CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
Authors: Karish Grover
Geoffrey J. Gordon
Christos Faloutsos
Venue: ICML 2025 poster (2025)
Keywords: Graph anomaly detection
Riemannian geometry
Mixed-curvature GNN
Spectral graph theory
Ollivier-Ricci curvature
Ricci flow
Primary area: deep_learning->graph_neural_networks
TL;DR: Mixed-curvature autoencoder for unveiling curvature-based (geometric) anomalies in graphs.

Abstract: Does the intrinsic curvature of complex networks hold the key to unveiling graph anomalies that conventional approaches overlook? Reconstruction-based graph anomaly detection (GAD) methods overlook such geometric outliers, focusing only on structural and attribute-level anomalies. To this end, we propose CurvGAD - a mixed-curvature graph autoencoder that introduces the notion of curvature-based geometric anomalies. CurvGAD introduces two parallel pipelines for enhanced anomaly interpretability: (1) Curvature-equivariant geometry reconstruction, which focuses exclusively on reconstructing the edge curvatures using a mixed-curvature, Riemannian encoder and Gaussian kernel-based decoder; and (2) Curvature-invariant structure and attribute reconstruction, which decouples structural and attribute anomalies from geometric irregularities by regularizing graph curvature under discrete Ollivier-Ricci flow, thereby isolating the non-geometric anomalies. By leveraging curvature, CurvGAD refines the existing anomaly classifications and identifies new curvature-driven anomalies. Extensive experimentation over 10 real-world datasets (both homophilic and heterophilic) demonstrates an improvement of up to 6.5% over state-of-the-art GAD methods. The code is available at: https://github.com/karish-grover/curvgad.

CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
Karish Grover! Geoffrey J. Gordon'
Christos Faloutsos
Abstract
focus on identifying anomalies through structural irregular- ities attribute deviations (Ding al., 2019; Fan et al., 2020), such as abnormal connections or unusual feature val- ues. However, these approaches often overlook the underly- ing geometric properties of graphs, particularly the graph curvature, which encapsulates essential information about the global and local topology of the graph. Recent advances in graph representation learning and Riemannian geometry have delved deeply into constant curvature spaces (Bach- mann al., 2020) namely hyperbolic, spherical, and Euclidean geometries, characterized by negative, positive, and zero curvature, respectively - to learn distortion-fret graph representations. Each of these spaces offers beneficial inductive biases for specific structures. For instance, hyper- bolic space is ideal for hierarchical, tree-like graphs (Chami et al., 2019), whereas spherical geometry is optimal for rep- resenting eyclic graphs (Gu et al., 2019). However, current graph anomaly detection (GAD) methods fail to leverage such insights, resulting in several limitations:
Does the intrinsic curvature of complex net- works hold the key to unveiling graph anoma- lies that conventional approaches overlook? Reconstruction-base graph anomaly detection (GAD) methods overlook such geometric outliers, focusing only on structural and attribute-level anomalies. To this end, we propose CurvGAD - a mixed-curvature graph autoencoder that in- troduces the notion of curvature- based geomet- ric anomalies. CurvGAD introduces two parallel pipelines for enhanced anomaly interpretability; (I) Curvature Curvature-equiivaiann geometry reconstruc- tion, which focuses exclusively reconstruct- ing the edge curvatures using mixed-curven curvature, Riemannian encoder and Gaussian kernel-based decoder, and (2) servature-invarian structure and attribute reconstruction, which decouples structural and attribute anomalies from geomet- ric irregularities by regularizing graph curvature under discrete Ollivier-Ricci flow, thereby isolat- ing the non- -eommetet anomalies. By leveraging curvature, CurvGAD refines the existing anomaly classifications and identifies new curvature-dis driven anomalies. Extensive experimentation over 10 real-world datasets (both homophilic and het- erophilic) demonstrates improvement of up 6.5% over state-of-the-arr GAD methods. The code is available at: tttpp://githhr com/ karish- teath-groveeeee curvgad.
(L1) Inadequate Representation of Complex Topologies. Existing methods assume that graphs can be effectively represented in Euclidean space. However, real-world graphs often exhibit complex topologies with intrinsic curvature variations (Gu et al., 2019) that cannot be captured by single geometric space.
