Community detection of hypergraphs by Ricci flow

Jicheng Ma; Liang Zhao; Yulu Tian; Yunyan Yang

REVIEW 4 major objections 6 minor 69 references

Community detection of hypergraphs by Ricci flow

T0 review · 4 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper proves that a Ricci flow on hyperedge weights has a unique solution for all time, and that its discretization, HyperRCD, detects communities in hypergraphs.

desk verdict Genuinely new hyperedge-level Ricci flow with a fixable proof issue, but the community detection results are oracle-fitted because Algorithm 1 tunes its cutoff against ground truth; the theory deserves referee time, the experiments need rework. read the letter →

arxiv 2505.12276 v1 pith:CRGWYH5T submitted 2025-05-18 cs.SI

Yulu Tian , Jicheng Ma , Yunyan Yang , Liang Zhao This is my paper · ORCID

classification cs.SI MSC 05C6553E2068R10

keywords hypergraphRicciflowOllivier-Riccicurvaturecommunitydetectionhigher-orderinteractionsWassersteindistancelong-timeexistencehyperedgeweights

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 a Ricci-flow equation defined directly on hyperedges—not on a graph approximation of the hypergraph—is mathematically well-posed and practically useful for community detection. It proves that the hyperedge weights solving the flow exist, are unique, and remain bounded away from zero and infinity for all $t \geq 0$, and it packages a discrete version of the flow into an algorithm named HyperRCD. If the claim is right, community structure is revealed by the flow's own dynamics: hyperedges with negative curvature are stretched, and deleting the stretched, large-weight hyperedges separates communities. The experiments on synthetic and real hypergraphs are offered as evidence that this geometric prescription detects higher-order community structure without first flattening hyperedges into pairwise edges.

What carries the argument

The central object is the Ollivier–Ricci curvature of a hyperedge, $\kappa_\alpha(h_l) = 1 - W_{h_l}/d(h_l)$, with $W_{h_l}$ the sum of the $1$-Wasserstein distances between the probability measures $\mu^\alpha_{x_{l_i}}$ carried by the vertices of $h_l$ and $d(h_l)$ the hyperedge length. The flow $w'_{h_l} = -d(h_l)\,\kappa_\alpha(h_l)$ turns this curvature into a weight update: negative curvature stretches a hyperedge, positive curvature shrinks it. The existence proof is carried by a local Lipschitz estimate for the map $w \mapsto W_{h_l}(w) - d(h_l)(w)$, obtained through the optimal-transport duality of Proposition II.3, together with exponential bounds that prevent any weight from collapsing to zero or blowing up. These ingredients make the flow a well-defined dynamical system on the space of positive hyperedge weights.

What would settle it

Run HyperRCD on a degree-corrected stochastic block model hypergraph with two planted 50-node communities; if after the flow the heaviest hyperedges mostly lie inside communities rather than between them, the weight-separation premise fails.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem III.2: for any connected weighted hypergraph with initial weights $w_0$, the flow $$\frac{d}{dt} w_{h_l}(t) = -d(h_l)\,\kappa_\$\alpha$(h_l) = W_{h_l}(w(t)) - d(h_l)(w(t))$$ has a unique solution for all $t\in[0,\infty)$, where $W_{h_l}$ is the sum of the $1$-Wasserstein distances between the vertex measures on $h_l$ and $d(h_l)$ is the hyperedge length. The proof bounds each weight below by exponential decay and bounds the total weight above by exponential growth, so the solution cannot hit zero or diverge in finite time. The authors then discretize the flow with step size $\eta = 0.1$ and use it to define HyperRCD, which removes hyperedges whose evolved weights exceed a cutoff and outputs the resulting partition. On degree-corrected stochastic block model hypergraphs the method maintains high normalized mutual information as network size, degree, and community ambiguity vary, and on seven real-world datasets it beats graph-embedding baselines, reaching the best NMI on Zoo and NTU2012.

Load-bearing premise

The method's load-bearing premise is that the curvature-driven flow pushes hyperedges crossing between communities to larger weights than hyperedges inside communities, so deleting the heaviest hyperedges—with a cutoff selected using ground-truth labels—recovers the true clusters.

Editorial extensions

If this is right

The flow can be run for arbitrarily many discrete steps without degenerating: the continuous uniqueness and global-existence result rules out weight collapse or blow-up.
The discrete update with step size $\eta = 0.1$ gives a concrete clustering rule: evolve weights, delete hyperedges heavier than a cutoff, and read off connected components.
On the synthetic series, the method's NMI stays above 0.6 up to 1000 nodes while several compared methods fall below 0.2, and it keeps high NMI when intra-community connectivity is weak.
On seven real-world benchmarks the method beats graph-embedding baselines on every dataset and reaches the best NMI on Zoo (0.981) and NTU2012 (0.836).
The $O(ED^3)$ cost, dominated by solving linear programs for Wasserstein distances, means the method does not finish within 48 hours on the large-cardinality Mushroom hypergraph.

