Temporal Motif Participation Profiles for Analyzing Node Similarity in Temporal Networks

TL;DR

This work addresses the challenge of identifying node similarity in temporal networks by moving beyond global motif counts to capture node-level roles in temporal motifs. It introduces Temporal Motif Participation Profiles (TMPPs), a 104-dimensional, position-aware embedding that encodes how often a node occupies each motif position across all 3-edge temporal motifs, with a 36-dimensional positionless variant for comparison. Through simulations with the MULCH model and a case study on militarized interstate disputes, the authors demonstrate that including motif positions yields significantly more accurate and interpretable clustering of nodes by role than positionless counts, enabling clearer insight into network structure and actor behavior. The TMPP framework provides a human-interpretable, unsupervised embedding that can facilitate analysis, visualization, and cross-domain application to complex time-evolving networks, with future work extending to larger motifs and scalable computation.

Abstract

Temporal networks consisting of timestamped interactions between a set of nodes provide a useful representation for analyzing complex networked systems that evolve over time. Beyond pairwise interactions between nodes, temporal motifs capture patterns of higher-order interactions such as directed triangles over short time periods. We propose temporal motif participation profiles (TMPPs) to capture the behavior of nodes in temporal motifs. Two nodes with similar TMPPs take similar positions within temporal motifs, possibly with different nodes. TMPPs serve as unsupervised embeddings for nodes in temporal networks that are directly interpretable, as each entry denotes the frequency at which a node participates in a particular position in a specific temporal motif. We demonstrate that clustering TMPPs reveals groups of nodes with similar roles in a temporal network through simulation experiments and a case study on a network of militarized interstate disputes.

Figures (8)

• Figure 1: \ref{['fig:all_temporal_motifs']} All possible 3-edge temporal motifs involving 2 or 3 nodes. The edges are labeled in the order in which they appear. The green box in the bottom left denotes motifs with 2 nodes, and the grey boxes denote triangle motifs. Figure credit: paranjape2017motifs. \ref{['fig:mids_total_motif_counts']} Occurrence frequencies of all temporal motifs in a militarized interstate dispute network, arranged to match the motif ordering in \ref{['fig:all_temporal_motifs']}. Figure credit: do2022analyzing. \ref{['fig:motif_positions']} Node positions in a temporal motif used in this paper, shown on motif $M_{1,1}$. The green node always denotes position 1, the red node always denotes position 2, and the blue node always denotes position 3. There is no position 3 for the 2-node motifs.
• Figure 2: Toy example showing the construction of a temporal motif participation profile (TMPP) for node A from the example network shown in \ref{['fig:tmpp_example_graph']}. The example network contains 4 temporal motifs, and the counts of each of the 3 nodes appearing in the 4 motifs in each position are shown in \ref{['fig:tmpp_example_counts']}, which are then normalized by the column sums to form TMPP vectors, shown in \ref{['fig:tmpp_example_vectors']}. Each TMPP vector can be interpreted using heatmaps of temporal motifs and positions, as shown in \ref{['fig:tmpp_example_heatmaps']} for node A.
• Figure 3: Heatmaps of cluster centroids from Scenario 1 using \ref{['fig:sim_scenario_1_tmpp']} positioned and \ref{['fig:sim_scenario_1_positionless']} positionless TMPPs. The positioned TMPP more accurately recovers the true blocks.
• Figure 4: Heatmaps of cluster centroids from Scenario 2 using \ref{['fig:sim_scenario_2_tmpp']} positioned and \ref{['fig:sim_scenario_2_positionless']} positionless TMPPs. The positioned TMPP more accurately recovers the true blocks.
• Figure 5: Dendrogram displaying results from hierarchical clustering on the (positioned) TMPPs from the MIDs data. The dendrogram is cut into 10 clusters, as indicated by the colors. The orange cluster furthest to the left is denoted as cluster 0, the next cluster in green is denoted as cluster 1, and so on.