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

Edge Sparsification via Temporal Forman–Ricci Curvature for Dynamic Graph Learning POUPAK AZAD, University of Manitoba, Canada CUNEYT GURCAN AKCORA, University of Central Florida, United States KIARASH SHAMSI, University of Manitoba, Canada Temporal graph learning has become essential for analyzing real-world systems whose interactions continuously evolve over time, including financial transaction networks, communication systems, and online social platforms. However, learning from large-scale temporal graphs remains computationally challenging when networks are dense and rapidly changing. To address this limitation, we propose a network-curvature-inspired edge sparsification framework for dynamic graph learning. Our proposed method, TRicci, extends classical Forman–Ricci curvature to directed weighted temporal graphs by capturing structural support, temporal recency, and local interaction competition. Experiments on 9 transaction networks and 3 temporal graph benchmark datasets demonstrate that the proposed framework preserves predictive performance across multiple graph-level prediction tasks. The results show that TRicci sparsifies temporal graphs by approximately 80% while reducing end-to-end downstream training and inference time by an average of 55.94%, without substantial degradation in predictive performance. Our findings suggest that temporal curvature can serve as a principled basis for scalable temporal graph learning by preserving

arXiv:2608.07158v1 [cs.LG] 7 Aug 2026

predictive temporal-structural information under substantial sparsification. CCS Concepts: • Information systems → Data mining; • Computing methodologies → Machine learning algorithms; • Mathematics of computing → Graph algorithms; • Networks → Network structure. Additional Key Words and Phrases: Dynamic Graph Learning, Temporal Graphs, Graph Sparsification, Forman–Ricci Curvature

1

Introduction

Many real-world systems are represented as temporal graphs, where entities interact through time-stamped edges whose structure evolve continuously [13]. Examples include communication systems [14], financial transaction networks [16], online social platforms [19], and recommender systems [5]. Learning from such graphs is important because temporal interaction patterns often reveal changes in activity, influence, participation, and emerging structural roles. In blockchain transaction networks, for instance, the temporal organization of transfers can expose shifts in user activity, transaction flow, and the emergence or disappearance of influential addresses [32]. Despite their importance, temporal graphs pose computational challenges. As interactions accumulate over time, temporal networks can become large, dense, and rapidly changing [11]. Downstream graph learning pipelines must repeatedly process evolving edge sets, construct temporal snapshots, extract structural features, and train predictive models over sequences of graph states. This becomes especially expensive when the graph contains many edges that contribute little to the target prediction task. In such settings, using the full graph may increase memory consumption and runtime while also introducing noisy or weakly informative interactions into the learning pipeline. One way to address this challenge is graph sparsification, where the goal is to retain a compact subset of informative edges while preserving the predictive and structural properties of the original graph [23]. Existing sparsification and graph condensation methods have shown that smaller graph representations can support efficient learning [20, 31, 29]. However, many existing approaches are designed primarily for static graphs, rely on degree-based or topology-only criteria, or optimize synthetic graph representations without explicitly modeling the temporal role of edges [8, 25]. This is limiting for dynamic transaction networks, where the importance of an interaction depends on its weight, timing, and local temporal interaction context. Ricci curvature has recently been used as an effective geometric tool [1] for graph learning, as it provides an edge-level measure of local structural organization. Curvature-based graph methods can identify bottleneck edges, redundant connections, community-like regions, and structurally important interactions, making them useful for graph representation learning, robustness analysis, and sparsification. However, existing graph Ricci curvature formulations are mainly defined for static graphs and do not explicitly account for edge timestamps, temporal recency, or time-dependent competition among nearby interactions [1, 31]. To this end, we propose TRicci, a temporal Forman–Ricci curvature driven edge sparsification framework, for efficient dynamic graph learning. Building on the classical Forman–Ricci curvature formulation for discrete geometric structures [6] and its graphbased extensions [24], our method, TRicci, assigns each temporal edge an importance score by jointly considering structural support, temporal context, and local interaction competition. TRicci then retains task-relevant curvature-ranked edges selected 1

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Poupak Azad, Cuneyt Gurcan Akcora, and Kiarash Shamsi

through validation to construct a compact, sparse graph for downstream prediction. We evaluate the framework on multiple temporal graph prediction tasks across blockchain transaction networks and temporal benchmark datasets. The results show that the proposed method preserves strong predictive performance while retaining only approximately 20% of the original edges and reducing end-to-end runtime by an average of 55.94%, demonstrating that temporal curvature can serve as an effective edge-selection mechanism for scalable dynamic graph learning. The main contributions of this paper are summarized as follows: • To the best of our knowledge, we introduce the first temporal Forman–Ricci curvature formulation for directed weighted temporal graphs. • We apply curvature-based edge sparsification to discrete-time dynamic graphs by retaining task-relevant curvature-ranked edges to construct compact temporal graph representations. • We evaluate TRicci on three graph-level temporal prediction tasks across 12 datasets, including blockchain transaction networks and TGBL benchmark graphs, and compare it against three state-of-the-art sparsification methods. The results show that high-curvature sparse graphs retain 97.7 ± 2.2% of the full-graph ROC–AUC while using only approximately 20% of the original edges, compared to 89.4 ± 6.3% for the strongest competing method. To support reproducibility, the anonymized implementation and experimental artifacts are publicly available at: https://anonym ous.4open.science/r/TemporalRicci-4FE9/README.md. 2

Related Work

Temporal graph learning methods are commonly categorized into discrete-time dynamic graphs (DTDGs) and continuous-time dynamic graphs (CTDGs) [11]. DTDG methods represent temporal evolution using a sequence of graph snapshots constructed over fixed time intervals, while CTDG methods model interactions as continuous event streams with exact timestamps. Snapshot-based DTDG approaches are widely used in temporal prediction and graph evolution analysis because they provide stable structural representations and scalable processing for large dynamic networks [13, 18]. In contrast, CTDG methods typically focus on eventdriven temporal message passing and continuous interaction modeling [18, 26]. This distinction is important for sparsification because edge-importance scores may be defined either over snapshot-level graph structures or over individual timestamped events. Graph sparsification methods [10] have been widely studied to reduce graph size, improve computational efficiency, and filter noisy or redundant connectivity. Current graph sparsification methods aim to construct a smaller subgraph that approximates the original graph both structurally and spectrally. In graph neural networks, early sparsification strategies often focused on randomly or structurally removing edges during training. DropEdge randomly removes edges from the input graph at each training epoch, acting both as data augmentation and as a message-passing reducer to alleviate overfitting and oversmoothing [17]. DropEdge++ extends this idea with structure-aware samplers, including layer-dependent and feature-dependent edge sampling, to better control oversmoothing in deep GCNs [9]. While these methods are simple and effective, they mainly operate as training-time regularization mechanisms and do not provide a deterministic edge-importance ranking for constructing a reusable sparse graph. Several studies learn or refine graph structures directly from data. For instance, Franceschi et al. formulate graph structure learning as a bilevel optimization problem, jointly learning a discrete edge distribution and the parameters of a graph convolutional network [7]. This allows the model to infer a sparse dependency structure when the observed topology is noisy, incomplete, or unavailable. Pro-GNN further improves robustness by jointly learning a clean graph and a graph neural network model under sparsity, low-rank, and feature-smoothness constraints [12]. PTDNet introduces a parameterized topological denoising network that removes task-irrelevant edges during training while imposing a low-rank constraint on the sparsified graph [15]. These methods demonstrate that edge selection can improve robustness and predictive performance when the input graph contains noisy or unreliable connections. However, because the sparse structure is learned alongside a particular downstream model, the sparsification process is often task-dependent and computationally more expensive than a preprocessing-based edge-ranking strategy [34]. Another group of methods addresses scalability by sampling sparse computation graphs rather than constructing a persistent sparse graph. FastGCN uses importance sampling to reduce the cost of graph convolution [2]. LADIES improves this idea by constructing layer-dependent sampled subgraphs for deep graph convolutional network training [35]. GraphSAINT performs subgraph sampling for scalable inductive learning and normalizes the estimator to support efficient mini-batch training [28]. These