Reading between the lines

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

Editorial inference: An unsupervised variant could choose the deletion cutoff at the largest gap in the evolved weight distribution, removing the current reliance on ground-truth labels.
Editorial inference: The same ODE argument would extend to directed hypergraphs if the distance and measure definitions are made directed and remain Lipschitz in the weights; the paper does not prove this.
Editorial inference: The observed insensitivity to average degree suggests the flow regularizes hyperedge weights geometrically; comparing HyperRCD with spectral or modularity-based edge reweighting on the same synthetic series would test that mechanism directly.

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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

Two things you should know. The theoretical core is genuinely new: the flow (7) acts directly on hyperedge weights rather than on a clique-expanded graph, and Theorem III.2 proves long-time existence for positive initial weights. The proof is a fairly standard ODE Lipschitz argument, and aside from a notational slip it essentially works. The experimental section, though, does not yet support the headline claims. Algorithm 1 sweeps the cutoff and selects the partition with the best NMI against ground-truth labels, so the reported numbers are the best of a one-parameter family, not the output of a label-free rule. That makes the robustness claims oracle-fitted.

What the paper does well: it gives a clean curvature definition for hyperedges, a natural evolution equation, and an explicit complexity analysis. It honestly reports that Mushroom exceeds the time limit. The idea of avoiding clique expansion is worthwhile, and the authors are careful about the local Lipschitz estimates that underpin the existence theorem.

The weak spot is load-bearing. There is no argument, and no evidence beyond assertion, that after the flow inter-community hyperedges systematically carry larger weights than intra-community ones, which is what the removal rule needs. Section IV simply states that negative curvature stretches inter-community edges. Without a proof or a systematic empirical check, the algorithm is incomplete as an unsupervised method. Also, baseline scores are imported from Lee and Shin and from Hacquard rather than rerun under the same code paths, so the comparisons are not controlled.

On the proof itself: inequality (10) has each hyperedge with its own cardinality s, and moving to (13) uses a single s for the lower bound. That is fixable by replacing s with the maximum hyperedge cardinality. The same paragraph contains a confusing self-referential note about what s means; a referee should ask for a clean rewrite. The paper also imports two technical lemmas from the authors' own prior papers [47,48]. That is not damning, but those results should be checked, especially Remark II.2(ii), since the local Lipschitz continuity of the probability measures is doing real work.

Who is this for? People working on discrete curvature flows for hypergraphs will find the theory section useful, and the community detection angle is interesting despite the current problems. I would not cite the empirical numbers as evidence. A serious editor should send this to peer review: the theoretical contribution is enough to warrant referee time, and the empirical issues are identifiable and fixable. The revision needs a label-free cutoff rule, rerun baselines, and either a separation theorem or a careful experiment showing that the flow actually separates communities.

My recommendation: engage with it, but treat the experimental claims as unvalidated until the oracle sweep is removed.

Referee Report

4 major / 6 minor

Summary. The paper defines an Ollivier-type Ricci curvature for weighted hypergraphs and proposes a Ricci flow that evolves hyperedge weights over time, proving long-time existence of solutions to the ODE system (Theorem III.2). It then introduces HyperRCD, a community detection algorithm that discretizes the flow and removes hyperedges with large weights after evolution. Experiments on synthetic DCSBM hypergraphs and seven real-world datasets claim that HyperRCD is robust to topological variations and competitive with or better than graph- and hypergraph-based baselines.

Significance. If the theoretical and empirical claims hold, the paper would provide a geometrically motivated hypergraph community detection method that avoids lossy clique or star expansion, with a rigorous existence theorem as a foundation. The flow construction is clearly defined, and the ODE existence argument is a standard finite-dimensional Lipschitz/ODE argument once the identified exponent issue is corrected. The availability of source code is a positive feature. However, the empirical evaluation is currently not sufficient to support the central performance claims because Algorithm 1 selects its cutoff using ground-truth labels, and baseline scores are imported from prior papers rather than rerun under controlled conditions. The community-separation property that underlies the algorithm is asserted rather than demonstrated. With a label-free cutoff rule and corrected experiments, the contribution could be valuable, but the manuscript as written overstates its empirical support.

major comments (4)

[§IV, Algorithm 1; §V.C, Table III and Figures 1–3] Algorithm 1's inner loop selects the cutoff by maximizing community detection accuracy against ground-truth labels and then returns the best partition, so every reported NMI is the maximum over a one-parameter family whose parameter is chosen using the target labels. This is an oracle evaluation, not a test of an unsupervised community-detection rule. The paper should either define and evaluate a label-free cutoff rule (e.g., a fixed weight percentile or a data-driven threshold not involving the labels) or explicitly report the full cutoff-NMI curve and state that the selected value is an upper bound, not a prediction. As written, the central empirical claim in the Abstract and Conclusion is unsupported.
[§IV, paragraph before Algorithm 1; §VI] The community-separation mechanism is asserted rather than established. The flow guarantees long-time existence, but nothing in Theorem III.2 implies that inter-community hyperedges end with larger weights than intra-community hyperedges, which is the premise of the 'remove edges with large weights' rule. The paper should add either a theorem or an empirical diagnostic on synthetic data showing that the weight distributions separate by ground-truth community membership after the flow. Without such evidence, the design principle of the algorithm is a conjecture.
[§III, Theorem III.2 proof, Eqs. (10)–(13)] The lower bound in Eq. (10) uses exponent s(s−1)/2 for the specific hyperedge's cardinality, but to bound ψ(t) = min_l w_{h_l}(t) uniformly the exponent must be taken as max_l |h_l|(|h_l|−1)/2. The parenthetical statement that s here refers to the vertex count of the minimally weighted hyperedge does not resolve the non-uniformity, since the bound for each h_l is valid only with that hyperedge's own s. Additionally, the upper bound in Eqs. (11)–(12) uses an underspecified constant nm; it should be stated explicitly, for instance as E = Σ_l C(|h_l|, 2). These are local fixes that preserve the theorem's conclusion.
[§V.B, paragraph 'Implementation Details'] All baseline scores are taken from Lee and Shin [57] and Hacquard [46] rather than rerun in the authors' environment. Different hardware, software, hyperparameter settings, and preprocessing can materially affect NMI values, so the comparisons in Table III and Figures 1–3 are not controlled. The authors should rerun the baseline methods on the same clique-expanded hypergraphs or hypergraph representations and report variance, or clearly present the imported scores as approximate and justify why the comparison is still valid.

