Temporal Embedding Grouping (TEG)

• Temporal Embedding Grouping (TEG) is a method that encodes time-stamped events into low-dimensional embeddings, capturing both temporal dynamics and interaction motifs.
• It integrates edge functions like inter-event time and motif labeling to facilitate clustering, anomaly detection, and pattern discovery in dynamic networks.
• TEG preserves complete timing and relational information, enabling lossless reconstruction of temporal data and supporting scalable, multiscale analyses.

Temporal Embedding Grouping (TEG) encompasses a broad set of methodologies that extract, encode, and group temporal structures from dynamic data—such as time-stamped interaction networks, video sequences, or evolving attributed graphs—into compact representations that capture both timing and interaction motifs. TEG aims to enable downstream tasks, including clustering, classification, anomaly detection, and pattern discovery, by preserving temporal dependencies and structural nuances across different resolutions and modalities.

1. Core Definitions and Theoretical Foundations

TEG refers to approaches that distill temporal information from sequential or event-driven data streams into grouped, often low-dimensional, embeddings where shared temporal characteristics and patterns can be exploited. In temporal networks, each event is typically described as $e_i = (u_i, v_i, t_i)$, with $u_i$ and $v_i$ representing entities and $t_i$ a timestamp. TEG frameworks construct representations where both the timing of events and the nature of their connectivity are jointly encoded, enabling the discovery of temporally cohesive groups, motifs, or roles that would be obfuscated in static or purely structural embeddings.

A foundational instance of TEG is the Temporal Event Graph (TEG) formalism (Mellor, 2017). The TEG represents each event as a vertex and links vertices according to δ-adjacency—an edge is drawn from $e_i$ to $e_j$ if:

1. The events share at least one node (${u_i, v_i} \cap {u_j, v_j} \ne \emptyset$),
2. The events occur within a prescribed interval ($0 < t_j - t_i < \delta$).

This yields a directed acyclic graph where edge labels include both the inter-event time $\tau((e_i, e_j)) = t_j - t_i$ and a motif label $\mu((e_i, e_j))$ capturing the sequence and roles of participants, foundational for temporal grouping and motif-based clustering.

2. Edge Functions and Motif Distributions

Central to TEG is the explicit association of two edge functions:

• Inter-event time $\tau((e_i, e_j))$: Quantifies the temporal interval between linked events. This feature enables characterization of actors' response times and collective dynamics.
• Motif function $\mu((e_i, e_j))$: Encodes the interaction type by mapping the ordered concatenation $(u_i, v_i, u_j, v_j)$ to a discrete label (e.g., ABAB, ABBA, ABAC), distinguishing patterns such as reciprocity, broadcasting, or chain interactions.

By systematically cataloguing joint distributions $Pr(\tau, \mu)$ across temporal components, one obtains empirical characterizations of both fine-scale individual behavior (e.g., reciprocated exchanges are faster on average) and macro-scale group dynamics (e.g., the emergence of tightly coupled clusters or bursty temporal components). Analysis of synthetic and real temporal networks reveals structural transitions, such as the percolation of a giant component as $\delta$ increases, and motif frequency distributions can be predicted analytically in large-$N$ limits.

3. Lossless, Static Representation and Reconstructibility

The TEG constitutes a lossless, static encoding of the temporal network: every event and its timing, as well as the identity of involved nodes and the temporal topology, are uniquely recoverable from the graph structure and its labeled edges. An explicit reconstruction algorithm leverages the maximal path in TEG, using $\tau$ and $\mu$ labels to sequentially reconstitute event timings and participant mappings, ensuring no information is lost relative to the original set of time-stamped events.

This invertibility distinguishes TEG from time-aggregated (snapshot) models, which inherently discard higher-order timing and sequencing information critical for accurately grouping or embedding temporally dependent patterns.

4. TEG as a Platform for Temporal Embedding Grouping

TEG’s expressivity enables several downstream temporal grouping and embedding tasks:

• Component analysis: TEG decomposes the temporal network into connected components, each corresponding to clusters of densely interlinked events. These components can represent tightly interacting collectives (e.g., conversation clusters in social networks) or bursty cascades.
• Feature extraction for embedding: Motif frequency vectors and inter-event time distributions serve as robust features for clustering algorithms (e.g., $k$-means, hierarchical clustering, or deep metric learning), providing interpretable axes along which to group temporal events or actors.
• Temporal motif clustering: The multilabel edge information allows decomposition of interaction streams according to characteristic motifs, revealing functional groupings such as reciprocating pairs or broadcast initiators.
• Signature identification: Discriminating node- or group-level signatures by comparing the distributions $Pr(\tau|\mu)$ or motif co-occurrence frequencies, facilitating anomaly detection or behavioral classification.

The unique disentangling of time and relational structure allows TEG-derived features to be directly incorporated into higher-level temporal embedding models that seek to group nodes/events with similar temporal and structural profiles.

5. Applications and Empirical Illustrations

The analytical and empirical utility of TEG has been demonstrated through both synthetic and real-world data:

• Synthetic network analysis: Using a suite of generated networks, the progression of TEG component size and motif distributions as a function of $\delta$ elucidates transitions from local, fragmented interaction groups to global, interconnected structures. Closed-form motif frequency probabilities have been established for large-$N$ regimes.
• Social communication networks: In longitudinal messaging data from university students, TEG visualizations uncover recurring communication motifs and dynamic formation of group conversations. Short inter-event intervals cluster around motifs associated with fast reciprocation, whereas motifs involving three or more actors correlate with broader, sparser temporal patterns.
• Temporal barcodes and burst detection: By mapping connected components in the TEG, event barcoding visualizations highlight the timing and extent of temporal clusters, offering interpretable summaries of complex temporal interaction patterns.

These analyses confirm that combining $\tau$ and $\mu$ distributions yields detailed characterizations of both individual and collective temporal behaviors, directly supporting the grouping of nodes or events according to shared temporal embedding profiles.

6. Significance and Implications for Temporal Embedding Grouping

The TEG and its associated feature distributions provide a principled foundation for Temporal Embedding Grouping. By preserving all original temporal and relational details, TEG-based features can be utilized as direct inputs for unsupervised and supervised embedding algorithms aimed at clustering or classifying dynamic interaction patterns. Critical aspects include:

• Fine-resolution temporal decomposition without aggregation-induced loss, enabling investigation of burst dynamics and motif-specific clustering.
• Support for motif-conditioned embeddings, enhancing sensitivity to qualitative interaction type beyond simple frequency-based or topological metrics.
• Scalability and interpretability, as TEG’s representation lends itself to both efficient implementation (each event has at most two outgoing edges) and rigorous reconstruction guarantees.

These properties position TEG as a core object for modern TEG methodologies, facilitating the development of embedding and grouping schemes that respect and exploit the duality of timing and pattern structure inherent to temporal data.

7. Perspectives and Future Directions

The TEG framework sets the stage for advanced Temporal Embedding Grouping strategies across a range of domains, from social and biological networks to communication, transportation, and transactional systems. Possible extensions include:

• Development of motif- and time-aware dimensionality reduction techniques engineered for TEG features.
• Integrative models that combine TEG-derived features with node or event attributes to refine subgroup identification.
• Rigorous exploration of hierarchical or multiscale temporal clustering—leveraging TEG’s tunable $\delta$ parameter to paper temporal resolutions from instantaneous interactions to persistent groupings.

In sum, the TEG provides an actionable, mathematically grounded basis for constructing, analyzing, and grouping temporal network embeddings that faithfully reflect the full richness of dynamic relational data, laying a robust foundation for the broader field of Temporal Embedding Grouping (Mellor, 2017).