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Edge Sparsification via Temporal Forman-Ricci Curvature for Dynamic Graph Learning

This paper proposes TRicci, a network-curvature-inspired edge sparsification framework that extends Forman-Ricci curvature to directed weighted temporal graphs, achieving approximately 80% sparsification and a 55.94% reduction in training and inference time across various datasets while maintaining predictive performance.

Original authors: Poupak Azad, Cuneyt Gurcan Akcora, Kiarash Shamsi

Published 2026-08-25

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CC BY 4.0

Original authors: Poupak Azad, Cuneyt Gurcan Akcora, Kiarash Shamsi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The modern world runs on networks that never stand still. Financial markets, social media feeds, and communication systems are not static maps but living streams of interactions, where connections form, fade, and shift every second. To understand these systems, scientists build digital models called temporal graphs, which capture not just who is connected to whom, but exactly when those connections happened. The challenge is that these models can become overwhelmingly large and dense, filled with millions of fleeting interactions. Processing such massive, rapidly changing data requires immense computing power, often slowing down analysis to a crawl or making it impossible to run on standard machines. The core question for researchers is how to strip away the noise and redundancy in these data streams without losing the vital patterns that reveal how the system actually works.

A team of researchers has proposed a new way to tackle this problem by looking at the geometry of these connections. Instead of simply counting how many times nodes interact or removing connections at random, they developed a method that measures the "curvature" of each interaction. Imagine a landscape where some paths are wide, well-traveled highways and others are narrow, redundant footpaths that lead nowhere new. In the language of mathematics, this landscape has a shape, and the researchers adapted an ancient geometric concept—originally used to describe the curvature of surfaces—to measure the importance of every single edge in a time-based network. They call their method TRicci. It assigns a score to every connection based on three things: how active the two ends of the connection are, how recently the interaction occurred, and whether there are many other similar interactions happening at the same time that make this specific one less unique.

The researchers applied this scoring system to a wide variety of real-world data, including nine different blockchain transaction networks and three large benchmark datasets covering everything from cryptocurrency transfers to online product reviews. In these networks, a single transaction might be a critical signal of a shift in user behavior, while thousands of other transactions might be repetitive noise that adds no new information. By calculating the curvature score for every edge in these massive datasets, the team could rank the connections from most important to least important. They then tested a simple strategy: keep only the top 20 percent of connections—the ones with the highest curvature scores—and discard the remaining 80 percent.

The results were striking. When the researchers fed these trimmed-down, sparse graphs into standard prediction models, the systems performed almost as well as they did with the full, untrimmed data. In fact, across all the experiments, the simplified graphs preserved 97.7 percent of the predictive power of the original, massive networks. This means that by removing the vast majority of the edges, the researchers did not lose the ability to forecast future network activity, identify influential users, or detect changes in participation. The method proved particularly effective at spotting the "highways" of the network—those interactions that carry unique structural and temporal weight—while filtering out the redundant "footpaths" that clutter the view.

Beyond just maintaining accuracy, the method delivered a massive boost in speed. Because the models had to process far fewer connections, the time required to train the algorithms and make predictions dropped by an average of 55.94 percent. In some cases, the time savings were even higher, reaching nearly 77 percent for specific datasets. This efficiency gain is crucial for real-time applications where decisions must be made quickly, such as detecting fraud in financial transactions or monitoring the spread of information on social platforms. The researchers found that the specific timing of interactions mattered deeply; connections that happened close together in time often competed with each other, and the method successfully identified which of those competing interactions were the most significant.

The study also explored how different ways of selecting edges affected the outcome. They tested whether keeping the most curved edges was better than keeping the least curved ones or selecting them randomly. The data showed a clear pattern: the most curved edges consistently held the most predictive value. This suggests that in a dynamic network, the most important interactions are not necessarily the most frequent ones, but rather those that stand out against the local background of activity. The researchers verified this by testing their method against several existing techniques designed to simplify graphs, and their approach consistently outperformed the others in preserving the ability to predict future network states.

What makes this approach distinct is that it does not rely on a specific type of machine learning model to do the work. Instead, it acts as a universal filter that can be applied before any analysis begins. The researchers demonstrated that by understanding the local geometry of the network—how an edge fits into its immediate neighborhood of time and activity—one can identify the essential structure of the system. This allows for a much lighter, faster, and more efficient way to study complex systems without sacrificing the insights that come from the data. The findings suggest that for many dynamic networks, the vast majority of the connections are not needed to understand the whole picture, and that a careful, geometry-based selection of the remaining edges can reveal the true shape of the system's evolution.

Technical Summary: Edge Sparsification via Temporal Forman-Ricci Curvature for Dynamic Graph Learning

Problem Statement

Temporal graph learning is critical for analyzing real-world systems with continuously evolving interactions, such as financial transaction networks, communication systems, and social platforms. However, learning from large-scale temporal graphs presents significant computational challenges. As interactions accumulate, networks become dense and rapidly changing, causing downstream learning pipelines (involving snapshot construction, feature extraction, and model training) to incur high memory and runtime costs. Furthermore, dense graphs often contain numerous edges that contribute little to predictive tasks, introducing noise and redundancy.