minor comments (6)

[§IV, Algorithm 1] The loop 'for cutoff = w'_max, ..., w'_min' should specify whether the sweep is over all distinct weight values, a fixed grid, or a continuous interval; otherwise the stopping criterion and computational cost are ambiguous.
[§II, Remark II.2(i)] The phrase 'If k = 2' should be 'If |h| = 2', since k is not otherwise defined in that remark and the condition concerns the cardinality of a hyperedge.
[§IV, Eq. (14)] The hyperedge is written h_l = {x_l1, x_l2, ..., x_lk}, which reuses k from the iteration counter; use a separate symbol such as s or r for the hyperedge cardinality.
[Algorithm 1 and §V.B] Algorithm 1 says 'Calculate the accuracy of community detection' while the evaluation section defines NMI as the metric; replace 'accuracy' with 'NMI' or explicitly state that the two are used interchangeably.
[§V.A] Please describe how the DCSBM is used to generate hyperedges, since 'intra-community connectivity strength' alone does not specify the higher-order sampling process; this is important for reproducibility beyond the code link.
[Throughout] There are minor typographical issues such as 'intra-comminity' in §V.C.2 and inconsistent spacing in some equations; a careful proofreading pass would improve readability.

Circularity Check

1 steps flagged · score 6.0 of 10

Empirical results are oracle-fitted through the cutoff sweep in Algorithm 1, reducing reported NMI to a ground-truth-selected maximum; the ODE theorem is not circular.

fitted input called prediction [Section IV, Algorithm 1 (after Eq. (14))]
"for cutoff= w' max, · · ·, w' min do for h' l ∈ H' do if w' h' l > cutoff then Remove the hyperedge h' l end end Calculate the accuracy of community detection end end Output the best community detection result of Γ"

The partition reported by HyperRCD is chosen by sweeping the removal threshold over the full range of evolved weights and keeping the one with highest accuracy. In Section V.B the accuracy metric is NMI against ground truth. Therefore every reported NMI in Table III and Figures 1–3 is max_cutoff NMI(partition(cutoff), ground_truth), i.e., an oracle-selected best case. The paper presents this as evidence that Ricci flow separates communities ('By removing edges with large weights, we can identify communities effectively'), but the algorithm never tests a fixed label-free cutoff; it uses the target labels to choose the cutoff. The empirical claim is thus fitted to the evaluation labels rather than predicted from the flow.

full rationale

The analytical part is largely self-contained: Theorem III.2 is a standard ODE existence argument once the right-hand side W_h - d(h) is shown locally Lipschitz, and Lemma III.1 supplies that regularity with a proof in the text. The citations to the authors' previous preprints [47,48] for Lemmas II.1 and Remark II.2(ii) are load-bearing in the sense that the proof does not reproduce those lemmas, but they are independent technical statements rather than restatements of Theorem III.2, so they do not by themselves make the derivation circular. The significant circularity is empirical. Algorithm 1's inner loop evaluates every possible cutoff against ground truth and outputs the best partition, so the claimed 'remarkable enhanced robustness and competitive performance' is the maximum of a one-parameter family fitted to the labels, not the performance of a parameter-free rule. This is a form of fitted input presented as prediction. The comparison is further weakened by importing baseline scores from other papers rather than rerunning them, though that is a control issue, not circularity. Overall score reflects one load-bearing oracle-fit in the empirical claim; the theoretical existence result is not circular.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The theory rests on standard optimal-transport machinery plus two lemmas imported from the authors' own earlier papers [47,48]. The algorithm rests on an unproved community-separation property and, in the reported experiments, on oracle selection of the cutoff via ground-truth labels. The main free parameters are alpha, eta, K, and the per-dataset cutoff.