(L2) Neglect of Geometric Anomalies. Current ap- proaches largely ignore geometric anomalies mani- fested through task-specific irregularities in the cur- vature of the graph, missing critical insights into the inherent geometry of the graph (Chatterjee et al., 2021).
(L33 Homophily Trap. Several methods predominantly op- erate under the homophily assumption (i.e., low-pass filters) and fail in heterophilic graphs where connected nodes tend to be dissimilar, missing out on anomalies that arise in such settings (He et al., 2023).
1. Introduction
Detecting anomalies in graph-structured data pivotal task across diverse fields, including social networks (Has- sanzadeh al., 2012), cybersecurity (Wang & Zhu, 2022), transportation systems (Hu et al., 2020), and biological net- works (Singh & Vig, 2017). Traditional methods primarily
Geometric anomalies, as we define them, are irregularities in graph structure revealed through deviations in task-specific curvature patterns. Depending on the application, different curvature regimes negative, positive, zero curvature may indicate anomalies. These anomalies occur when the curvature at certain nodes or edges significantly devi- ates from expected patterns, signaling unusual structural properties or interactions not detectable through traditional methods. There are numerous real-world instances where
'Camegie Mellon University. Correspondence to: Karish Grover www.hhhwssscccc.u......
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
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Figure 1. Curvature matters. The effect of curvature information on node-level anomaly scores for CurvGAD across Weibo, Reddit, Cornell and Chameleon datasets (Zhao et al., 2020). The plots show anomaly score distributions against curvature values, where the blue region highlights potentially new anomalies due to curvature deviations, and the red region highlights the effect of curvature on existing outliers. W/ Curv refers to anomaly scores calculated without curvature. Observe that interesting anomaly-curvature patterns arise- e.g- in (a), known anomalies exhibit predominantly positive curvature, whereas in (d), they mostly negative. In (h), some non-anomalous nodes exhibit irregular curvature deviations, forming an isolated cluster (i.e. potential curvature anomalies).
curvature can potentially serve as a critical heuristic for de- tecting anomalies in graph-structured data, e.g. (a) Fake News Propagation: Hierarchical cascade-like diffusion pat- terns in social networks, often the result of fake news dis- semination by malicious users (anomalies), exhibit high negative curvature due to rapid branching and sparse con- nections (Ducci et al., 2020; Xu et al., 2021). (b) Biological Networks: Bottleneck proteins in protein-protein interaction (PPI) networks, acting as critical hubs between functional modules, signal disruptions that can affect biological pro- cesses, and have positive curvature (Topping et al., 2021).
spectrum (addresses L3). Using the node embeddings, the decoder applies a Gaussian kernel-based approximation to reconstruct the edge curvatures. Geometric anomalies mani- fest as irregular curvature values, typically associated with a relatively larger reconstruction loss, providing insights into anomalies such as bottlenecks and hubs within the graph.
(b) Curvature-invariant Structure and Attribute Recon- struction. Reconstructs the adjacency and feature matrices, ensuring that the process remains invariant to the curvature of the graph. To achieve this, the input graph first regular- ized by deforming under the discrete Ollivier-Ree Ricci flow, which standardizes the curvature of the graph, converging to uniform value. This allows the subsequent structure and attribute reconstructions to focus solely on non -eoomeeet anomalies. The encoder operates on the regularized graph, while the decoder reconstructs the adjacency and feature matrices. Decoupling geometric irregularities from the non- geometric ones, ensures a dedicated focus on the former (addresses L2). This unified framework improves the ability to detect broader spectrum of graph anomalies. Our key contributions can be summarized as follows.
We propose CurvGAD, novel framework for detecting graph anomalies by integrating geometric insights through a mixed-curvature lens. Intuitively, CurvGAD addresses the limitations of current GAD methods in two key ways: (1) refines the classification of pre-existing anomalies by incorporating curvature information, improving the de- tection of structural or attribute-level anomalies by lever- aging geometric insights; and (2) It uncovers previously undetected anomalies driven by curvature irregularities anomalies that may not be labeled in the original dataset but emerge through curvature deviations. The core idea behind CurvGAD is the decomposition of graph anomalies into two parallel pipelines described below.
(C1) Novelty: To the best of our knowledge, this the first work to study curvature-based anomalies and approach GAD from mixed-curvature perspective.