Existing graph sparsification methods primarily target static graphs, relying on degree-based or topology-only criteria that fail to account for the temporal role of edges (e.g., timestamps, recency, and time-dependent competition). While some recent works address spatio-temporal settings, they often focus on specific tasks like node classification or forecasting and lack a model-independent mechanism to rank edges in directed, weighted temporal graphs based on both structural support and temporal context.

Methodology: TRicci

The authors propose TRicci, a network-curvature-inspired edge sparsification framework designed specifically for directed weighted temporal graphs. The method extends classical Forman–Ricci curvature to the temporal domain to assign an importance score to each edge, reflecting its structural support, temporal relevance, and local interaction competition.

Core Components

Temporal Forman–Ricci Curvature Formulation:
The framework adapts the classical "support-minus-penalty" structure of Forman–Ricci curvature. For a temporal edge $e = (u, v, w_e, t_e)$ , the score is calculated as:
$\text{TRicci}(e) = S_e - D_e$
- Endpoint Support ( $S_e$ ): Measures the distinctiveness of the edge relative to the activity of its endpoints. The authors use log-smoothed node activity ( $s(x) = \log(1 + \text{strength}(x))$ ) to prevent highly active nodes from dominating the score. The support term is inversely proportional to the smoothed activity of the source and destination nodes.
- Temporal Competition Penalty ( $D_e$ ): Quantifies the redundancy of the edge within its local temporal neighborhood. This component considers outgoing edges from the source and destination nodes. Crucially, it employs a temporal decay kernel $K(e, e', \tau) = \exp(-|t_e - t_{e'}|/\tau)$ to weight neighboring edges. Edges occurring close in time contribute more strongly to the penalty, reflecting that temporally proximate interactions often provide overlapping structural information.
Snapshot-Based Computation:
To avoid mixing interactions from different temporal contexts, the temporal edge stream is partitioned into non-overlapping discrete-time snapshots. Curvature scores are computed independently within each snapshot, ensuring that edge importance is relative to its local temporal context without future information leakage.
Task-Guided Edge Selection:
TRicci serves as a model-independent scoring mechanism. After computing curvature scores, a task-guided policy selects which edges to retain. The authors evaluate different policies (retaining high-curvature, low-curvature, or intermediate bands) via validation. Their experiments indicate that retaining high-curvature edges (the top-ranked edges) yields the best predictive preservation.

Key Contributions

The paper outlines three primary contributions:

Novel Formulation: The introduction of the first Temporal Forman–Ricci curvature formulation specifically for directed weighted temporal graphs, incorporating endpoint activity, temporal recency, and local interaction competition.
Sparsification Framework: The application of this curvature metric to construct compact, sparse temporal graph representations by retaining task-relevant, curvature-ranked edges.
Comprehensive Evaluation: An extensive evaluation across 12 datasets (9 blockchain transaction networks and 3 TGBL benchmarks) and three graph-level prediction tasks. The study compares TRicci against three state-of-the-art sparsification methods (MoG, TEDDY, SEM).

Experimental Results

The framework was evaluated on three graph-level prediction tasks:

Network Activity Growth Prediction: Predicting if the number of edges increases in the next snapshot.
Network Participation Expansion Prediction: Predicting if the number of active nodes increases.
Influential Node Turnover Prediction: Predicting significant changes in the set of top-degree nodes.

Key Findings:

Predictive Preservation: Under an 80% edge removal ratio (retaining only ~20% of edges), TRicci preserved 97.7% ± 2.2% of the full-graph ROC–AUC on average. This significantly outperformed the strongest competing method (TEDDY), which preserved 89.4% ± 6.3%.
Efficiency: The framework reduced end-to-end training and inference time by an average of 55.94% across all datasets.
Curvature Analysis: Sensitivity analysis revealed that high-curvature edges (Bin 5) consistently retained the most predictive information, supporting a Pareto-style concentration where a small subset of edges carries the majority of the signal.
Robustness: The method showed reasonable robustness to the temporal decay parameter $\tau$ , with optimal performance typically observed around $\tau = 1$ day.

Significance and Claims

The authors claim that TRicci demonstrates that temporal curvature can serve as a principled basis for scalable temporal graph learning. By preserving predictive temporal-structural information under substantial sparsification, the method addresses the computational bottleneck of dense dynamic networks without requiring task-specific retraining of the sparsification logic.

The paper emphasizes that unlike previous methods that optimize synthetic representations or rely on static topology, TRicci explicitly models the temporal role of edges (timestamps and recency) and local interaction competition. This allows for the construction of sparse graphs that are not only computationally efficient but also retain the specific structural dynamics necessary for downstream prediction tasks in evolving networks. The authors position this work as a step toward scalable learning in domains where interaction data is massive, dense, and time-sensitive, such as blockchain analysis and online social platforms.