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The increasing availability of temporal network data is calling for moreresearch on extracting and characterizing mesoscopic structures in temporalnetworks and on relating such structure to specific functions or properties ofthe system. An outstanding challenge is the extension of the results achievedfor static networks to time-varying networks, where the topological structureof the system and the temporal activity patterns of its components areintertwined. Here we investigate the use of a latent factor decompositiontechnique, non-negative tensor factorization, to extract the community-activitystructure of temporal networks. The method is intrinsically temporal and allowsto simultaneously identify communities and to track their activity over time.We represent the time-varying adjacency matrix of a temporal network as athree-way tensor and approximate this tensor as a sum of terms that can beinterpreted as communities of nodes with an associated activity time series. Wesummarize known computational techniques for tensor decomposition and discusssome quality metrics that can be used to tune the complexity of the factorizedrepresentation. We subsequently apply tensor factorization to a temporalnetwork for which a ground truth is available for both the community structureand the temporal activity patterns. The data we use describe the socialinteractions of students in a school, the associations between students andschool classes, and the spatio-temporal trajectories of students over time. Weshow that non-negative tensor factorization is capable of recovering the classstructure with high accuracy. In particular, the extracted tensor componentscan be validated either as known school classes, or in terms of correlatedactivity patterns, i.e., of spatial and temporal coincidences that aredetermined by the known school activity schedule.

Detecting the community structure and activity patterns of temporal networks: a non-negative tensor factorization approach