free parameters (4)

alpha (mixing parameter in curvature) = 0.5
Mixing parameter in the probability measure (5); set to 0.5 based on earlier graph experiments in [34,48], not derived in this paper. Curvature values and the flow trajectory depend on it.
eta (step size) = 0.1
Discretization step size in (14) and Algorithm 1; chosen to balance accuracy and efficiency, no sensitivity analysis or convergence test provided.
K (maximum iterations) = not specified
Maximum iteration count in Algorithm 1; no default value or criterion is given, and the inner accuracy loop effectively tunes over iterations and cutoffs.
cutoff threshold per dataset = dataset-dependent, not reported
Algorithm 1 sweeps cutoff from w'_max to w'_min and keeps the result with highest accuracy. In the experiments accuracy is NMI against ground truth, so the cutoff is a fitted selection parameter.

assumptions (7)

standard math Kantorovich-Rubinstein duality for finite metric spaces
Used in Proposition II.3 to rewrite Wasserstein distance as a supremum over 1-Lipschitz functions, and in Lemma III.1 to estimate Lipschitz continuity. Cited to [49,50].
standard math Local Lipschitz continuity of the shortest-path distance in hyperedge weights (Lemma II.1)
Imported from the authors' earlier paper [47] by Ma and Yang, two co-authors. The current paper does not prove it, so the long-time existence proof leans on a self-cited result.
standard math Local Lipschitz continuity of the probability measures in hyperedge weights (Remark II.2(ii))
Imported from the authors' earlier paper [48], also by Ma and Yang. Used to establish that the flow vector field is locally Lipschitz.
standard math Picard-Lindelof existence and uniqueness for ODE systems
Classical theorem invoked in the proof of Theorem III.2 to obtain local solutions and unique continuation.
ad hoc to paper The flow separates communities
Algorithm 1 removes hyperedges with weights above a cutoff, based on the claim in Section IV that negative-curvature edges are stretched between communities. This property is asserted and never derived from the flow equations.
domain assumption Input hypergraph is connected
Stated in Section II: 'Unless otherwise specified, any mention of hypergraphs refers to undirected, connected hypergraphs', which is needed to define the distance function d(u,v).
domain assumption Ground-truth community labels are available to select the cutoff in Algorithm 1
The reported NMI values are obtained by maximizing accuracy over cutoffs, so the evaluation assumes access to the labels that the algorithm is supposed to discover.

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Pith. "Pith review of Community detection of hypergraphs by Ricci flow." pith.science (2026). https://pith.science/paper/CRGWYH5T

@misc{pith2026250512276,
  author       = {Pith},
  title        = {Pith review of: Community detection of hypergraphs by Ricci flow},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CRGWYH5T}},
  note         = {Machine review of arXiv:2505.12276}
}

read the original abstract

Community detection in hypergraphs is both instrumental for functional module identification and intricate due to higher-order interactions among nodes. We define a hypergraph Ricci flow that directly operates on higher-order interactions of hypergraphs and prove long-time existence of the flow. Building on this theoretical foundation, we develop HyperRCD-a Ricci-flow-based community detection approach that deforms hyperedge weights through curvature-driven evolution, which provides an effective mathematical representation of higher-order interactions mediated by weighted hyperedges between nodes. Extensive experiments on both synthetic and real-world hypergraphs demonstrate that HyperRCD exhibits remarkable enhanced robustness to topological variations and competitive performance across diverse datasets.

Figures

Figures reproduced from arXiv: 2505.12276 by the authors.

**Figure 2.** FIG. 2. NMI performance on synthetic network D2 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** FIG. 3. NMI performance on synthetic network D3 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 6.** Figure 6: Since the hyperedges may obscure the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 4.** Figure 4: FIG. 4. HyperRCD Community Detection on D1: Original vs. Hyp [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: FIG. 5. HyperRCD Community Detection on D3: Original vs. Hyp [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: FIG. 6. HyperRCD Community Detection on Zoo Dataset: Origin [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

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Reference graph

Works this paper leans on

69 extracted references · 59 canonical work pages

[57]

I’m me, we’re us, and i’m us : Tri-directional contrastive learning on hypergraphs,

Dongjin Lee and Kijung Shin, “I’m me, we’re us, and i’m us : Tri-directional contrastive learning on hypergraphs,” in Proceedings of the AAAI conference on artiﬁcial intelligen ce, V ol. 37 (2023) pp. 8456–8464

work page 2023
[46]

Hypergraph clustering using ricci curvature: an edge transport perspective,

Olympio Hacquard, “Hypergraph clustering using ricci curvature: an edge transport perspective,” arXiv: 2412.15695 (2024)

work page arXiv 2024
[1]

Hypergraph laplace op erators for chemical reaction networks,

Jürgen Jost and Raffaella Mulas, “Hypergraph laplace op erators for chemical reaction networks,” Adv. Math. 351, 870–896 (2019)

work page 2019
[2]

Self- supervised multi-channel hypergraph convolutional netwo rk for social recommendation,

Junliang Y u, Hongzhi Yin, Jundong Li, Qinyong Wang, Nguy en Quoc Viet Hung, and Xiangliang Zhang, “Self- supervised multi-channel hypergraph convolutional netwo rk for social recommendation,” in Proceedings of the web conference 2021 (2021) pp. 413–424

work page 2021
[3]

Self-supervised learning for recom- mender systems: A survey,

Junliang Y u, Hongzhi Yin, Xin Xia, Tong Chen, Jundong Li, and Zi Huang, “Self-supervised learning for recom- mender systems: A survey,” IEEE Trans. on Knowl. and Data Eng . 36, 335–355 (2023)

work page 2023
[4]