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

3

Fig. 1. Architecture of the proposed Temporal Forman–Ricci curvature-based sparsification framework in three components: (a) the curvature scoring module, (b) the snapshot ranking and binning module, and (c) the feature extraction and prediction module. The left column summarizes the overall workflow. Gray blocks indicate input graph representations and intermediate data structures, purple blocks denote temporal curvature components, yellow blocks represent scoring, ranking, aggregation, and prediction-processing stages, blue blocks indicate feature extraction and learning modules, and green blocks denote the retained sparse edge set and final prediction output. The retained high-curvature bin is shown as the task-specific setting used in this study; for other downstream tasks, a different curvature range may be selected if it better preserves task-relevant information.

approaches are effective for reducing training cost, especially on large static graphs. Nevertheless, their main goal is efficient mini-batch optimization, and the sampled graphs are not necessarily designed to provide a stable sparsified representation that can be reused across multiple downstream temporal prediction tasks. Recent work also studies sparsification through graph lottery tickets and adaptive pruning. Chen et al. jointly prune both graph edges and model weights, showing that sparse subgraphs and sparse subnetworks can preserve the performance of dense graph neural networks [3]. TEDDY reduces the cost of graph lottery ticket discovery by using a one-shot degree-based edge sparsification strategy [20]. Zhang et al. extend this direction by combining multiple sparsifier experts with different pruning criteria and sparsity levels, enabling node-adaptive graph sparsification [30]. These methods provide strong evidence that a large portion of graph edges can be removed without substantial performance loss. However, they are still mainly developed for static graph settings, where edge importance is estimated from a fixed topology rather than from a sequence of timestamped interactions. A smaller body of work addresses sparsification in temporal or spatio-temporal settings. Zheng et al. jointly learn a stochastic edge-sampling network and a temporal-structural convolutional classifier, showing that task-driven sparsification can reduce overfitting while preserving discriminative temporal interaction patterns [33]. Duan et al. introduce an adaptive sparsification strategy for spatial-temporal graph neural networks by pruning spatial adjacency matrices in traffic forecasting models [4]. STGS formulates spatio-temporal graph sparsification as a reinforcement learning problem that removes edges while preserving structural and dynamic properties relevant to downstream forecasting [21]. dyGRASS studies dynamic spectral graph sparsification from a graph maintenance perspective by maintaining spectral sparsifiers under streaming edge insertions and deletions using localized random walks on GPUs [27]. Although these studies extend sparsification beyond static graphs, they are primarily developed for specific settings such as temporal node classification, spatiotemporal forecasting, spatiotemporal prediction, or dynamic spectral maintenance. They do not directly provide a model-independent edge-ranking mechanism for directed weighted temporal graphs that jointly accounts for structural support and temporal interaction context before downstream learning. Nevertheless, sparsifying temporal interaction networks remains a distinct challenge. Existing temporal and spatio-temporal sparsification methods address important settings such as temporal node classification, spatial-temporal forecasting, skeleton-based action recognition, and dynamic spectral maintenance. However, they are not primarily designed to provide an edge-level ranking

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Poupak Azad, Cuneyt Gurcan Akcora, and Kiarash Shamsi

for directed weighted temporal graphs across different downstream prediction tasks. This leaves a gap for a model-independent temporal sparsification strategy that can score edges before downstream learning while accounting for both structural support and temporal interaction contexts. 3

Methodology

This section presents the proposed Temporal Forman–Ricci curvature framework for edge sparsification in directed weighted temporal graphs. Our goal is to assign each temporal edge an importance score that reflects its structural support, temporal relevance, and local interaction competition, and then use this score to construct a compact, sparse graph for downstream dynamic graph learning. 3.1

Problem Setup

Let G = (V, E) denote a directed weighted temporal graph, where each edge 𝑒 ∈ E is represented as a tuple 𝑒 = (𝑢, 𝑣, 𝑤𝑒 , 𝑡𝑒 ), with source node 𝑢 ∈ V, destination node 𝑣 ∈ V, positive edge weight 𝑤𝑒 > 0, and timestamp 𝑡𝑒 . The edge weight may represent the strength of interaction, such as transaction value, review intensity, or interaction frequency. e| ≪ |E |, such e = (V, E), e where E e ⊂ E and | E Given a temporal graph G, our objective is to construct a sparse temporal graph G e preserves the most informative temporal interactions for downstream prediction. We sparsify at the edge level: edges are that G ranked by their Temporal Forman–Ricci curvature scores, and the retained curvature range is selected through validation for the downstream task. Figure 1 illustrates the proposed framework. We organize the framework into three components: a curvature-scoring module, a ranking-and-selection module, and a downstream feature-extraction-and-prediction module. We first partition the temporal edge stream into discrete-time snapshots, compute curvature-based edge importance scores within each snapshot, select task-relevant curvature-ranked edges to construct sparse temporal graphs, and then apply the same graph-level feature extraction and sequential prediction pipeline used in the downstream evaluation. 3.2

Temporal Forman–Ricci Curvature

Classical Forman–Ricci curvature provides an edge-centered geometric measure of local graph structure. For a static undirected weighted graph, the graph-based Forman–Ricci curvature of an edge 𝑒 = (𝑢, 𝑣) can be written [6, 24] as: ! ∑︁ 𝑤 𝑣 𝑤𝑢 𝑤 𝑣 ∑︁ 𝑤𝑢 𝐹 (𝑒) = 𝑤𝑒 + − − , √ √ 𝑤𝑒 𝑤𝑒 𝑒 ∼𝑒 𝑤𝑒 𝑤𝑒𝑢 𝑒 ∼𝑒 𝑤𝑒 𝑤𝑒 𝑣 𝑢

