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

Cuneyt Gurcan Akcora; Kiarash Shamsi; Poupak Azad

REVIEW 4 major objections 6 minor 35 references

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

T0 review · 4 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A temporal curvature score ranks which edges in a time-stamped network matter, allowing about 80% of edges to be deleted while retaining 97.7% of full-graph ROC-AUC on average.

desk verdict A plausible temporal curvature sparsifier with a broad evaluation, but the missing random-deletion baseline and an internal formula inconsistency keep the central claim from being fully established. read the letter →

arxiv 2608.07158 v1 pith:LSRFZALE submitted 2026-08-07 cs.LG

Poupak Azad , Cuneyt Gurcan Akcora , Kiarash Shamsi This is my paper · ORCID

classification cs.LG

keywords TemporalgraphlearningForman-RiccicurvatureEdgesparsificationDynamicpredictionSnapshot-basedgraphsTransactionnetworksGraph-level

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that an edge's local temporal geometry, measured by a temporal extension of Forman–Ricci curvature, is a reliable signal of which interactions matter for graph-level prediction on dynamic networks. It introduces TRicci, a score that combines how distinctive an edge is relative to its endpoints' activity with a temporally weighted penalty from nearby competing edges, computed inside each snapshot. If the claim holds, large and dense temporal graphs can be aggressively sparsified before any downstream model is trained, retaining roughly a fifth of the edges while keeping almost all predictive accuracy and cutting end-to-end runtime.

What carries the argument

The load-bearing object is the Temporal Forman–Ricci curvature score $TRicci(e)$, a support-minus-competition functional over snapshot-level directed weighted temporal graphs. For an edge $e=(u,v,w_e,t_e)$, the support term is $S_e=w_e(1/s(u)+1/s(v))$ with $s(x)=\log(1+\sum w_{e'})$ the log-smoothed endpoint activity, and the competition term averages penalties from outgoing edges at $u$ and $v$, each weighted by a temporal kernel $K(e,e',\tau)=\exp(-|t_e-t_{e'}|/\tau)$. The kernel makes temporally nearby interactions compete more strongly, so the score ranks edges by how isolated they are in both structure and time; the framework then selects a curvature band through validation and deletes the rest per snapshot.

What would settle it

Run TRicci on a temporal graph whose prediction labels are engineered to depend on rare, low-competition, temporally isolated edges: if the high-curvature retention policy must delete those edges by construction, the sparsified graph's ROC-AUC should fall well below the full graph, and low-curvature retention should instead match it.

Watch

Extended reading notes

Core claim

The paper's central claim is that high temporal curvature marks task-relevant edges: across nine blockchain transaction networks and three TGBL benchmarks, retaining only the highest-curvature approximately 20% of edges preserves 97.7±2.2% of the full-graph ROC-AUC on average, outperforming three sparsification baselines under the same 80% deletion budget, and reduces end-to-end training plus inference time by 55.94% on average. The curvature score is $TRicci(e)=S_e - D_e$, where $S_e$ weights the edge by log-smoothed inverse endpoint activity and $D_e$ averages temporally decayed penalties from outgoing neighboring edges around both endpoints. The paper frames the result as a Pareto-style concentration: most predictive temporal-structural information lives in the high-curvature slice of the edge set.

Load-bearing premise

The framework assumes that the choices made on validation data—a one-day decay scale, an 80% deletion rate, and keeping only the highest-curvature edges—work on the test data and on other prediction tasks, even though the paper notes that other tasks may need a different curvature range.

Editorial extensions

If this is right

A practitioner can delete roughly 80% of temporal edges before feature extraction and prediction with only about 2% average ROC-AUC loss, so dense dynamic graphs can be processed in a fraction of the time.
High-curvature retention is the effective default policy across activity-growth, participation-expansion, and influential-node-turnover tasks; low or intermediate curvature bins consistently lose more predictive signal.
Runtime savings are end-to-end: they include sparsification, feature processing, and prediction, and they hold on every dataset tested, averaging 55.94%.
Because selection is per snapshot, the sparse graph keeps coverage over the whole observation period instead of concentrating edges in dense windows.
The framework is model-independent: it ranks edges before training, so the same sparse representation can feed any downstream graph-level predictor.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the same curvature ranking could be used as a deterministic edge-pruning schedule for continual or streaming learning, since each snapshot is scored independently and the retained set is reusable.
A testable extension is to replace the validation-chosen high-curvature bin with a learned threshold per task, which could close the remaining 2-3% AUC gap while keeping the runtime benefit.
The sensitivity trend suggests that on graphs with very different time scales, the decay parameter should be set relative to snapshot length or interaction frequency; treating one day as universal would likely underperform on hourly or yearly interaction streams.
If labels are driven by rare, temporally isolated events, the high-curvature policy is the wrong end of the ranking; a fair comparison on such tasks would need the band-selection variant the paper describes but does not evaluate.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper is a clean, incremental extension of Forman–Ricci edge scoring to directed weighted temporal graphs, and the empirical study is broad and useful. But the headline result—97.7% ROC-AUC preservation at 80% sparsification—doesn't yet prove that temporal curvature is doing the work, because there is no random-edge-deletion baseline and the three prediction tasks are aggregate count statistics that random sampling may preserve almost as well.