Ricci curvature of markov chains on metr ic spaces,

Yann Ollivier, “Ricci curvature of markov chains on metr ic spaces,” J. Funct. Anal. 256, 810–864 (2009)

work page 2009
[5]

Ricci curvature and eigenvalue estimate on locally ﬁnite graphs,

Y ong Lin and Shing-Tung Yau, “Ricci curvature and eigenvalue estimate on locally ﬁnite graphs,” Math. Res. Lett. 17, 343–356 (2010)

work page 2010
[6]

Ricci curvatu re of graphs,

Y ong Lin, Linyuan Lu, and Shing-Tung Yau, “Ricci curvatu re of graphs,” Tohoku Math. J. 63, 605–627 (2011)

work page 2011

Show all 69 references

[7]

On the sum of ricci-curvatures for weighted graphs,

Shuliang Bai, An Huang, Linyuan Lu, and Shing-Tung Yau, “ On the sum of ricci-curvatures for weighted graphs,” Pure Appl. Math. Q. 17, 1599–1617 (2021)

2021
[8]

Ollivier–ri cci curvature and the spectrum of the normalized graph laplace operator,

Frank Bauer, Jürgen Jost, and Shiping Liu, “Ollivier–ri cci curvature and the spectrum of the normalized graph laplace operator,” Math. Res. Lett. 19, 1185–1205 (2012)

2012
[9]

V olume gr owth, curvature, and buser-type inequalities in graphs,

Brian Benson, Peter Ralli, and Prasad Tetali, “V olume gr owth, curvature, and buser-type inequalities in graphs,” Int. Math. Res. Not. IMRN 2021, 17091–17139 (2021)

2021
[10]

Discrete curva ture on graphs from the effective resistance,

Karel Devriendt and Renaud Lambiotte, “Discrete curva ture on graphs from the effective resistance,” J. Phys. Complex. 3, 025008 (2022)

2022
[11]

Ollivier’s ricci curvatu re, local clustering and curvature-dimension inequalitie s on graphs,

Jürgen Jost and Shiping Liu, “Ollivier’s ricci curvatu re, local clustering and curvature-dimension inequalitie s on graphs,” Discrete Comput. Geom. 51, 300–322 (2014)

2014
[12]

Ollivi er ricci curvature for general graph laplacians: heat equation, laplacian comparison, non-explosion and diamet er bounds,

Florentin Münch and Radosław K Wojciechowski, “Ollivi er ricci curvature for general graph laplacians: heat equation, laplacian comparison, non-explosion and diamet er bounds,” Adv. Math. 356, 106759 (2019)

2019
[13]

The ricci curvature on directed graphs,

Taiki Yamada, “The ricci curvature on directed graphs, ” J. Korean Math. Soc. 56, 113–125 (2019). 16

2019
[14]

Geo metric and spectral properties of directed graphs under a lower ricci curvature bound,

Ryunosuke Ozawa, Y ohei Sakurai, and Taiki Yamada, “Geo metric and spectral properties of directed graphs under a lower ricci curvature bound,” Calc. V ar. Partial. Differ.Equ. 59, 1–39 (2020)

2020
[15]

Ollivier ricci curvatur e of directed hypergraphs,

Marzieh Eidi and Jürgen Jost, “Ollivier ricci curvatur e of directed hypergraphs,” Sci. Rep. 10, 12466 (2020)

2020
[16]

Edge-based analysis of net- works: curvatures of graphs and hypergraphs,

Marzieh Eidi, Amirhossein Farzam, Wilmer Leal, Areeji t Samal, and Jürgen Jost, “Edge-based analysis of net- works: curvatures of graphs and hypergraphs,” Theory Biosc i. 139, 337–348 (2020)

2020
[17]

Forman–ricci curvature for hypergraphs,

Wilmer Leal, Guillermo Restrepo, Peter F Stadler, and J ürgen Jost, “Forman–ricci curvature for hypergraphs,” Adv. Complex Syst. 24, 2150003 (2021)

2021
[18]

Ricci curv ature of random and empirical directed hypernetworks,

Wilmer Leal, Marzieh Eidi, and Jürgen Jost, “Ricci curv ature of random and empirical directed hypernetworks,” Appl. Netw. Sci. 5, 1–14 (2020)

2020
[19]

Ollivier-ricci curvature for hypergraphs: A uniﬁed framework,

Corinna Coupette, Sebastian Dalleiger, and Bastian Ri eck, “Ollivier-ricci curvature for hypergraphs: A uniﬁed framework,” arXiv: 2210.12048 (2022)

2022 arXiv
[20]

Three-manifolds with positive ri cci curvature,

Richard S Hamilton, “Three-manifolds with positive ri cci curvature,” J. Differ. Geom. 17, 255–306 (1982)

1982
[21]

The entropy formula for the ricci ﬂow and its geometric applications,

Grisha Perelman, “The entropy formula for the ricci ﬂow and its geometric applications,” arXiv: math/0211159 (2002)

2002 arXiv
[22]

Finite extinction time for the solut ions to the ricci ﬂow on certain three-manifolds,