𝑣

where 𝑤𝑒 is the edge weight, 𝑤𝑢 and 𝑤 𝑣 are the weights of the endpoint nodes, and 𝑒𝑢 and 𝑒 𝑣 denote neighboring edges incident to 𝑢 and 𝑣, respectively. This formulation follows a support-minus-penalty structure: the endpoint terms increase the curvature of 𝑒, while neighboring edges around the endpoints reduce it. The classical graph formulation is static and does not directly model edge direction, timestamps, or time-dependent local interaction patterns. We therefore introduce a temporal directed adaptation. Our Temporal Forman–Ricci curvature score keeps the support-minus-penalty principle, but modifies the two components as follows. First, we replace static node weights with log-smoothed endpoint activity computed inside each temporal snapshot. Second, we define directed outgoing neighborhoods around the endpoints of each edge. Third, and most importantly, we weight neighboring-edge penalties by temporal proximity, since temporally adjacent interactions often provide overlapping local structural information and should therefore contribute more strongly to the competition penalty. This follows the Forman-Ricci support-minus-penalty principle, where neighboring interactions around an edge reduce its curvature when they represent local competition. We first define the endpoint-support component of the proposed temporal curvature score. For a node 𝑥 ∈ V, let E (𝑥) denote the set of edges incident to 𝑥 within the current temporal snapshot. We define the weighted activity of 𝑥 as ∑︁ strength(𝑥) = 𝑤𝑒 ′ . 𝑒 ′ ∈ E (𝑥 )

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

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Since temporal interaction networks often contain heavy-tailed edge weights and highly active nodes, we use a log-smoothed node activity term, 𝑠 (𝑥) = log (1 + strength(𝑥)) . This transformation reduces the dominance of highly active nodes while preserving relative activity differences. Using this smoothed activity, we define the endpoint support of edge 𝑒 = (𝑢, 𝑣, 𝑤𝑒 , 𝑡𝑒 ) as   1 1 𝑆𝑒 = 𝑤 𝑒 + . 𝑠 (𝑢) 𝑠 (𝑣) The support term assigns larger values to edges whose endpoints are not dominated by many high-weight incident interactions. In this sense, 𝑆𝑒 measures how distinctive the edge is relative to the local activity around its endpoints. We next define the neighboring-edge penalty. To make this penalty time-aware, we use a temporal decay kernel between edge 𝑒 and a neighboring edge 𝑒 ′ :   |𝑡𝑒 − 𝑡𝑒 ′ | , 𝐾 (𝑒, 𝑒 ′, 𝜏) = exp − 𝜏 where 𝜏 > 0 is a temporal scale parameter. When two edges occur close in time, 𝐾 (𝑒, 𝑒 ′, 𝜏) is close to one; when they are far apart, the influence of 𝑒 ′ on 𝑒 decays toward zero. Thus, the kernel allows nearby interactions to contribute more strongly to the neighboring-edge penalty than temporally distant interactions. Because the graph is directed, we construct the neighboring-edge penalty from outgoing temporal neighborhoods around the two endpoints of the target edge. For edge 𝑒 = (𝑢, 𝑣, 𝑤𝑒 , 𝑡𝑒 ), we define the outgoing neighborhood of the source endpoint 𝑢 as N𝑢 (𝑒) = {𝑒 ′ = (𝑢, 𝑥, 𝑤𝑒 ′ , 𝑡𝑒 ′ ) ∈ E | 𝑥 ≠ 𝑣 }, and the outgoing neighborhood of the destination endpoint 𝑣 as N𝑣 (𝑒) = {𝑒 ′ = (𝑣, 𝑥, 𝑤𝑒 ′ , 𝑡𝑒 ′ ) ∈ E | 𝑥 ≠ 𝑢}. Edges in N𝑢 (𝑒) represent source-side alternative interactions, while edges in N𝑣 (𝑒) capture destination-side continuation interactions. Together, these neighborhoods describe the local temporal context of 𝑒. Temporally close neighboring interactions increase the penalty, indicating that the target edge is less isolated in its local temporal context. We compute the temporal neighboring-edge penalty from endpoint 𝑢 as ∑︁ 𝐾 (𝑒, 𝑒 ′, 𝜏)  𝑤𝑒   , √   |N𝑢 (𝑒)|  𝑤𝑒 𝑤𝑒 ′ 𝑒 ′ ∈ N𝑢 (𝑒 ) 𝐷𝑢 (𝑒) =     0,  We compute the corresponding penalty from endpoint 𝑣 as ∑︁ 𝐾 (𝑒, 𝑒 ′, 𝜏)  𝑤𝑒   , √   |N𝑣 (𝑒)|  𝑤𝑒 𝑤𝑒 ′ 𝑒 ′ ∈ N𝑣 (𝑒 ) 𝐷 𝑣 (𝑒) =     0, 

|N𝑢 (𝑒)| > 0, |N𝑢 (𝑒)| = 0.

|N𝑣 (𝑒)| > 0, |N𝑣 (𝑒)| = 0.

The total temporal competition penalty is then 𝐷𝑢 (𝑒) + 𝐷 𝑣 (𝑒) . 2 Finally, we define the Temporal Forman–Ricci curvature score of edge 𝑒 as TRicci(𝑒) = 𝑆𝑒 − 𝐷𝑒 . Equivalently,   1 1 1 TRicci(𝑒) = 𝑤𝑒 + − [𝐷𝑢 (𝑒) + 𝐷 𝑣 (𝑒)] . 𝑠 (𝑢) 𝑠 (𝑣) 2 𝐷𝑒 =

TRicci is a real-valued curvature score, with TRicci(𝑒) ∈ R. A high TRicci value indicates that an edge has strong endpoint support and a limited temporally proximate neighboring-edge penalty. A low TRicci value indicates that the edge has weak endpoint support or appears in a crowded temporal neighborhood where several nearby interactions provide similar local information. TRicci is intended as a temporal edge-importance functional inspired by the support-minus-competition structure of Forman– Ricci curvature. We do not claim that the proposed score preserves all geometric interpretations associated with classical graph curvature. Instead, the formulation adapts the Forman-style support-minus-penalty principle to rank directed weighted temporal interactions for downstream sparsification and prediction.

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Poupak Azad, Cuneyt Gurcan Akcora, and Kiarash Shamsi

local competing neighborhood of 𝒆𝟗

g

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𝑒3

c 𝑒2 b

𝑒10

𝑒7

𝑒4

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𝑒12 v 𝑒13 𝑒11

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Edge thickness = weight (𝑤𝑒 ) High Weight Low Weight

Edge color = temporal proximity to target edge Close in time Farther in time

Edge role Target / removed edge Other edges

Fig. 2. Single temporal snapshot. Illustration of low-curvature edge removal within a single temporal snapshot. Edge thickness indicates interaction weight, while edge color indicates temporal proximity to the target edge. The red dashed edge is the target edge selected for removal because it has weak curvature relative to its local competing neighborhood.

