithm for multi-graphs that is based on a principled constrained eigenvalue optimization. MultiSynch is a general solution that can cope with any linear group and we show to be profitably usable both on synthetic and real problems.