Analytical Models for Motifs in Temporal Networks

Abstract

Dynamic evolving networks capture temporal relations in domains such as social networks, communication networks, and financial transaction networks. In such networks, temporal motifs, which are repeated sequences of time-stamped edges/transactions, offer valuable information about the networks' evolution and function. However, calculating temporal motif frequencies is computationally expensive as it requires: First, identifying all instances of the static motifs in the static graph induced by the temporal graph. And second, counting the number of subsequences of temporal edges that correspond to a temporal motif and occur within a time window. Since the number of temporal motifs changes over time, finding interesting temporal patterns involves iterative application of the above process over many consecutive time windows. This makes it impractical to scale to large real temporal networks. Here, we develop a fast and accurate model-based method for counting motifs in temporal networks. We first develop the Temporal Activity State Block Model (TASBM), to model temporal motifs in temporal graphs. Then we derive closed-form analytical expressions that allow us to quickly calculate expected motif frequencies and their variances in a given temporal network. Finally, we develop an efficient model fitting method, so that for a given network, we quickly fit the TASMB model and compute motif frequencies. We apply our approach to two real-world networks: a network of financial transactions and an email network. Experiments show that our TASMB framework (1) accurately counts temporal motifs in temporal networks; (2) easily scales to networks with tens of millions of edges/transactions; (3) is about 50x faster than explicit motif counting methods on networks of about 5 million temporal edges, a factor which increases with network size.