(a) Curvature-equivariant Geometry Reconstruction. Detects geometric anomalies by learning representations in mixed-curvatun spaces and reconstructing the graph's 's curvature matrix (addresses limitation L2). The encoder em-
(C2) Interpretability: Offers interpretable detection by dis-
entangling curvature survatureiiducedd anomalies from struc- tural and attribute-level irregularities.
(C3) Universality: CurvGAD performs well in detecting geometric, structural, and attribute-based anomalies across heterophilic and homophilic networks.
ploys a mixed-curvature Chebyshev polynomials-based fil- ter bank, which encodes graph signals into representations that adapt to the curvature of the underlying graph topol- ogy (addresses L1). This includes multiple low-pass and high-pass filters operating in a product manifold, ensuring that capture signals from multiple bands in the eigen-
(C4) Effectiveness:
Experimentation with 10 real-world datasets for node-level GAD shows that CurvGAD achieves up 6.5 .55 gain over SOTA methods.
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CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
2. Previous Works
Product Manifolds. A product manifold (Gu et al., 2019) PP is defined the Cartesian product of P constant-curvature manifolds, i.e. P x x M where Mp E { H. S} represents a component manifold with curvature Kip and dimension dz. The total dimension of the product mani- fold is the sum of the component dimensions. The above decomposition of PP is called its signature (Appendix B.
Graph Anomaly Detection. GAD aims to identify ab- normal patterns or instances in graph data. Traditional node-level GAD methods focus primarily on detecting (a) structural (b) attribute-bann based anomalies within graph. Recent advances have introduced proaches for GAD. These methods employ autoencoders to reconstruct graph structures (adjacency matrices) and node attributes, characterizing anomalies with a higher reconstruc- tion error as anomalous nodes or substructures are relatively difficult to reconstruct. For example, DOMINANT (Ding et al., 2019) and AnomalyDAE (Fan al., 2020) use dual discriminative mechanism to simultaneously detect struc- tural and attribute anomalies by minimizing reconstruction loss in the adjacency and feature matrices.
ceastruuctionbased ap-
K Stereogrrrpphic Model. In this work, we adopt the k- Stereographic Model (Bachmann et al., 2020) to define Rie- mannian algebraic operations across both positively and negatively curved spaces within unified framework. This model eliminates the need for separate mathematical formu- lations for different geometries. In particular, .M'dd is the stereographic sphere model for spherical manifold (k >> O), while it is the Poincaré ball model (Ungar, 2001) for hyper- bolic manifold (x < 0) (See Appendix B.2).
Riemannian Graph Neural Networks. Non-Euclidcan manifolds, particularly hyperbolic (Sala et al., 2018) and spherical (Liu et al., 2017) geometries, have proven effective for learning distortion- miniima graph representations. Two main approaches dominate this domain: (a) Single Mani- fold GNNs: Models like HGAT (Zhang et al., 2021) and HGCN (Chami et al., 2019) achieve state-of-the-ae thr- perfor- mance on hierarchical graphs by embedding them in hyper- bolic spaces. (b) Mixed- Saxed-Curvature GNNs: Recognizing that single-geometry manifolds fall short in representing com- plex, real-world topologies, mixed-curvature GNNs embed graphs in product manifolds combining spherical, hyper- bolic, and Euclidean components, Pioneered by (Gu et al., 2019), this idea has been extended by models like k k-GCN (Bachmann et al., 2020), which uses the r-stereograppic model, and Q-GCN (Xiong et al., 2022), which operates on pseudo-Riemannian manifolds. Despite these efforts, GAD methods overlook geometrie information (curvature), spec- tral properties (e.g., heterophily), and fail to leverage Rie- mannian embeddings, limiting their ability to detect more nuanced anomalies that arise in complex graph structures.
Discrete Laplace- Beltrami (LB) Operator. The Laplace- Beltrami operator (Urakawa, 1993) is generalization of the Laplace operator to functions defined on Riemannian mani- folds. We define the discrete Laplace-Beltrami operator, Lp, for graph discretized over the product manifold P, based on the cotangent discretization scheme (Belkin et al., 2008; Crane, 2019). For two connected vertices Us and vj. the off- diagonal element of Lp is: Lp = [cnt 0: +oot 1652 where 0, and dij represent the angles opposite to the edge (i, in the adjacent triangles. and A; is the Voronoi area of vertex o (or Heron's area in the case of obtuse triangles). The diagonal element is computed as: Lppi = - E - LP, This operator highlights the geometric properties by incor- porating the manifold curvature.
