Efficient Sampling Algorithms for Approximate Temporal Motif Counting

Abstract

A great variety of complex systems ranging from user interactions in communication networks to transactions in financial markets can be modeled as temporal graphs, which consist of a set of vertices and a series of timestamped and directed edges. Temporal motifs in temporal graphs are generalized from subgraph patterns in static graphs which take into account edge orderings and durations in addition to structures. 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. In this paper, we focus on approximate temporal motif counting via random sampling. We first propose a generic edge sampling (ES) algorithm for estimating the number of instances of any temporal motif. Furthermore, we devise an improved EWS algorithm that hybridizes edge sampling with wedge sampling for counting temporal motifs with 3 vertices and 3 edges. We provide comprehensive analyses of the theoretical bounds and complexities of our proposed algorithms. Finally, we conduct extensive experiments on several real-world datasets, and the results show that our ES and EWS algorithms have higher efficiency, better accuracy, and greater scalability than the state-ofthe-art sampling method for temporal motif counting.

An efficient procedure for mining egocentric temporal motifs

“…Finally, Paranjape et al (2017) propose a mining strategy that extracts static motifs from the aggregate network (obtained collapsing all the temporal layers together and thus dropping the temporal information) and expands them into temporal motifs by considering the order of appearance of edges within a given temporal gap. Other studies investigated approximate methods for counting temporal motifs (Liu et al 2019;Wang et al 2020). None of these approaches tries to capture the temporal evolution of the interactions of a single node, which is the focus of our work.…”

Faster and Generalized Temporal Triangle Counting, via Degeneracy Ordering

“…Although, the definition in [20] is more restrictive and requires that all edges incident to a node are consecutive events of that node. Most existing temporal triangle counting literature uses these definitions [25,27,46,53]. We describe a simple example to see how Definition 1.1 offers richer temporal information.…”

“…There has also been recent progress on approximating the counts of temporal motifs and triangles [46,53]. Particularly , Liu et al presented a sampling framework for approximating the counts of 𝛿 1,3 -temporal motifs [25].…”

“…Temporal triangle and motif counting has applications in graph representation learning [48], expressivity of graph neural networks (GNNs) [7], network classification [47], temporal text network analysis [52], computer networks [50], and brain networks [9]. Recently, there has been significant interest in temporal triangle and motif counting algorithms [6,7,25,27,35,[46][47][48]53].…”

Quantifying Uncertainty for Temporal Motif Estimation in Graph Streams under Sampling

“…Several methods have been designed to count the occurrences of temporal motifs in graph streams (see [14] for a survey), with recent work focusing on estimating the count under various sampling schemes along with establishing concentration properties [19,36,32]. However, little attention has been given to the problem of uncertainty quantification and the asymptotic statistical properties of these temporal motif count estimators.…”