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Cliques, groups of fully connected nodes in a network, are often used to study group dynamics of complex systems. In real-world settings, group dynamics often have a temporal component. For example, conference attendees moving from one group conversation to another. Recently, maximal clique enumeration methods have been introduced that add temporal (and frequency) constraints, to account for such phenomena. These methods enumerate so called (delta,gamma)-maximal cliques. In this work, we introduce an efficient (delta,gamma)-maximal clique enumeration algorithm, that extends gamma from a frequency constraint to a more versatile weighting constraint. Additionally, we introduce a definition of (delta,gamma)-cliques, that resolves a problem of existing definitions in the temporal domain. Our approach, which was inspired by a state-of-the-art two-phase approach, introduces a more efficient initial (stretching) phase. Specifically, we reduce the time complexity of this phase to be linear with respect to the number of temporal edges. Furthermore, we introduce a new approach to the second (bulking) phase, which allows us to efficiently prune search tree branches. Consequently, in experiments we observe speed-ups, often by several order of magnitude, on various (large) real-world datasets. Our algorithm vastly outperforms the existing state-of-the-art methods for temporal networks, while also extending applicability to weighted networks. |