24,
Ollivier-Rieci Curvature. In graphs, the lack of in- herent manifold structure necessitates the use of discrete curvature analogs, such as Ollivier- Ricci curvature (ORC) (Ollivier, 2007), which extends the concept of continous manifold curvature (Tanno, 1988) to networks. ORC is defined as a transportation- -baed curvature along an edge, where the curvature between neighborhoods of two nodes measured via the Wasserstein-1 distance (Piccoli & Rossi, 20.6). For an unweighted graph, each node æ is assigned neighborhood measure m == [
3. Preliminaries
Riemannian Geometry. A smooth manifold M general- izes surfaces to higher dimensions. At each point x E M, the tangent space Tx is locally Euclidean. The Rie- mannian metric (x(:))) TxMM x T&M + I equips TxM with an inner product that enables the definition of distances and angles, forming a Riemannian manifold (Do Carmo& & Flaherty Francis, 1992). The exponential map expx(v): T&M + M maps tangent vectors to the man- ifold, while the logarithmic map lygx(y)= .M- + TxM maps points back to the tangent space. The curvature (k) at each point describes the geometry, with three common types: positively curved spherical
1-6 for 2 E (xx).
and m z. (z) = 8. ORC for an edge (x. 3) is then calculated as: Rey := 1 Wi (m .n
dz(z.g) We approximate ORC in linear
time using combinitorial bounds (Jost & Liu, 2014). See Appendix B.3 for details on computational considerations. We denote the continuous manifold curvature using k and the discrete Ollivier-Rica curvature using K.
Ollivier-Ricci Flow. The Ricci flow, introduced by Hamil- ton (Chow et al., 2023), is a process that smooths the cur- vature of a manifold by deforming its metric over time. In the graph domain, this concept is adapted to the discrete Ollivier-Ricci flow. In each iteration of this evolving process,
(S) (K > 0), negatively curved hyperbolic (H) (I < 0), and flat Euclidean (E) (ı- = 0).
3

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CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
Ricci flow generates time-dependent family of weighted graphs (V. E, w()) such that the weight way on edge zy changes proportionally to the ORC of the edge ay at time (u()) t, Kay. Ollivier (Ollivier, 2009) defined the Ricci flow for (t) continuous time as at W zy = - W zy Then (Ni al., 2018) proposed Ricci flow for discrete time & (for step size my (++1) = (1 ( way- For an unweighted graph, as: Wa
Spherica (a > 0)
Hyperbolic (x 0)
Eaclidean (* 0)
(t)
Cuzvalure Eqsivarinnt Ressunicnclion
(Product Maaniiold)
Original Graph
=1, 1, y E V. Typically, ORC for all edges con- verges to zero curvature as 1. + 00.
(0)
Curvature
ayy
Eacodes
Decodee
Setup. We consider graphs G = A), where V is a set of [ = n vertices, & is a set of edges, and A E R is the adjacency matrix. The nodes are associated with the node feature matrix X E Raxdx (dx is the feature dimension). A graph signal x V + I may be re- garded as vector X R" where H, is the value of x at the 7th node. The Laplace-Beltrami Beltrami operator Lp (Sec- tion 3) extends the notion of the traditional graph Lapla- cian to mixed-curvature spaces by incorporating geometric clues, As a real, symmetric, and positive semidefinite ma- trix, Lp admits a complete set of orthonormal eigenvectors Up- [(w]]=]]
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X, MLP(o,)
R2xn, and their associated ordered real nonnegative eigenvalues {{))]=0) E IR", identified as the frequencies of the graph. Similar to the graph Lapla- cian, the Laplace-Beltrane matrix can be diagonalized as
Regularized Graph
(Buclidesa Mantfold)
Mixed-curvattree Clacbyshev Eucodor
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z
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4. Proposed Approach: CurvGAD
z 91
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In this section, we provide an in-depth exploration of the CurvGAD architecture. We first introduce the curvature-equivariint pipeline (Section 4.1), which recon- structs the curvature matrix to detect geometric anoma- lies. This involves mixed-curvature encoder (common to both pipelines) equipped with spectral graph filters (Section 1.). followed by a Gaussian kernel-based de- coder (Section 4.2. that predicts curvature values. Next, we extend our framework
Tiihhpa
(2 1 Lp Ba,, 1-13) Me
N
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Figure 2. Architecture of CurvGAD. The proposed model employs two synergistic pipelines for anomaly detection: (1) Curvature- Equivariant Reconstruction embedding the input graph into a mrixed-curvatie product manifold using spectral Chebyshev filter bank, with a Gaussian kernel-based decoder reconstructing the curvature matrix to capture curvature irregularities; (2) Curvature- Invoriant Reconstruction - regularizing the graph under Ollivier- Ricci flow to uniformize edge curvatures, followed by Fuclidean manifold- based encoder to learn representations for the decoupled reconstruction of adjacency and feature matrices.