Figure 2 illustrates how the proposed curvature score identifies a low-curvature edge within a single temporal snapshot (We report curvature scores of all nodes in App. A). The target edge 𝑒 9 connects nodes 𝑢 and 𝑣, but it is surrounded by several local alternatives around the same endpoints. In particular, outgoing edges from 𝑢 and 𝑣, such as 𝑒 7 , 𝑒 8 , 𝑒 12 , and 𝑒 13 , act as direct competing interactions in the local neighborhood. When these competing edges are temporally close to the target edge, they increase the temporal competition penalty. In addition, incident activity around 𝑢 and 𝑣 contributes to endpoint strength, which can reduce the support term of the target edge. As a result, 𝑒 9 is selected for removal during sparsification. 3.3

Snapshot-Level Curvature Computation

Temporal graphs evolve continuously, and computing curvature over the entire history may mix interactions from different temporal contexts. We avoid this issue by representing the temporal edge stream as a sequence of non-overlapping snapshots. Let Δ denote the snapshot length. We define the edge set of snapshot 𝑟 as E𝑟 = {𝑒 ∈ E | 𝑡𝑒 ∈ [𝑇𝑟 ,𝑇𝑟 + Δ)}. Each snapshot defines a local temporal graph G𝑟 = (V𝑟 , E𝑟 ), where V𝑟 contains the nodes incident to edges in E𝑟 . We compute Temporal Forman–Ricci curvature independently within each snapshot. This prevents information leakage across future snapshots and ensures that each edge is scored relative to its local temporal context. 3.4

Task-Guided Curvature-Based Edge Selection

We use TRicci as a model-independent scoring mechanism for temporal edge selection. Algorithm 1 summarizes the proposed task-guided sparsification procedure. For each temporal snapshot, TRicci assigns a curvature score to every edge by combining endpoint support, temporal proximity, and local outgoing interaction competition. The resulting scores define an ordering over temporal interactions within the snapshot. The sparsification policy is separated from the curvature definition. After computing the snapshot-level scores, the downstream task determines which curvature range should be retained. A policy may retain high-curvature edges, low-curvature edges, or an

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

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intermediate curvature band, depending on the validation criterion for task T . This distinction is important because curvature provides an edge ranking, while the optimal retained range can vary across prediction tasks. Let 𝜌 ∈ [0, 1) denote the edge-deletion ratio. For snapshot E𝑟 , let q𝑟 = {𝑞𝑒 : 𝑒 ∈ E𝑟 } denote the TRicci score vector, and let 𝑘𝑟 = ⌈(1 − 𝜌)|E𝑟 |⌉ denote the number of retained edges. We define a task-guided curvature-selection policy Π 𝜌 that retains 𝑘𝑟 e𝑟 = Π 𝜌 (E𝑟 , q𝑟 , T ). edges according to their curvature ranks for task T . The sparse snapshot is then given by E high

Several policies can be obtained from this definition. A high-curvature policy retains the top-ranked edges, Π 𝜌

(E𝑟 , q𝑟 , T ) =

Top𝑘𝑟 (E𝑟 ; q𝑟 ), where Top𝑘𝑟 returns the 𝑘𝑟 edges with the largest TRicci scores. A low-curvature policy retains the bottom-ranked edges, Π low 𝜌 (E𝑟 , q𝑟 , T ) = Bottom𝑘𝑟 (E𝑟 ; q𝑟 ), where Bottom𝑘𝑟 returns the 𝑘𝑟 edges with the smallest TRicci scores. Instead of highor low-selection, an alternative approach could use a band-selection policy that first partitions edges into curvature-ranked bins and then retains the bin or set of bins selected for task T . Since edge selection is performed independently within each snapshot, the retained sparse graph preserves temporal coverage across the full observation period. This avoids selecting edges only from dense or high-activity snapshots and ensures that each time interval contributes to the final sparse temporal representation. Algorithm 1 TRicci: Temporal Forman–Ricci Sparsification Require: Directed weighted temporal graph G = (V, E), snapshot length Δ, temporal scale 𝜏, edge-deletion ratio 𝜌, downstream task T , curvature-selection policy Π 𝜌 e = (V, E) e Ensure: Sparse temporal graph G 1: Split E into non-overlapping E𝑟 using Δ e←∅ 2: Initialize E 3: for each temporal snapshot E𝑟 do 4: Construct local graph G𝑟 = (V𝑟 , E𝑟 ) 5: Compute log-smoothed node strengths 𝑠 (𝑥) for all 𝑥 ∈ V𝑟 6: for each edge 𝑒 = (𝑢, 𝑣, 𝑤𝑒 , 𝑡𝑒 ) ∈ E𝑟 do 7: Compute endpoint support 𝑆𝑒 8: Identify outgoing competing neighborhoods N𝑢 (𝑒) and N𝑣 (𝑒) 9: Compute temporal penalties 𝐷𝑢 (𝑒) and 𝐷 𝑣 (𝑒) using 𝐾 (𝑒, 𝑒 ′, 𝜏) 𝑣 (𝑒 ) 10: Compute the TRicci score 𝑞𝑒 = 𝑆𝑒 − 𝐷𝑢 (𝑒 )+𝐷 2 11: end for 12: Let q𝑟 = 𝑞𝑒 : 𝑒 ∈ E𝑟 e𝑟 = Π 𝜌 (E𝑟 , q𝑟 , T ) 13: Select task-relevant curvature-ranked edges E e 14: Retain E𝑟 as the sparse snapshot for task T e←E e∪E e𝑟 15: E 16: end for e = (V, E) e 17: return G Analytical properties. The proposed curvature score has two elementary monotonicity properties that formalize its intended interpretation. The first property shows that the support term decreases when either endpoint becomes more active. The second property shows that the temporal penalty is strongest for nearby interactions and weakens as neighboring edges move farther away in time. Together, these properties justify the use of TRicci as a support-minus-competition score for directed weighted temporal graphs. Proposition 3.1 (endpoint activity monotonicity). Holding 𝑤𝑒 and all penalty terms fixed, the support term decreases strictly when the endpoint activity at 𝑢 or 𝑣 increases. Proof. Let 𝑎𝑢 = strength(𝑢) and 𝑎 𝑣 = strength(𝑣), with 𝑎𝑢 , 𝑎 𝑣 > 0. Since 𝑠 (𝑥) = log(1 + strength(𝑥)), the support term can be written as 𝑆𝑒 = 𝑤𝑒 (log(1+𝑎𝑢 ) −1 +log(1+𝑎 𝑣 ) −1 ). Differentiating with respect to 𝑎𝑢 gives 𝜕𝑆𝑒 /𝜕𝑎𝑢 = −𝑤𝑒 ((1+𝑎𝑢 )(log(1+𝑎𝑢 )) 2 ) −1 < 0. The same calculation gives 𝜕𝑆𝑒 /𝜕𝑎 𝑣 = −𝑤𝑒 ((1 + 𝑎 𝑣 )(log(1 + 𝑎 𝑣 )) 2 ) −1 < 0. Hence any increase in endpoint activity lowers the support assigned to 𝑒. This proves that edges become less distinctive when their endpoints are crowded by other weighted interactions.