Grisha Perelman, “Finite extinction time for the solut ions to the ricci ﬂow on certain three-manifolds,” arXiv: math/0307245 (2003)

2003 arXiv
[23]

Ricci ﬂow with surgery on three-mani folds,

Grisha Perelman, “Ricci ﬂow with surgery on three-mani folds,” arXiv: math/0303109 (2003)

2003 arXiv
[24]

Discret e surface ricci ﬂow,

Miao Jin, Junho Kim, Feng Luo, and Xianfeng Gu, “Discret e surface ricci ﬂow,” IEEE Trans. Vis. Comput. Graphics 14, 1030–1043 (2008)

2008
[25]

Greedy routing with guaranteed delivery using ricci ﬂows,

Rik Sarkar, Xiaotian Yin, Jie Gao, Feng Luo, and Xianfen g David Gu, “Greedy routing with guaranteed delivery using ricci ﬂows,” in 2009 International Conference on Information Processing i n Sensor Networks (IEEE, 2009) pp. 121–132

2009
[26]

Feature-aware reconstruction of trimmed splines using ricci ﬂow with metric optimization,

Kendrick M Shepherd, Xianfeng David Gu, and Thomas JR Hu ghes, “Feature-aware reconstruction of trimmed splines using ricci ﬂow with metric optimization,” Comput. Methods Appl. Mech. Engrg. 402, 115555 (2022)

2022
[27]

The four dimensions of social network analysis: An overview of r esearch methods, applications, and software tools,

David Camacho, Angel Panizo-LLedot, Gema Bello-Orgaz , Antonio Gonzalez-Pardo, and Erik Cambria, “The four dimensions of social network analysis: An overview of r esearch methods, applications, and software tools,” Inf. Fusion 63, 88–120 (2020)

2020
[28]

Community detection in graphs,

Santo Fortunato, “Community detection in graphs,” Phy s. Rep. 486, 75–174 (2010)

2010
[29]

A survey of community search over big graphs,

Yixiang Fang, Xin Huang, Lu Qin, Ying Zhang, Wenjie Zhan g, Reynold Cheng, and Xuemin Lin, “A survey of community search over big graphs,” The VLDB Journal 29, 353–392 (2020)

2020
[30]

Empi rical comparison of algorithms for network community detection,

Jure Leskovec, Kevin J Lang, and Michael Mahoney, “Empi rical comparison of algorithms for network community detection,” in Proceedings of the 19th international conference on W orld wide web (2010) pp. 631–640

2010
[31]

Community-diversiﬁed inﬂuence maximization in social networks,

Jianxin Li, Taotao Cai, Ke Deng, Xinjue Wang, Timos Sell is, and Feng Xia, “Community-diversiﬁed inﬂuence maximization in social networks,” Inf. Syst. 92, 101522 (2020)

2020
[32]

Community discover y in dynamic networks: a survey,

Giulio Rossetti and Rémy Cazabet, “Community discover y in dynamic networks: a survey,” ACM Comput. Surv. 51, 1–37 (2018)

2018
[33]

A comprehensive survey on community detection with deep l earning,

Xing Su, Shan Xue, Fanzhen Liu, Jia Wu, Jian Yang, Chuan Z hou, Wenbin Hu, Cecile Paris, Surya Nepal, Di Jin, et al. , “A comprehensive survey on community detection with deep l earning,” IEEE Trans. Neural Netw. Learn. Syst. (2022)

2022
[34]

Commun ity detection on networks with ricci ﬂow,

Chien-Chun Ni, Y uyao Lin, Feng Luo, and Jie Gao, “Commun ity detection on networks with ricci ﬂow,” Sci. Rep. 9, 9984 (2019)

2019
[35]

Olliv ier-ricci curvature-based method to community detection in complex networks,

Jayson Sia, Edmond Jonckheere, and Paul Bogdan, “Olliv ier-ricci curvature-based method to community detection in complex networks,” Sci. Rep. 9, 9800 (2019)

2019
[36]

Normalized discre te ricci ﬂow used in community detection,

Xin Lai, Shuliang Bai, and Y ong Lin, “Normalized discre te ricci ﬂow used in community detection,” Phys. A Stat. Mech. Appl. 597, 127251 (2022)

2022
[37]

Ollivier ricci-ﬂow on weighted graphs,

Shuliang Bai, Y ong Lin, Linyuan Lu, Zhiyu Wang, and Shin g-Tung Yau, “Ollivier ricci-ﬂow on weighted graphs,” Am. J. Math. 146, 1723–1747 (2024)

2024
[38]

Bakry–émery curvature sharpness and curvature ﬂow in ﬁnit e weighted graphs: implementation,

David Cushing, Supanat Kamtue, Shiping Liu, Florentin Münch, Norbert Peyerimhoff, and Ben Snodgrass, “Bakry–émery curvature sharpness and curvature ﬂow in ﬁnit e weighted graphs: implementation,” Axioms 12, 577 (2023)

2023
[39]

Bakry-émery curvature sharpness and curvature ﬂow in ﬁnite weighted graphs: theory,