survature-invarian anomaly de- tection (Section 4.2) by deforming the graph under Ollivier- Ricci flow (Section 4.2. thereby regularizing curvature distortions. While the encoder remains identical, we replace the product manifold with a Euclidean manifold to ensure that structural and attribute anomalies are reconstructed in- dependently of curvature.
En dp. We determine the task-specific signature of p by examining the distribution of the Ollivier-Ricci curva- ture within the graph and identifying the most significant curvature bands (see Appendix F). Next, we project the original Euclidean input node features onto the mixed- curvature manifold as X' = expo" (fo(X) where So()) : Rdx -> Rd represents neural network with pa- rameter set {0} that generates the hidden state euclidean features of dimension dp. Here, cxpo" Fpp : Rdd + M is the exponential map (Section 3) to project the node fea- tures X the th manifold. The projected node features are
4.1. Curvature-equivariint Reconstruction
Product manifold Construction. The product manifold P can have multiple hyperbolic or spherical components with distinct learnable curvatures. This allows us to accommo- date broader spectrum of curvatures. Consequently, the product manifold can be succinetly described as par (x H H .d..d)) x (x__) (k) x Edd,
x XP_I
with total dimension of dp Σ dr 1 ds
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CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
utilized in the encoder to learn node representations, which subsequently used to reconstruct the curvature.
Here, N is the unfiltered case, where we pass the identity matrix I instead of Lg. This approach ensures that the filter bank captures both high-frequency (heterophilic) and low- frequency (homophilic) signals within the mixed-curvature manifold, enabling CurvGAD to generalize effectively across heterophilic and homophilic graphs.
4.1.1. MIXED-CURVATURE CHEBYSHEV ENCODER
Once the input features are projected into the product mani- fold, we introduce a filterbank of Chebyshev approximation- based spectral filters. Let & represent the graph filter op- erator, the filtering operation for a signal x is defined as:
Final Representations. Filtered node representations are aggregated hierarchically to synthesize information across both spectral filters and manifold components. Unlike naive concatenation, this aggregation mechanism assigns learn- able importance weights to both filters and manifolds:
v6 (Lp)) = 1 (UAAA x = UA+(AA) x. (1)
Chebyshev Approximation. Direct eigen- decomposi- Beltrami operator L is computationally prohibitive, especially for large graphs. To address this we employ Chebyshev polynomial approximation of the filters in the mixed-curvature space. Recall that the Cheby- shev polynomial z() of order f may be computed by the following stable recurrence relation:
tion of the Laplace-Belrrami
nj =
M
Bp
Bay
- f=0
a Bxp (N ()
m==
Here is the concatenation operator and nj E TPde is final node representation for node j. 3n and oy are learnable weights which assert the relative importance of the pd com- ponent manifold embedding and fth filter.
(x) = (z) - 2 (z),
with _ = 1 and 2 x. In our framework, we gener- alize this recurrence to accommodate the geometry of the mixed-curvature manifold by replacing the standard opera- tions with their curvature-awary counterparts. The filtering process on the 7dhh manifold proceeds as follows:
4.1.2. DECODER FOR CURVATURE RECONSTRUCTION
The decoder aims to reconstruct the Ollivier-Riie Ricci curvature Czy for nodes x and y. by leveraging their latent embed- dings nz, ny E TpdM, learned through the mixed-curvature encoder. We propose using a Gaussian kernel defined on the manifold distance between the node embeddings.
z (0)
Marnt expo Ap (X)( (Feature mapping)
(3) Definition 4.1 (Curvature Decoder). Let Da ny E ppd
(()
Morro = Lp BR expo Kr (X)(
(4) be the latent embeddings of nodes and y. The predicted curvature Caz is defined as: Cay = 2 (5) sigmoid (1 - K(nn, ny)) - 1, where K(nz, ny) is a Gaussian kernel: K(nn, ny) = exp (-> "mmiinnn)))
z(1)
MMpiie (2 2x, Lp Br z ((111) Manile)
Ox2 (2)2
Ma
((F)
z(1)
65 Bky M2
and (n,, V EMM DMI, (n,,ny) aggregates geodesic distances D DM,, over manifold components. Here, 7 is a fixed kernel width and T is a sensitivity parameter.