□

Proposition 3.2 (temporal decay and limiting regimes). Fix the snapshot, edge weights, and directed neighborhood sets. Then each individual penalty contribution decreases strictly with temporal distance. Moreover, as 𝜏 → 0+ , only exactly synchronous

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Poupak Azad, Cuneyt Gurcan Akcora, and Kiarash Shamsi

neighbors contribute to the penalty. If no timestamp ties occur inside the snapshot, then TRicci(𝑒) → 𝑆𝑒 . As 𝜏 → ∞, the score converges to a time-agnostic directed support-minus-average-competition score. Proof. Consider a nonempty source-side neighborhood 𝑁𝑢 (𝑒) and a neighboring edge 𝑒 ′ ∈ 𝑁𝑢 (𝑒). Let 𝑑 = |𝑡𝑒 −𝑡𝑒 ′ |. The penalty √︁ √︁ contribution of 𝑒 ′ to 𝐷𝑢 (𝑒) is |𝑁𝑢 (𝑒)| −1 𝑤𝑒 /𝑤𝑒 ′ exp(−𝑑/𝜏). Its derivative with respect to 𝑑 is −|𝑁𝑢 (𝑒)| −1𝜏 −1 𝑤𝑒 /𝑤𝑒 ′ exp(−𝑑/𝜏) < 0, since 𝑤𝑒 > 0, 𝑤𝑒 ′ > 0, and 𝜏 > 0. The same argument applies to every destination-side contribution in 𝑁 𝑣 (𝑒). Thus, each individual neighboring-edge penalty decreases strictly as temporal distance increases. For the first limit, exp(−𝑑/𝜏) → 1𝑑=0 as 𝜏 → 0+ . Therefore only neighbors with 𝑡𝑒 = 𝑡𝑒 ′ contribute. If no timestamp ties occur inside the snapshot, then every neighboring edge has 𝑑 > 0, so 𝐷𝑢 (𝑒) → 0 and 𝐷 𝑣 (𝑒) → 0, which gives TRicci(𝑒) → 𝑆𝑒 . For the second limit, exp(−𝑑/𝜏) → 1 as 𝜏 → ∞ for every fixed 𝑑. Hence, temporal distances no longer affect the penalty, and TRicci(𝑒) converges to the directed weighted score √︁ Í obtained by replacing every temporal kernel factor with one. As 𝜏 → ∞, TRicci(𝑒) → 𝑆𝑒 − 12 (|𝑁𝑢 (𝑒)| −1 𝑒 ′ ∈𝑁𝑢 (𝑒 ) 𝑤𝑒 /𝑤𝑒 ′ + √︁ Í |𝑁 𝑣 (𝑒)| −1 𝑒 ′ ∈𝑁 𝑣 (𝑒 ) 𝑤𝑒 /𝑤𝑒 ′ ). □ Computational complexity. Let 𝑚𝑟 = |E𝑟 | denote the number of edges in snapshot 𝑟 . For each snapshot, computing node activities and constructing outgoing adjacency lists requires 𝑂 (𝑚𝑟 ) time. Ranking edges by their curvature scores requires 𝑂 (𝑚𝑟 log 𝑚𝑟 ) time. The main additional cost comes from computing the neighboring-edge penalty, where each edge is compared with outgoing neighboring edges around its endpoints. The per-snapshot complexity is therefore 𝑂 (𝑚𝑟 log 𝑚𝑟 + 𝐶𝑟 ), where 𝐶𝑟 is the total cost of scanning local outgoing neighborhoods in snapshot 𝑟 . Since curvature is computed independently across snapshots, the total runtime is the sum of this cost over all snapshots. After the scores are computed, the selection step is lightweight because it applies the policy Π 𝜌 to the ranked edge list. 4

Evaluation

We evaluate TRicci through experiments designed to assess predictive preservation, curvature-selection behavior, runtime reduction, and sensitivity to temporal parameters. We first describe the datasets, tasks, baselines, and evaluation protocol. We then analyze curvature-ranked edge bins to determine which curvature ranges preserve the strongest downstream signal. Based on this analysis, we select the default sparsification policy used in the baseline comparison and runtime experiments. 4.1

Experimental Setup

Prediction tasks. We evaluate sparsification quality using three graph-level temporal prediction tasks. These tasks are designed to measure whether a sparse graph preserves information about future changes in network activity, influential nodes, and participation breadth. Task 1: Network Activity Growth Prediction. This task predicts whether the number of edges increases in the next temporal snapshot. Let E𝑡 and E𝑡 +1 denote the edge sets in the current and next snapshots, respectively. The binary label is defined as 𝑦𝑡 =

   1,    0, 

if |E𝑡 +1 | > |E𝑡 |, otherwise.

This task evaluates whether the sparse graph preserves signals associated with future expansion in network activity. Task 2: Network Participation Expansion Prediction. This task predicts whether the number of active nodes increases in the next temporal snapshot. Let V𝑡𝑎𝑐𝑡 and V𝑡𝑎𝑐𝑡 +1 denote the sets of nodes that participate in at least one edge in the current and next snapshots, respectively. The binary label is defined as    1,  𝑦𝑡 =   0, 

𝑎𝑐𝑡 if |V𝑡𝑎𝑐𝑡 +1 | > |V𝑡 |,

otherwise.

This task evaluates whether the sparsified graph preserves information about future changes in the breadth of network participation. Task 3: Influential Node Turnover Prediction. This task predicts whether the set of influential nodes changes substantially in the next temporal snapshot. For each snapshot, nodes are ranked by degree, and the top-𝑘 nodes are treated as influential nodes. Let

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

9

Table 1. Statistics of the temporal graph datasets.

Dataset

Nodes

Edges

Start

End

Days

ADX 14,567 BAG 11,860 BEPRO 26,521 DERC 24,277 DINO 15,837 ETH2x-FLI 11,008 EVERMOON 7,552 GLM 53,385 HOICHI 5,075 tgbl-review 352,637 tgbl-coin 638,486 tgbl-comment 994,790

123,755 122,634 120,261 111,205 94,140 199,088 79,868 234,912 77,361 4,873,540 22,809,486 44,314,507

2020-08-03 2023-01-10 2020-09-28 2021-08-02 2022-11-11 2021-03-14 2023-05-25 2020-11-19 2022-08-25 1999-06-13 2022-04-01 2005-12-12

2023-11-04 2023-11-04 2023-11-04 2023-11-04 2023-11-04 2023-11-04 2023-11-04 2023-11-04 2023-11-04 2018-10-04 2022-11-01 2010-12-31

1,189 299 1,133 825 359 966 164 1,081 437 7,054 214 1,845

T𝑡 and T𝑡 +1 denote the top-𝑘 node sets in the current and next snapshots, respectively. The turnover ratio is defined as 𝑟𝑡 =

|T𝑡 +1 \ T𝑡 | . 𝑘

The binary label is 𝑦𝑡 =

   1,    0, 

if 𝑟𝑡 > 𝜃, otherwise.