What's new: the TRicci score combines log-smoothed endpoint activity, directed outgoing neighborhoods, and an exponential time-decay penalty. That specific temporal extension isn't in the cited static Forman-Ricci, MoG, TEDDY, SEM, or dyGRASS work. The evaluation is also broad: 12 datasets, three graph-level tasks, and a comparison to three sparsifiers under the same retention budget. The runtime saving is real, and the sensitivity analysis for tau and rho is a nice touch. Code and artifacts are provided.

The soft spots are real but fixable. The missing random baseline is the most important. The tasks predict next-snapshot edge-count growth, active-node growth, and top-k degree turnover. A random 20% edge sample estimates total edge count and active-node coverage with small relative error on large snapshots, and retains high-degree nodes with high probability. So 97.7% preservation might reflect sampling fidelity rather than curvature. Comparing to TEDDY, MoG, and SEM doesn't settle it—none is a random control. Second, the penalty term in Section 3.2 uses sqrt(w_e w_e') in the sum, but Proposition 3.2's proof and the tau->infty limit use sqrt(w_e / w_e'). That's a discrepancy that affects the actual score if implemented as written. Third, Tables 3 and 4 show single runs with no per-dataset error bars; only cross-dataset std is reported. Fourth, tau=1, rho=0.8, and the high-curvature policy are chosen from the validation/sensitivity analysis, and the Figure 1 caption explicitly says a different curvature range might be better for other tasks—so the method's generality is partly self-selected.

Who this is for: practitioners who need a cheap pre-processing step for dense temporal graphs, and researchers working on curvature-based graph measures. The paper deserves a serious referee; the missing random baseline and the formula inconsistency should be addressed, and the claims should be softened accordingly.

My recommendation: engage with it, but require the random-deletion control and consistency between definition and proof before it's citable as evidence for curvature-based edge importance.

Referee Report

4 major / 6 minor

Summary. The paper proposes TRicci, an edge-level sparsification score for directed weighted temporal graphs that combines log-smoothed endpoint activity, a temporal decay kernel, and outgoing-neighborhood competition in the style of Forman-Ricci curvature. Edges are scored independently within each snapshot, ranked, and a task-chosen curvature band (here, the high-curvature band) is retained at a deletion ratio rho; the sparse snapshots are then fed into the GraphPulse-style LSTM/GRU prediction pipeline. Experiments on nine blockchain transaction networks and three TGBL datasets across three graph-level prediction tasks report that retaining about 20% of edges preserves 97.7 +/- 2.2% of full-graph ROC-AUC on average and reduces end-to-end runtime by 55.94%.

Significance. If the reported preservation figures are robust, TRicci would be a useful model-independent preprocessing tool for snapshot-based graph-level prediction, with the practical virtues of being deterministic, per-snapshot local, and accompanied by released code. The paper is also honest that the score is an adaptation rather than a geometric theorem. However, the significance of the central claim---that high temporal curvature is the mechanism behind the preserved signal---is currently limited by the absence of a random-deletion control, single-run results for key comparisons, and validation-based selection of the policy parameters. These issues are fixable and do not, in my view, invalidate the methodology.

major comments (4)

[Section 3.2 and Proposition 3.2]
[Section 4.1, Table 2]
[Tables 2, 3, and 4]
[Section 4.2.1, Figure 1 caption, Figure 5, Figure 6]

minor comments (6)

[Figure 2 caption and Appendix A, Table 5]
[Section 4.1]
[Section 4.2.1, Table 3]
[Section 3.2]
[Section 3.2]
[Table 4]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported preservation ratios are empirical test-split evaluations of a hand-defined score; validation-based selection of hyperparameters and the GraphPulse pipeline citation are standard methodological choices, not circular reductions.

full rationale

The paper does not present a derivation chain in which a target quantity is constructed from its own definition. TRicci is an explicitly defined score (Section 3.2, with equations for S_e, D_u(e), D_v(e), and TRicci(e)=S_e-D_e), and its ranking is then used to select edges. The central preservation claim, 97.7±2.2% of full-graph ROC-AUC at roughly 80% edge removal, is measured on a chronological test split after the retention policy, rho=0.8, and tau=1 day are chosen on validation data (Section 4.2.1, Figure 5, Figure 6, Table 3). Choosing hyperparameters on a validation split and then reporting test performance is standard practice, and the test metric is not algebraically forced by the selected parameter values. The self-citation to GraphPulse [22] supplies the evaluation pipeline and datasets, but GraphPulse is a published, externally available framework, and all methods are compared under the same snapshot-level protocol and edge-retention budget; this is a methodological reuse rather than a load-bearing unverified premise. The paper also explicitly disclaims geometric equivalence of its score ('We do not claim that the proposed score preserves all geometric interpretations associated with classical graph curvature') and acknowledges that the retained curvature band is task-specific (Figure 1 caption), which weakens any appearance of a universal forced conclusion. The absence of a random-edge-deletion control is a legitimate experimental-design concern about whether curvature is the causal mechanism, but it is not a circular reduction of the reported numbers to their inputs. No equation in the paper is equal to another by construction, and no fitted parameter is renamed as a prediction; therefore no circular steps are identified.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. TRicci is a scalar scoring function, not a new particle, force, or conserved quantity. The score's behavior depends on several free parameters chosen by validation, and the empirical claims rest on the domain assumptions listed above.

