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A great variety of complex systems, from user interactions in communicationnetworks to transactions in financial markets, can be modeled as temporalgraphs consisting of a set of vertices and a series of timestamped and directededges. Temporal motifs are generalized from subgraph patterns in static graphswhich consider edge orderings and durations in addition to topologies. Countingthe number of occurrences of temporal motifs is a fundamental problem fortemporal network analysis. However, existing methods either cannot supporttemporal motifs or suffer from performance issues. Moreover, they cannot workin the streaming model where edges are observed incrementally over time. Inthis paper, we focus on approximate temporal motif counting via randomsampling. We first propose two sampling algorithms for temporal motif countingin the offline setting. The first is an edge sampling (ES) algorithm forestimating the number of instances of any temporal motif. The second is animproved edge-wedge sampling (EWS) algorithm that hybridizes edge sampling withwedge sampling for counting temporal motifs with $3$ vertices and $3$ edges.Furthermore, we propose two algorithms to count temporal motifs incrementallyin temporal graph streams by extending the ES and EWS algorithms referred to asSES and SEWS. We provide comprehensive analyses of the theoretical bounds andcomplexities of our proposed algorithms. Finally, we perform extensiveexperimental evaluations of our proposed algorithms on several real-worldtemporal graphs. The results show that ES and EWS have higher efficiency,better accuracy, and greater scalability than state-of-the-art sampling methodsfor temporal motif counting in the offline setting. Moreover, SES and SEWSachieve up to three orders of magnitude speedups over ES and EWS while havingcomparable estimation errors for temporal motif counting in the streamingsetting.

Efficient Sampling Algorithms for Approximate Motif Counting in Temporal Graph Streams