Scalable Temporal Motif Densest Subnetwork Discovery

TL;DR

This work designs two novel randomized approximation algorithms with rigorous probabilistic guarantees that provide high-quality solutions to the novel problem of identifying the temporal motif densest subnetwork, i.e., the densest subnetwork with respect to temporal motifs.

Abstract

Finding dense subnetworks, with density based on edges or more complex structures, such as subgraphs or $k$-cliques, is a fundamental algorithmic problem with many applications. While the problem has been studied extensively in static networks, much remains to be explored for temporal networks. In this work we introduce the novel problem of identifying the temporal motif densest subnetwork, i.e., the densest subnetwork with respect to temporal motifs, which are high-order patterns characterizing temporal networks. This problem significantly differs from analogous formulations for dense temporal (or static) subnetworks as these do not account for temporal motifs. Identifying temporal motifs is an extremely challenging task, and thus, efficient methods are required. To this end, we design two novel randomized approximation algorithms with rigorous probabilistic guarantees that provide high-quality solutions. We perform extensive experiments showing that our methods outperform baselines. Furthermore, our algorithms scale on networks with up to billions of temporal edges, while baselines cannot handle such large networks. We use our techniques to analyze a financial network and show that our formulation reveals important network structures, such as bursty temporal events and communities of users with similar interests.

Figures (12)

• Figure 1: \ref{['subfig:TNandTM']}: Representation of a temporal network $T$ with $n=7$ vertices, $m=18$ edges (edge labels denote the timings, and commas denote multiple edges), and a temporal motif $M$ with its ordering $\sigma$ (i.e., $\sigma$ captures the temporal dynamics of the motif $M$). \ref{['subfig:deltaInsts']}: for $\delta=10$ only the green sequences ($S_1, S_2$) are $\delta$-instances of $M$ in $T$; the red sequences are not $\delta$-instances, since $S_3$ cannot be mapped on the motif $M$ following $\sigma$, and $S_4$ exceeds the timing constraint on $\delta$.
• Figure 2: Temporal motifs ($\mathsf{M}_i,i\in[10]$) used in the experimental evaluation. Motifs with blue vertices are not used on EquinixChicago since this network is bipartite. For each motif $t_i, i=1,\ldots$ denotes the ordering of its edges in $\sigma$.
• Figure 3: For each configuration we report (top): the quality of the solution compared to the best empirical solution (i.e., $\hat{\mathrm{OPT}\xspace}$), and (bottom): the average running times to achieve such solution.
• Figure 4: Directed static network of $T[W^*_{\delta}]$ according to different values of $\delta$ on $\mathsf{M}_5$, we color edges according to the number temporal edges that map on each static edge. Below each static network we report the temporal support of $T[W^*_{\delta}]$, i.e., we place a bar in correspondence of the timings of the events in $T[W_\delta^*]$ over the timespan of observation of the network. (Left): $\delta_1=7\,200$. (Right): $\delta_2=172\,800$.
• Figure 5: Dependency graph for the main results we obtain, i.e., Theorem \ref{['lemma:ratioSampling']} and Theorem \ref{['lemma:ratioHybrid']}.

Theorems & Definitions (13)

• definition 1: Paranjape2016
• definition 2: $\delta$-instance of a temporal motif
• lemma 1
• theorem 1
• theorem 2
• lemma 2
• lemma 3
• corollary 1
• lemma 4
• lemma 5