Monid
(E(O,F)
Observe how Equations 3 to generalise the Chebyshev recurrence in Equation 2 to the mixed-curvature setting, for the t component manifold. Further, as detailed in previous section, the Equation 3 transforms the original Euclidean node features X to the product manifold. Finally, in Equation 6 the filtered signals from different Chebyshev orders (till F) are aggregated to form the final node rep- resentation on the manifold p. Here, Dx, ON. 2x and & denote mobius addition, mobius subtraction, m-riiht- multiplication and x-left-matrix-ii mat--mmattiiiiiieea mutttiiiimuliiiiaatiion respectively (Appendix B.2). These operations generalize vector algebra to k- stereographic model. di are the learnable weights.
See Appendix D.I for more intuition. The decoder mini- mizes the loss of the Frobenius norm: Le = ||C C where C and C denote the predicted and original curvature matrices, respectively. This objective ensures accurate cur- vature reconstruction while adapting to geodesic distances.
4.2. Curvafure-invariant Reconstruction
This pipeline focuses on reconstructing the adjacency matrix A and the feature matrix X independently of the graph's underlying geometry (curvature). To achieve this, the graph is first deformed under the Ollivier-Rieci flow, followed by curvature- invariiat encoding and decoding.
Filter Bank. To effectively capture spectral informa- tion across diverse graph frequencies, we adopt the above construction to get a filter bank of multiple spectral filters:
1.2. RICCI FLOW AND CURVATURE REGULARIZATION
INI
Monddy (1) Mardd (T)
Ollivier- Riciii flow is applied to the original graph to regu- larize edge curvatures by iteratively updating edge weights
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CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
based on their curvature values. This process, outlined in Algorithm 4.2.1, transforms the graph into constant- curvature space, thereby neutralizing curvature-induced dis- tortions. Refer to Figure 3 for better intuition.
Definition 4.2 (Adjacency Decoder). The predicted adja-
ceney value Azy for nodes x; y is modeled as the likelihood of an edge existing between them: p(Azy 1 nay ny) a (n.. ny), where nz. By E E are Euclidean latent embed- dings, and (.) is the sigmoid activation.
Definition 4.3 (Attribute Decoder). The reconstructed fea- tures for a node are given by: X₂ = faee (n,),, where fas-x is multi-layer perceptron (MLP) mapping latent embeddings n2 E R to feature space.
Oliivier Rioci Flow
Existing Attzlbute Stracture Outliers
Upelaled Edge
The decoders are trained using reconstruction losses: LA A A for adjacency and Lx = =|X X| | for features.
Weights
Deviations
Glbvie-RRood Currottore
4.3. Objective Function
Figure 3. Intuition on Ollivier-Riet flow. Curvature regular- ization of Karate Club graph (Rozemberezki al., 2020) under Rliivi flow. Observe how the flow increases (decreases) the weight negatively (positively) curved edges. Red nodes indi- cate curvature- based outliers near edges with extreme curvatures.
The objective function balances curvature reconstruction, structural and attribute reconstruction, and supervised clas- sification loss to detect both geometric and non non-geometric anomalies. The total loss is:
Liotal = ActsCols + (1- Aas) (Acfe + AACA Xx
Algorithm Discrete Ollivier-Rice Flow
where Cc, denote reconstruction losses for the cur- vature, adjacency, and feature matrices, while Lass represents the cross-entropy classification loss. The anomaly score for each node is computed as a weighted sum of reconstruction errors:
Require: An undirected graph g (V, small thresh- old A >> 0, learning rate >> 0
Ensure: Edge weights of G as Ricci flow metrics
Score, Xc ß -c, 2 +A (ā - az + [ā, xz 22- where G ã; 2i are the reconstructed curvature, adjacency, and feature values, respectively. Learnable tradeoff parame- ters Ax, Àc, AA, Acts dynamically adjust loss contributions. The final anomaly detection integrates this score with classi- fication logits, ranking anomalies based on both geometric and non-geometric deviations. An ablation study evaluating the impact of Las and Lc is presented in Section 5.5. In the following section, we lay out the empirical results to validate the efficacy of CurvGAD.