Datasets. We evaluate TRicci on 12 directed weighted temporal graphs: nine blockchain token-transfer networks from GraphPulse [22] and three TGBL temporal graph benchmarks [11]. Table 1 summarizes the dataset statistics. In the blockchain networks, nodes are wallet addresses and directed temporal edges are token transfers. The TGBL datasets are tgbl-coin, tgbl-review, and tgbl-comment, covering cryptocurrency transfers, product reviews, and comment interactions. These datasets test whether TRicci preserves graph-level predictive signals across temporal graphs with different sizes, densities, durations, and interaction semantics. Baselines. We compare TRicci with three recent and prominent graph sparsification baselines: MoG [30], TEDDY [20], and SEM [31]. MoG uses a mixture-of-experts strategy to select pruning criteria and sparsity levels for local graph regions. TEDDY performs one-shot edge sparsification through degree-based structural discrimination. SEM uses Ricci-curvature-based sparsification to preserve informative topological structure for continual graph representation learning. These methods provide strong topology-based comparison points, but their edge-scoring rules do not explicitly use timestamps or temporal interaction context. For a controlled comparison, we apply all baselines under the same snapshot-level protocol and edge-retention budget. TRicci differs by defining edge importance directly on directed weighted temporal graphs through endpoint support, temporal proximity, and local outgoing interaction competition. Extending the baselines with native timestamp-dependent objectives or temporal neighborhood definitions would require separate temporal sparsification models, which is beyond the scope of this work. Evaluation setting. The downstream learning pipeline follows the temporal graph property prediction setting introduced by GraphPulse [22], since this setting directly matches our objective of evaluating whether sparse temporal graphs preserve graph-level predictive signals over time. For each sparsified 7-day weekly snapshot, graph-level task features are extracted from the corresponding graph representation and passed to a sequential (i.e., LSTM and GRU-based) prediction framework [22]. To preserve temporal ordering and avoid information leakage, the data were divided using a chronological split, with 70% used for training, 15% for validation, and 15% for testing. All methods are evaluated under the same edge-retention setting, where approximately 𝜌 = 80% of the original edges are deleted per snapshot. We set 𝑘 to the top 10% of active nodes and use 𝜃 = 0.30 to identify meaningful changes in influential-node composition. The same feature extraction and prediction pipeline is applied to the full graph, the proposed sparse graph, and all baseline sparsification methods to ensure a fair comparison. Our code is available at https://anonymous.4open.science/r/TemporalRicci-4FE9/README.md. Evaluation metrics. We evaluate predictive performance using ROC–AUC for each graph-level temporal prediction task. To compare sparse and full graphs across datasets, we also report the full-graph-normalized ROC–AUC, defined as 𝑅AUC = e AUC( G)/AUC(G). Values close to 1 indicate that the sparse graph preserves the predictive performance of the full graph. e|/|E |) and 100| E e|/|E |, Sparsification is measured using edge removal and edge retention percentages, defined as 100(1 − | E

10

Poupak Azad, Cuneyt Gurcan Akcora, and Kiarash Shamsi Table 2. Average ROC–AUC preservation (𝑅AUC ) relative to the full graph on all datasets and tasks. Method

Avg. Preservation Ratio (↑)

Std. Preservation

0.977 0.894 0.886 0.884

0.022 0.063 0.077 0.080

TRicci (ours) TEDDY MoG SEM

respectively. The edge-removal ratio 𝜌 ∈ [0, 1) controls the fraction of deleted edges, so each sparse graph removes approximately 100𝜌% of edges and retains approximately 100(1 − 𝜌)%. Runtime efficiency is measured as end-to-end reduction relative to the full-graph pipeline, 100(1 − 𝑇sparse /𝑇full ), where 𝑇 is end-to-end and includes edge selection, feature extraction, and downstream prediction. Hyperparams. The default temporal decay parameter is set to 𝜏 = 1 day. We train the prediction model using the Adam optimizer with a learning rate of 10−4 and binary cross-entropy loss. The output layer uses a sigmoid activation for binary prediction. All experiments were run for 100 epochs without learning-rate decay or early stopping. Computational Environment. Runtime measurements were conducted under a fixed single-machine environment to ensure a controlled comparison across methods. The experiments were executed on an Intel(R) Core(TM) Ultra 9 285H CPU with 16 physical cores, 31.4 GB RAM, and an NVIDIA RTX PRO 1000 GPU with 8 GB memory. 4.2

Experimental Results

We report the blockchain transaction network results in Figure 3 and the TGBL benchmark results in Figure 4. TRicci uses the high

redΠ 𝜌

curvature-selection strategy (the rationale behind this strategy, along with the broader analysis of curvature-ranked

deletion policies, is examined shortly in Section 4.2.1). Figure 3 and Figure 4 show that the proposed method consistently preserves strong predictive performance across the evaluated temporal graph prediction tasks under substantial edge sparsification. Across all reported dataset-task combinations, TRicci achieves the best ROC–AUC result in 27 out of 30 cases. On the TGBL benchmarks, TRicci achieves the strongest performance in all 6 evaluated cases. The improvements are especially consistent on the blockchain transaction networks, where interaction patterns evolve rapidly and many temporal edges are structurally or temporally redundant. Task 3 evaluates influential-node turnover prediction and is particularly important for newly developing transaction networks, as it predicts the future prospects of the underlying asset. In more established or stable networks however, the influential-node set changes much less across snapshots, producing highly imbalanced or nearly degenerate labels. This is why Task 3 results are not reported for the TGBL datasets and for several blockchain datasets where the influential-node composition remains relatively stable throughout the observation period. We aggregate the figure-level results in Table 2, which further confirms that the proposed method achieves the strongest predictive preservation relative to the original full graphs. Across all valid dataset-task pairs, TRicci preserves 97.7% of the full-graph ROC–AUC on average, outperforming TEDDY, MoG, and SEM under the same sparsification budget. These results show that the proposed Temporal Forman–Ricci curvature ranking retains substantially more predictive temporal-structural information after removing approximately 80% of the edges. 4.2.1 Curvature Selection Policy. We evaluate whether different curvature-ranked edge groups preserve different levels of predictive information. To this end, we analyze the impact of the edge-removal ratio 𝜌 and the edge-selection strategy 𝜋 𝜌 . Figure 5 shows the sensitivity of the proposed method to different edge-removal ratios on Task 1, averaged over datasets. We see a similar trend in other tasks. The ROC–AUC remains relatively stable from 0% to 80% removal, decreasing only slightly from 0.789 to 0.768 even when 80% of the edges are removed. In contrast, performance drops sharply beyond 80% removal. Based on this trend, we use 80% edge removal in the main experiments (𝜌 = 0.8) because it provides a strong balance between graph compression and predictive preservation. For each temporal snapshot, we partition edges into five equal-frequency bins according to their TRicci scores, where Bin 1 contains the lowest-curvature edges, and Bin 5 contains the highest-curvature edges.