David Cushing, Supanat Kamtue, Shiping Liu, Florentin Münch, Norbert Peyerimhoff, and Ben Snodgrass, “Bakry-émery curvature sharpness and curvature ﬂow in ﬁnite weighted graphs: theory,” Manuscripta mathematica 176, 11 (2025). 17

2025
[40]

A version of bakry-Éme ry ricci ﬂow on a ﬁnite graph,

Bobo Hua, Y ong Lin, and Tao Wang, “A version of bakry-Éme ry ricci ﬂow on a ﬁnite graph,” arXiv: 2402.07475 (2024)

2024 arXiv
[41]

Community de tection in hypergraphs: Optimal statistical limit and efﬁcient algorithms,

I Chien, Chung-Yi Lin, and I-Hsiang Wang, “Community de tection in hypergraphs: Optimal statistical limit and efﬁcient algorithms,” in International Conference on Artiﬁcial Intelligence and St atistics (PMLR, 2018) pp. 871– 879

2018
[42]

Community detec tion for hypergraph networks via regularized tensor power iteration,

Zheng Tracy Ke, Feng Shi, and Dong Xia, “Community detec tion for hypergraph networks via regularized tensor power iteration,” arXiv: 1909.06503 (2019)

2019 arXiv
[43]

Random walks and community detection in hyper- graphs,

Timoteo Carletti, Duccio Fanelli, and Renaud Lambiott e, “Random walks and community detection in hyper- graphs,” J. Phys. Complex. 2, 015011 (2021)

2021
[44]

Community detection in large hypergraphs,

Nicolò Ruggeri, Martina Contisciani, Federico Battis ton, and Caterina De Bacco, “Community detection in large hypergraphs,” Sci. Adv. 9, eadg9159 (2023)

2023
[45]

Co mmunity detection in hypergraphs via mutual information maximization,

Jürgen Kritschgau, Daniel Kaiser, Oliver Alvarado Rod riguez, Ilya Amburg, Jessalyn Bolkema, Thomas Grubb, Fangfei Lan, Sepideh Maleki, Phil Chodrow, and Bill Kay, “Co mmunity detection in hypergraphs via mutual information maximization,” Sci. Rep. 14, 6933 (2024)

2024
[47]

A modiﬁed ricci ﬂow on arbit rary weighted graph,

Jicheng Ma and Y unyan Yang, “A modiﬁed ricci ﬂow on arbit rary weighted graph,” arXiv: 2408.09435 (2024)

2024 arXiv
[48]

Evolution of weights on a co nnected ﬁnite graph,

Jicheng Ma and Y unyan Yang, “Evolution of weights on a co nnected ﬁnite graph,” arXiv: 2411.06393 (2024)

2024 arXiv
[49]

Ollivier-ricci idleness func- tions of graphs,

David P Bourne, David Cushing, Shiping Liu, F Münch, and Norbert Peyerimhoff, “Ollivier-ricci idleness func- tions of graphs,” SIAM J. Discrete Math. 32, 1408–1424 (2018)

2018
[50]

58 (American Mathematical Soc., 2021)

Cédric Villani, Topics in optimal transportation, V ol. 58 (American Mathematical Soc., 2021)

2021
[51]

Efﬁcient monte carlo and greedy heur istic for the inference of stochastic block models,

Tiago P . Peixoto, “Efﬁcient monte carlo and greedy heur istic for the inference of stochastic block models,” Phys. Rev. E 89, 012804 (2014)

2014
[52]

Fast unfolding of commu- nities in large networks,

Vincent D Blondel, Jean-Loup Guillaume, Renaud Lambio tte, and Etienne Lefebvre, “Fast unfolding of commu- nities in large networks,” Journal of statistical mechanic s: theory and experiment 2008, P10008 (2008)

2008
[53]

Community structu re in social and biological networks,

Michelle Girvan and Mark EJ Newman, “Community structu re in social and biological networks,” Proceedings of the national academy of sciences 99, 7821–7826 (2002)

2002
[54]

Community detect ion via semi-synchronous label propagation algorithms,

Gennaro Cordasco and Luisa Gargano, “Community detect ion via semi-synchronous label propagation algorithms,” in 2010 IEEE international workshop on: business application s of social network analysis (BASNA) (IEEE, 2010) pp. 1–8

2010
[55]

Maps of random wal ks on complex networks reveal community structure,

Martin Rosvall and Carl T Bergstrom, “Maps of random wal ks on complex networks reveal community structure,” Proceedings of the national academy of sciences 105, 1118–1123 (2008)

2008
[56]

Computing communiti es in large networks using random walks,

Pascal Pons and Matthieu Latapy, “Computing communiti es in large networks using random walks,” in Computer and Information Sciences-ISCIS 2005: 20th International S ymposium, Istanbul, Turkey, October 26-28, 2005. Proceedings 20 (Springer, 2005) pp. 284–293

2005
[58]

Uci machine learning repository,

Kelly Markelle, Rachel Longjohn, and Kolby Nottingham , “Uci machine learning repository,” The UCI Machine Learning Repository (2017)

2017
[59]

Collective classiﬁcation in network data,

Prithviraj Sen, Galileo Namata, Mustafa Bilgic, Lise G etoor, Brian Galligher, and Tina Eliassi-Rad, “Collective classiﬁcation in network data,” AI magazine 29, 93–93 (2008)