1: Initialize edge weights (0) for all (x, v) E
ay
2: repeat
3: Normalize edge weights:
(1) ay
18/-1(1)
Enoies w!y
4: Compute ORC: Kiy W1 (mm2.m) m
D (2.w)
5: Update edge weight under Ricci flow:
1++1) w
w (t) my 2y
e ((()
((())
2y
6: Check convergence:
(1++1)
Kuy
Kay
(t)
A
7:
until convergence condition is met for all edges
5. Experimentation
5.1. Datasets
4.2.2. CURFORM-CURVATURE RECONSTRUCTION
We evaluate CurvGAD on 10 datasets, each containing or- ganic node-level anomalies, assess its effectiveness across homophilic and heterophilic settings. These datasets span multiple domains, including social media, e-commerce, and financial networks. Specifically, (1) Weibo (Zhao et al., 2020; Liu et al., 2022), (2) Reddit (Kumar et al., 2019; Liu et al., 2022), (3) Questions (Platonov et al., 2023), and (4) T-Social (Tang et al., 2022) focus on identifying anomalous user behaviors on social media platforms. In the context of crowdsourcing and e-commerce, (5) Tolokers (Platonov et al., 2023), (6) Amazon (McAuley & Leskovec, 2013; Dou et al., 2020), and (7) YelpChi (Rayana & Akoglu, 2015; Dou et al., 2020) are used to detect fraudulent workers, reviews, and reviewers. Mcanwhile, (8) T-Finance (Tang et al., 2022), (9) Elliptic (Weber et al., 2019), and (10)
Under Ollivier-Ricci flow, the edge curvatures of the regu- larized graph approach near zero, allowing reconstructions to operate within a Euclidean manifold. We reuse the Cheby- shev filterbank encoder (Section 4.1.1) to map the nodes in the curvature-regularizee graph (Section 4.2.1) to latent Euclidean embeddings, ensuring curvature invariance in the learned representations. Unlike the mixed-curvature en- coder (Section 4.1). this encoder simplifies the geometric complexity by restricting operations to a Euclidean space.
Decoders. We employ separate decoders for adjacency and feature reconstruction tasks. These decoders ensure that structural and attribute information is effectively recovered without being influenced by curvature irregularities.
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We conduct experiments in transductive, supervised setting for node-level graph anomaly detection, following standard data splits where available. If splits are not provided, we adopt the strategy from (Tang et al., 2022), partitioning nodes into 40%/20%%/400 /40% for training, validation, and test- ing, as detailed in Table 6. To ensure robustness, we perform ten random splits per dataset and report the average perfor- mance. As per established anomaly detection benchmarks (Han et al., 2022; Liu et al., 2022), we evaluate models using CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
Model
Reddit
Weibo
Amazon
YelpChi
T-Finance
Elliptic
Tolokers
Questions
DGraph
T-Social
Av. Gain
GCN
65.25++a0
97.03+ta57
82,48+043
57.81 toe
92.41+235
81,42+-ass
75.63++12
70.93+134
75.98+a2s
80.21
10.02
GAT
65.38
95.35-1147
96.75+1.5
19.
92.81:119
84,93
19.18:1106
71.17:143
75.96
75.42
6.29
GraphSAGE
62.86.t
94.94.11m
89,651sa
85.25
82.89
85.35
79.32.1n94
72.15
79.29 75.63.tmm
7.20
BernNet
65.82.1175
94.96.1.56
96,21
83.07
Last
92.81
83.18.1..