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

Full graph

TRicci (Ours)

MoG

11

TEDDY

SEM

ROC-AUC

0.8 0.6 0.4 0.2 0.0

ADX

BAG

BEPRO

DERC

DINO

ETH2x

EVER

GLM

HOICHI

EVER

GLM

HOICHI

ROC-AUC

(a) Task 1: network activity growth prediction

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

ADX

BAG

BEPRO

DERC

DINO

ETH2x

ROC-AUC

(b) Task 2: network participation expansion prediction

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

ADX

BEPRO

DERC

ETH2x

GLM

HOICHI

(c) Task 3: influential node turnover prediction Fig. 3. ROC–AUC comparison on token transaction networks across the three evaluated graph-level temporal prediction tasks. Results for Task 3 are reported only for datasets with non-degenerate labels.

Table 3 reports the ROC–AUC results across datasets and prediction tasks for each curvature-ranked bin. Although the bestperforming curvature range is not identical for every dataset-task pair, the highest-curvature group provides the strongest and most consistent performance overall. In particular, Bin 5 achieves the best ROC–AUC result in the majority of evaluated settings.

12

Poupak Azad, Cuneyt Gurcan Akcora, and Kiarash Shamsi

ROC-AUC

0.8

Full graph

TRicci (Ours)

MoG

TEDDY

SEM

0.6 0.4 0.2 0.0

TGBL-COIN

TGBL-COMMENT

TGBL_REVIEW

(a) Task 1: network activity growth prediction

0.8 ROC-AUC

0.6 0.4 0.2 0.0

TGBL-COIN

TGBL-COMMENT

TGBL_REVIEW

(b) Task 2: network participation expansion prediction Fig. 4. ROC–AUC comparison on TGBL benchmark datasets across two graph-level temporal prediction tasks.

These results indicate that high-curvature temporal interactions contain a large fraction of the predictive information preserved in the full graph. Specifically, the observed behavior follows a Pareto-style concentration pattern: approximately 20% of the temporal edges, selected from the highest-curvature range, retain most of the downstream performance obtained from the complete graph. This finding is consistent with prior evidence from graph sparsification and graph lottery-ticket studies, where a relatively small subset of edges was found to preserve most of the predictive signal of the full graph [23, 3, 20, 30]. In our setting, this concentration emerges from the curvature ranking itself, where high-curvature edges are less redundant in their local temporal neighborhoods high

and therefore carry a larger share of task-relevant information. Based on these validation results, we use Π 𝜌

as the default

retention policy under the 80% edge-deletion setting. 4.2.2 Sensitivity Analysis. To evaluate the robustness of the proposed framework with respect to temporal influence modeling, we conduct a sensitivity analysis on the temporal decay parameter 𝜏, which controls how rapidly the influence between temporally neighboring edges decays during curvature computation. Smaller 𝜏 values restrict the interaction range to nearly simultaneous transactions, while larger 𝜏 values allow temporally distant interactions to contribute more strongly to the curvature calculation. We analyze 𝜏 ∈ {0.25, 0.5, 1, 2, 4, 7, 14, 30} in Figure 6. The results indicate that TRicci achieves its strongest predictive performance around 𝜏 = 1 across most tasks. When 𝜏 becomes excessively small, the curvature computation becomes highly localized and sensitive to short-term fluctuations, leading to reduced predictive stability. In contrast, large 𝜏 values weaken the temporal discrimination capability of TRicci by allowing a broad range of temporally distant interactions to contribute similarly during local neighboring-edge penalty modeling. The performance degradation remains relatively moderate over a broad range of 𝜏 values, indicating that the proposed framework is reasonably robust to the choice of temporal decay scale.

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

13

0.85

Mean ROC-AUC

0.80 0.75 0.70 0.65 0.60 0.55

Mean ROC-AUC Std. across datasets

0

20

40

60

80

Removed Edges (%)

90

Fig. 5. Sensitivity of the proposed framework to the edge-removal ratio on Task 1.

= 1 day

Average ROC-AUC

0.74

Task 1: Network Activity Growth Task 2: Participation Expansion Task 3: Influential Node Turnover

0.72 0.70 0.68 0.66 0.64

0.25

0.5

1

2

4

7

Temporal decay parameter (days)

14

30

Fig. 6. Sensitivity of the proposed framework to the temporal decay parameter 𝜏.

Runtime analysis. We evaluate end-to-end runtime saving relative to the full-graph pipeline, including sparsification, feature processing, and prediction. Table 4 reports the saving percentage for each method and dataset, computed as 100(1 − 𝑇sparse /𝑇full ). TRicci reduces runtime on every dataset and achieves a 55.94% average saving across all datasets. MoG, SEM, and TEDDY achieve average savings of 50.17%, 52.17%, and 57.24%, respectively. Although TEDDY has a slightly higher average runtime saving, TRicci provides nearly the same efficiency gain while preserving substantially stronger predictive performance, as shown in the AUC results of Table 2.

14

Poupak Azad, Cuneyt Gurcan Akcora, and Kiarash Shamsi

Table 3. ROC–AUC results of the proposed method across curvature-ranked bins. The Full column reports the original graph performance, while Bin1–Bin5 represent increasingly higher-curvature edge groups. The best-performing bin for each dataset/task is highlighted. Dataset / Task

Full Bin1 Bin2 Bin3 Bin4 Bin5

ADX / task1 ADX / task2 ADX / task3

0.76 0.80 0.71

0.56 0.57 0.44

0.66 0.50 0.56

0.70 0.65 0.49

0.73 0.75 0.63

0.75 0.77 0.68

BAG / task1 BAG / task2

0.78 0.82

0.44 0.58

0.34 0.37

0.63 0.76

0.75 0.81

0.73 0.80

BEPRO / task1 BEPRO / task2 BEPRO / task3

0.82 0.68 0.75

0.67 0.55 0.52

0.69 0.59 0.56

0.72 0.65 0.64

0.74 0.67 0.68

0.80 0.65 0.72

DERC / task1 DERC / task2 DERC / task3

0.75 0.77 0.84

0.56 0.58 0.60

0.58 0.55 0.59

0.60 0.66 0.77

0.68 0.77 0.69

0.75 0.77 0.83

DINO / task1 DINO / task2

0.70 0.81

0.44 0.50

0.56 0.36

0.50 0.69

0.72 0.67

0.69 0.79

ETH2X-FLI / task1 ETH2X-FLI / task2 ETH2X-FLI / task3

0.71 0.70 0.75

0.37 0.51 0.67

0.49 0.54 0.62

0.39 0.39 0.69

0.62 0.65 0.62

0.68 0.72 0.74

EVERMOON / task1 0.68 EVERMOON / task2 0.65

0.36 0.35

0.55 0.66

0.26 0.44

0.64 0.51

0.67 0.62

GLM / task1 GLM / task2 GLM / task3

0.84 0.78 0.75

0.60 0.56 0.67

0.65 0.69 0.63

0.71 0.70 0.54

0.77 0.70 0.66

0.80 0.76 0.69

HOICHI / task1 HOICHI / task2 HOICHI / task3

0.86 0.70 0.79

0.65 0.56 0.42

0.61 0.45 0.67

0.62 0.63 0.44

0.71 0.69 0.76

0.83 0.68 0.76

TGBL-CMT / task1 TGBL-CMT / task2 TGBL-COIN / task1 TGBL-COIN / task2 TGBL-REV / task1 TGBL-REV / task2

