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Efficient Sampling Algorithms for Approximate Motif Counting in Temporal Graph Streams
A great variety of complex systems, from user interactions in communication
networks to transactions in financial markets, can be modeled as temporal
graphs consisting of a set of vertices and a series of timestamped and directed
edges. Temporal motifs are generalized from subgraph patterns in static graphs
which consider edge orderings and durations in addition to topologies. Counting
the number of occurrences of temporal motifs is a fundamental problem for
temporal network analysis. However, existing methods either cannot support
temporal motifs or suffer from performance issues. Moreover, they cannot work
in the streaming model where edges are observed incrementally over time. In
this paper, we focus on approximate temporal motif counting via random
sampling. We first propose two sampling algorithms for temporal motif counting
in the offline setting. The first is an edge sampling (ES) algorithm for
estimating the number of instances of any temporal motif. The second is an
improved edge-wedge sampling (EWS) algorithm that hybridizes edge sampling with
wedge sampling for counting temporal motifs with vertices and edges.
Furthermore, we propose two algorithms to count temporal motifs incrementally
in temporal graph streams by extending the ES and EWS algorithms referred to as
SES and SEWS. We provide comprehensive analyses of the theoretical bounds and
complexities of our proposed algorithms. Finally, we perform extensive
experimental evaluations of our proposed algorithms on several real-world
temporal graphs. The results show that ES and EWS have higher efficiency,
better accuracy, and greater scalability than state-of-the-art sampling methods
for temporal motif counting in the offline setting. Moreover, SES and SEWS
achieve up to three orders of magnitude speedups over ES and EWS while having
comparable estimation errors for temporal motif counting in the streaming
setting.Comment: 27 pages, 11 figures; overlapped with arXiv:2007.1402
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