free parameters (4)

tau (temporal decay scale) = 1 day
Appears in K(e,e',tau)=exp(-|t_e-t_e'|/tau); chosen via validation sensitivity (Figure 6), not derived.
rho (edge-deletion ratio) = 0.8
Selected from Figure 5 to maximize compression while maintaining AUC; used in all main experiments.
Delta (snapshot length) = 7 days
Weekly snapshots used for all datasets; not tuned, but determines the local temporal context for curvature.
k (top influential nodes) and theta (turnover threshold) = top 10% of active nodes; theta=0.30
Task-3 parameters chosen by hand; do not affect sparsification but affect label definition and which datasets are reported.

assumptions (4)

domain assumption The support-minus-penalty principle of Forman-Ricci curvature is a valid proxy for task-relevant edge importance in temporal graphs.
Section 3.2: the score is 'inspired by' this principle; no theorem connects it to predictive performance.
domain assumption Discrete-time snapshot partitioning with length Delta does not leak future information and preserves local temporal context.
Section 3.3: edges are scored only within their snapshot; the choice of Delta=7 days is a modeling assumption.
domain assumption The validation-selected high-curvature bin and tau=1 generalize to the test split and to other graph-level tasks.
Section 4.2.1: default policy is chosen based on validation; the paper admits other tasks may need other curvature ranges.
ad hoc to paper Log-smoothed node activity s(x)=log(1+strength(x)) is an appropriate normalizer for heavy-tailed transaction weights.
Section 3.2: introduced to 'reduce dominance of highly active nodes'; no comparison to other normalizers.

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Cite this review

Pith. "Pith review of Edge Sparsification via Temporal Forman-Ricci Curvature for Dynamic Graph Learning." pith.science (2026). https://pith.science/paper/LSRFZALE

@misc{pith2026260807158,
  author       = {Pith},
  title        = {Pith review of: Edge Sparsification via Temporal Forman-Ricci Curvature for Dynamic Graph Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LSRFZALE}},
  note         = {Machine review of arXiv:2608.07158}
}

read the original abstract

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 predictive temporal-structural information under substantial sparsification.

Figures

Figures reproduced from arXiv: 2608.07158 by the authors.

**Figure 1.** Architecture of the proposed Temporal Forman–Ricci curvature-based sparsification framework in three components: (a) the curvature [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 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 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** ROC–AUC comparison on token transaction networks across the three evaluated graph-level temporal prediction tasks. Results for Task 3 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: ROC–AUC comparison on TGBL benchmark datasets across two graph-level temporal prediction tasks. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: Sensitivity of the proposed framework to the edge-removal ratio on Task 1. [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Sensitivity of the proposed framework to the temporal decay parameter [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Jialong Chen, Bowen Deng, Zhen Wang, Chuan Chen, and Zibin Zheng. 2025. Graph neural ricci flow: evolving feature from a curvature perspective. In International Conference on Learning Representations. Vol. 2025, 31083–31111

work page 2025

[2] [2]

Jie Chen, Tengfei Ma, and Cao Xiao. 2018. Fastgcn: fast learning with graph convolutional networks via importance sampling. InInternational Conference on Learning Representations

work page 2018

[3] [3]

Tianlong Chen, Yongduo Sui, Xuxi Chen, Aston Zhang, and Zhangyang Wang. 2021. A unified lottery ticket hypothesis for graph neural networks. InInternational conference on machine learning. PMLR, 1695–1706

work page 2021

[4] [4]

Wenying Duan, Xiaoxi He, Zimu Zhou, Lothar Thiele, and Hong Rao. 2023. Localised adaptive spatial-temporal graph neural network. InProceedings of the 29th acm sigkdd conference on knowledge discovery and data mining, 448–458

work page 2023

[5] [5]

Ziwei Fan, Zhiwei Liu, Jiawei Zhang, Yun Xiong, Lei Zheng, and Philip S Yu. 2021. Continuous-time sequential recommendation with temporal graph collaborative transformer. InProceedings of the 30th ACM international conference on information & knowledge management, 433–442

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work page 2003

[7] [7]

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[8] [8]

Xinyi Gao, Guanhua Ye, Tong Chen, Wentao Zhang, Junliang Yu, and Hongzhi Yin. 2025. Rethinking and accelerating graph condensation: a training-free approach with class partition. InProceedings of the ACM on Web Conference 2025, 4359–4373

work page 2025