Temporal hypergraph motifs

Abstract

Group interactions arise in our daily lives (email communications, on-demand ride sharing, and comment interactions on online communities, to name a few), and they together form hypergraphs that evolve over time. Given such temporal hypergraphs, how can we describe their underlying design principles? If their sizes and time spans are considerably different, how can we compare their structural and temporal characteristics? In this work, we define 96 temporal hypergraph motifs (TH-motifs) and propose the relative occurrences of their instances as an answer to the above questions. TH-motifs categorize the relational and temporal dynamics among three connected hyperedges that appear within a short time. For scalable analysis, we develop THyMe $^{+}$, a fast and exact algorithm for counting the instances of TH-motifs in massive hypergraphs, and we show that THyMe $^{+}$ is up to 2,163$\times$ faster while requiring less space than baseline approaches. In addition to exact counting algorithms, we design three versions of sampling algorithms for approximate counting. We theoretically analyze the accuracy of the proposed methods, and we empirically show that the most advanced algorithm, , is up to $11.1\times$ more accurate than baseline approaches. Using the algorithms, we investigate 11 real-world temporal hypergraphs from various domains. We demonstrate that TH-motifs provide important information useful for downstream tasks and reveal interesting patterns, including the striking similarity between temporal hypergraphs from the same domain.

Appendices

Appendix A: Details of Dynamic Programming (DP)

In this subsection, we provide details of the Dynamic Programming (DP), a straightforward extension of temporal network motif counting [33]. As shown in Algorithm 6, it utilizes a dynamic programming approach to reduce the redundant enumeration of every instance of TH-motifs of the input temporal hypergraph T. The procedure count (lines 9–19) counts the instances of TH-motifs that induce a set of $\ell$ connected static hyperedges. That is, given a set $s=\{\tilde{e}_1,\dots ,\tilde{e}_{\ell }\}$ of $\ell$ connected static hyperedges, count first constructs a time-sorted sequence e(s) of temporal hyperedges whose nodes is one of s (line 10). It also introduces a map C that maintains the counts of ordered hyperedges of length at most $\ell$. Then count scans through the temporal hyperedges in e(s) and tracks the subsequences that occur within the temporal window that spans temporal hyperedges within $\delta$ time units. As the temporal window slides through the temporal hyperedges e(s), the count of the sequences are computed based on the subsequences counted in C. Refer to [33] for more intuition behind this dynamic programming formulation.

Appendix B: Details of Discrete Time Partitioning (DTP)

In this subsection, we provide details of the Discrete Time Partitioning (DTP), a straightforward extension of sampling algorithm for approximate temporal network motif counting [30]. As shown in Algorithm 7, it begins with randomly drawing a shift s from $\{-c\delta +1,\cdots ,0\}$ for the predefined input integer $c>0$ that controls the size of the sampling windows (line 3). Then, based on the selected shift s, the time is discretely partitioned into $c\delta$-sized intervals:

$$\begin{aligned} \mathscr {I}_s = \{[s+(j-1)c\delta , s + j\cdot c\delta - 1], j=1,2,...\}. \end{aligned}$$

The probability of an instance $\langle e_i, e_j, e_k \rangle$ of TH-motif to be completely contained within an interval $\mathscr {I}_s$ is:

$$\begin{aligned} 1 - \frac{\bigtriangleup (\langle e_i, e_j, e_k \rangle )}{c\delta }. \end{aligned}$$

Then, for each interval $I\in \mathscr {I}_s$, DTP constructs a sub-temporal hypergraph that consists of temporal hyperedges within I (line 6). From the partial temporal hypergraph, it uses THyMe to exhaustively discover the instances of TH-motifs. It associates a weighted count of the number of instances of TH-motifs by incrementing the counts by the inverse of the probability to be completely contained in the interval I. By adding up the weighted counts of all intervals, it yields an unbiased estimation of the number of instances of each TH-motif.

To speed up the estimation, DTP incorporates importance sampling to pick only a subset of intervals and combines the counts from them. Let $I_j$ be the j-th interval. Specifically, it samples an interval $I_j\in \mathscr {I}_s$ independently with the predefined probability $z_j$ (line 5). Here, DTP uses the ratio of the number of temporal hyperedges that arrived within the interval, which can be easily obtained from the input temporal hypergraph. Formally, the ratio $z_j$ is

$$\begin{aligned} z_j = r \cdot \frac{|\{e=(\tilde{e}, t)\in \mathscr {E} : t\in I_j\}|}{|\mathscr {E}|} \end{aligned}$$

where r is a constant that controls the magnitude of the probability. Then, the number of instances counted in the interval is additionally weighted by $\frac{1}{z_j}$ (line 9). These computations are repeated b times to reduce the variance of the estimation. Refer to [30] for details of the original algorithm.

Appendix C: Sub-algorithms of THyMe and THyMe+

In this subsection, we provide pseudocode of four sub-algorithms of THyMe and THyMe $^{+}$. These algorithms are used for analyzing the time complexity of THyMe and THyMe $^{+}$ in Sect. 4.