2008
[60]

The network data reposit ory with interactive graph analytics and visualization,

Ryan Rossi and Nesreen Ahmed, “The network data reposit ory with interactive graph analytics and visualization,” in Proceedings of the AAAI conference on artiﬁcial intelligen ce, V ol. 29 (2015)

2015
[61]

On visual similarity based 3d model retrieval,

Dingyun Chen, Xiaopei Tian, Y ute Shen, and Ming Ouhyoun g, “On visual similarity based 3d model retrieval,” in Computer graphics forum, V ol. 22 (Wiley Online Library, 2003) pp. 223–232

2003
[62]

node2vec: Scalable f eature learning for networks,

Aditya Grover and Jure Leskovec, “node2vec: Scalable f eature learning for networks,” in Proceedings of the 22nd ACM SIGKDD international conference on Knowledge discover y and data mining (2016) pp. 855–864

2016
[63]

Deep graph infomax,

Petar V eli ˇckovi´c, William Fedus, William L Hamilton, Pietro Liò, Y oshua Ben gio, and R Devon Hjelm, “Deep graph infomax,” arXiv:1809.10341 (2018)

2018 arXiv
[64]

Deep graph contrastive representation learning,

Yanqiao Zhu, Yichen Xu, Feng Y u, Qiang Liu, Shu Wu, and Li ang Wang, “Deep graph contrastive representation learning,” arXiv:2006.04131 (2020)

2020 arXiv
[65]

Double-scale self-supervised hypergraph learning for group recommendation,

Junwei Zhang, Min Gao, Junliang Y u, Lei Guo, Jundong Li, and Hongzhi Yin, “Double-scale self-supervised hypergraph learning for group recommendation,” in Proceedings of the 30th ACM international conference on 18 information & knowledge management (2021) pp. 2557–2567

2021
[66]

Community dete ction algorithm using hypergraph modularity,

Bogumił Kami ´nski, Paweł Prałat, and François Théberge, “Community dete ction algorithm using hypergraph modularity,” in International Conference on Complex Networks and Their App lications (Springer, 2020) pp. 152– 163

2020
[67]

Robust data clustering,

LN Fred Ana and Anil K Jain, “Robust data clustering,” in 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings., V ol. 2 (IEEE, 2003) pp. II–II

2003
[68]

Finding and evalua ting community structure in networks,

Mark EJ Newman and Michelle Girvan, “Finding and evalua ting community structure in networks,” Phys. Rev. E 69, 026113 (2004)

2004
[69]

Community detection algorithms: a comparative analysis,

Andrea Lancichinetti and Santo Fortunato, “Community detection algorithms: a comparative analysis,” Phys. Rev. E 80, 056117 (2009). 19

2009

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [57]

I’m me, we’re us, and i’m us : Tri-directional contrastive learning on hypergraphs,

Dongjin Lee and Kijung Shin, “I’m me, we’re us, and i’m us : Tri-directional contrastive learning on hypergraphs,” in Proceedings of the AAAI conference on artiﬁcial intelligen ce, V ol. 37 (2023) pp. 8456–8464

work page 2023

[2] [46]

Hypergraph clustering using ricci curvature: an edge transport perspective,

Olympio Hacquard, “Hypergraph clustering using ricci curvature: an edge transport perspective,” arXiv: 2412.15695 (2024)

work page arXiv 2024

[3] [1]

Hypergraph laplace op erators for chemical reaction networks,

Jürgen Jost and Raffaella Mulas, “Hypergraph laplace op erators for chemical reaction networks,” Adv. Math. 351, 870–896 (2019)

work page 2019

[4] [2]

Self- supervised multi-channel hypergraph convolutional netwo rk for social recommendation,

Junliang Y u, Hongzhi Yin, Jundong Li, Qinyong Wang, Nguy en Quoc Viet Hung, and Xiangliang Zhang, “Self- supervised multi-channel hypergraph convolutional netwo rk for social recommendation,” in Proceedings of the web conference 2021 (2021) pp. 413–424

work page 2021

[5] [3]

Self-supervised learning for recom- mender systems: A survey,

Junliang Y u, Hongzhi Yin, Xin Xia, Tong Chen, Jundong Li, and Zi Huang, “Self-supervised learning for recom- mender systems: A survey,” IEEE Trans. on Knowl. and Data Eng . 36, 335–355 (2023)

work page 2023

[6] [4]

Ricci curvature of markov chains on metr ic spaces,

Yann Ollivier, “Ricci curvature of markov chains on metr ic spaces,” J. Funct. Anal. 256, 810–864 (2009)

work page 2009

[7] [5]

Ricci curvature and eigenvalue estimate on locally ﬁnite graphs,

Y ong Lin and Shing-Tung Yau, “Ricci curvature and eigenvalue estimate on locally ﬁnite graphs,” Math. Res. Lett. 17, 343–356 (2010)

work page 2010

[8] [6]

Ricci curvatu re of graphs,

Y ong Lin, Linyuan Lu, and Shing-Tung Yau, “Ricci curvatu re of graphs,” Tohoku Math. J. 63, 605–627 (2011)

work page 2011