0.78 0.71 0.76 0.74 0.69 0.71

0.50 0.49 0.37 0.40 0.49 0.51

0.60 0.47 0.68 0.62 0.52 0.63

0.49 0.49 0.68 0.58 0.48 0.53

0.69 0.51 0.71 0.71 0.62 0.70

0.75 0.69 0.72 0.73 0.68 0.69

Table 4. Runtime saving percentage across datasets. For each dataset and method, Save% is computed as 100(1 − 𝑇sparse /𝑇full ), where 𝑇sparse = 𝑇spars. + 𝑇proc. + 𝑇pred. includes sparsification, feature processing, and prediction runtime, and 𝑇full is the runtime of the original full-graph pipeline. The Avg. column reports the arithmetic mean across datasets.

Method

ADX

BAG

BEPRO

DERC

DINO

ETH2x

EVER

GLM

HOICHI

tgbl-cmt

tgbl-coin

tgbl-rev

Avg. Saved (%)

TRicci MoG SEM TEDDY

47.12 33.25 2.94 38.92

48.60 45.63 49.23 56.88

50.73 47.62 47.88 52.33

57.22 45.50 59.14 52.71

51.70 47.24 58.71 57.96

56.91 52.10 60.56 58.84

49.06 51.50 52.16 52.34

56.28 54.20 56.77 61.77

48.31 40.12 56.90 48.90

68.74 67.40 66.82 69.89

59.33 41.70 42.39 63.42

77.04 75.78 72.58 76.91

55.94 50.17 52.17 57.24

5

Conclusion

We have introduced TRicci, a graph-curvature-based edge sparsification framework for dynamic graph learning. The method extends Forman–Ricci curvature to directed weighted temporal graphs by combining endpoint support, temporal proximity, and local outgoing interaction competition into a single edge-level score. By computing curvature within temporal snapshots, TRicci ranks edges according to their local temporal-structural importance and constructs sparse graph representations that preserve coverage across the full observation period. The experiments show that high-curvature edge retention is effective across blockchain transaction networks and TGBL benchmark datasets. TRicci removes approximately 80% of the edges while preserving 97.7% of the full-graph ROC–AUC on average. It achieves the best ROC–AUC in 27 out of 30 reported dataset-task settings and in all 6 TGBL settings, outperforming SOTA competitors under the same sparsification budget. TRicci reduces end-to-end runtime by 55.94% on average.

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

15

These results indicate that temporal curvature is an effective criterion for selecting informative interactions in large dynamic graphs. Future work can study adaptive temporal windows, task-specific curvature thresholds, and extensions to continuous-time temporal graph learning. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

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Poupak Azad, Cuneyt Gurcan Akcora, and Kiarash Shamsi

Table 5. Temporal Forman–Ricci curvature (TRicci) and standard weighted Forman–Ricci curvature (FRC) scores for the edges shown in Figure 2. Edge Date

𝑤𝑒

Jan. 1 Jan. 3 Jan. 4 Jan. 5 Jan. 6 Jan. 7 Jan. 8 Jan. 9

0.144 0.265 0.229 0.225 0.238 0.292 0.281 0.252

𝑒6 𝑒7 𝑒8 𝑒9 𝑒 10 𝑒 11 𝑒 12 𝑒 13

[31] [32] [33]

[34] [35]

A

FRC TRicci -1.004 -2.200 -1.839 -5.331 -3.610 -4.373 -3.894 -3.565

1.276 0.914 0.714 0.496 0.828 0.919 1.280 1.252

Xikun Zhang, Dongjin Song, and Dacheng Tao. 2023. Ricci curvature-based graph sparsification for continual graph representation learning. IEEE Transactions on Neural Networks and Learning Systems, 35, 12, 17398–17410. Lin Zhao, Sourav Sen Gupta, Arijit Khan, and Robby Luo. 2021. Temporal analysis of the entire ethereum blockchain network. In Proceedings of the Web Conference 2021 (WWW ’21). Association for Computing Machinery, Ljubljana, Slovenia, 2258–2269. isbn: 9781450383127. doi:10.1145/3442381.3449916. Cheng Zheng, Bo Zong, Wei Cheng, Dongjin Song, Jingchao Ni, Wenchao Yu, Haifeng Chen, and Wei Wang. 2020. Node classification in temporal graphs through stochastic sparsification and temporal structural convolution. In Joint European Conference on Machine Learning and Knowledge Discovery in Databases. Springer, 330–346. Yanqiao Zhu, Weizhi Xu, Jinghao Zhang, Yuanqi Du, Jieyu Zhang, Qiang Liu, Carl Yang, and Shu Wu. 2021. A survey on graph structure learning: progress and opportunities. arXiv preprint arXiv:2103.03036. Difan Zou, Ziniu Hu, Yewen Wang, Song Jiang, Yizhou Sun, and Quanquan Gu. 2019. Layer-dependent importance sampling for training deep and large graph convolutional networks. Advances in neural information processing systems, 32.

Analysis of Curvature Scores in a Local Neighborhood

To examine the behavior of curvature-based edge scoring in a local temporal neighborhood, Figure 2 presents a directed transaction subgraph with assigned edge weights and relative temporal positions. The highlighted edge 𝑒 9 represents the target interaction selected for curvature analysis. In the figure, edge color intensity indicates temporal proximity relative to the target edge: darker edges correspond to interactions occurring closer in time to 𝑒 9 , while lighter edges represent temporally more distant interactions. Therefore, the temporal encoding is defined relative to the inspected edge rather than as a globally ordered timestamp axis. Table 5 reports both the proposed Temporal Forman–Ricci curvature scores and the standard weighted Forman–Ricci curvature scores for the edges shown in Figure 2. In this example, the target edge 𝑒 9 receives the lowest score under both formulations, indicating that it is weakly supported relative to the surrounding interactions and is therefore a natural candidate for removal. However, this agreement should not be interpreted as a general property of the two methods. Standard weighted FRC evaluates only weighted local topology and does not explicitly model temporal proximity or directed temporal competition. In contrast, the proposed TRicci formulation incorporates relative temporal distance and outgoing neighboring-edge competition into the edge scoring process. Consequently, the two formulations may rank edges differently in more complex temporal settings where strong static connectivity does not necessarily imply strong temporal relevance.

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