76.95.Luss
68.82.1295
73.25.1am
66.85_1ssa
7.74
ChebyNet
64.11+144
95.02+tu9
96.59+201
80.23+t21
93.12+t098
39.34++307
19.81+114
69.23+11s
74.09++144
+8.11
7.97
HGCN
64.13:
94.23-
79,94
76.46
92.32::41s
78.6642222
75.78:t101
67.16:2174
71.27
49.43-+422
10.00
HIGAT
63.45.Luzs
92.24.Lu9s
85.98.L
81.19 Liss
92.86.Laa5
85.45.L..2
76.34.1
68.93.L127
72.45 La.s
71.13
8.93
KGON
61.77:1134
97.87-1124
97,34:1119
83.76.11.44
92.54.1434
76.13
79.64-1007
73.155..34
78.02:1.44
85.65-13.1
5.35
QGCN
66.45-t122
97.34:tu.4
96.71
13
92.23+t-0s
85.55+
11.46:t019
71.74
18.455t1
43.32
5.30
DCI
66.5443338
94,25--173
94.63+ta2
77.89
86.81=
82.86+
75.52+t0..
69.21
74,74
80.83-t6.04
7.61
PCGNN
53.2512219 90.27:1151
97,31
79.76.11.77
93.34.110
85.89
72.83.12077
69.9111143
72.08.iax
69.27.11.9
9.54
BWGNN
65.43:::35
97.38:t09
97.05=ta77
84,96.2733
94,181:.55
15.2
80.43:1:054
71.85#222
76.341:044
4.43
DOMINANT
62.32:1205
87,43:12m
76.23:1xw
65.03.14.
19,43.1121
79.34:1usa
76.43.1u9s
64.92.1124
70.76.1.
76.09
13.14
AnomalyDAE
65.75+:16
93
91.99:t007
81.34
91.33+t002
83.69+0..9
17.43+t100
62.45
778. 18.66+11%
82,22
7.09
GADNR
67.01 96,92:1mm
93.46.1203
79.241:264
95,9911153
82.57
76.43.1165
70.11
75.03-1am7
85,64
5.69
ADAGAD
66.45-tow
98.01
96.05tt:.9
88.777+1.m
92.01-t211
86.01-toas
80.99tto..
71.06.1104
75.99-t2.5
84.54+22m
4.25
CurvGAD
70.42
99.04.l.3
99.62
89.33
98.13
9
85.22.1om1
74.45
83.77.1244
89.23
A Imp.
5.09%
1.05%
2.34%
4.15%
2.23%
4.79%
4.61%
1.78%
6.50%
4.18%
Table CurvGAD wins. AUROC Score (Mean + 95% Confidence Interval) for all baselines vs. CurvGAD. First best performing models for each dataset have been highlighted. Av. Gain shows the average absolute improvement of Cur over each model across all datasets, and A Imp, indicates the percentage gain of CurvGAD over the second best for every dataset.
Second and Third
DGraph-Fin (Huang et al., 2022) are employed to identify fraudulent users and illicit activities financial networks. Refer to Table (Appendix E. E. for dataset statistics.
the Area Under the Receiver Operating Characteristic Curve (AUROC). ORC is computed with = 0.5, distributing equal probability mass between node and its neighbors. Given the prohibitive cost of exact ORC computation on large graphs, we employ linear-time approximation via combinatorial bounds (Appendix B.3). The manifold signa- ture is decided heuristically using the ORC distribution of the datasets (Algorithm 2, Appendix F). For optimization, we leverage the k-stereographic product manifold imple- mentation from Geoopt' and use Riemannian Adam for gradient-based learning across product manifolds. All ex- periments are conducted on A6000 GPUs (48GB), using total manifold dimension of dp = 48, a learning rate of 0.01, and filterbank comprising F = 8 filters. Appendix E.2 enlists the hyperparameter configurations tried and we analyse the time complexity of CurvGAD in Appendix C.
5.2. Baselines
For a fair comparison, we evaluate CurvGAD against four types of baselines: (a) Conventional, including tradi- tional models such as GCN (Kipf& & Welling, 2017), GAT (Veličković et al., 2017), and SAGE (Hamilton et al., 2017). (b) Riemannian, comprising (i) Constant-curvattre GNNs
like HGCN (Chami et al., 2019) and HGAT (Zhang et al., 2021), and (üi Mixed-curvature GNNs such as xGCN (Bach- mann et al., 2020) and QGCN (Xiong et al., 2022). (c) Spectral, including ChebyNet (Defferrard al., 2016) and BernNet (He et al., 2021). (d) Specialized GAD models, including (i) Reconstruction-bee based approaches like DOM- INANT (Ding et al., 2019) and AnomalyDAE (Fan et al., 2020), and (ii) GNNs like DCI (Wang et al.. 2021), PC-GNN (Liu et al., 2021), BWGNN (Tang et al., 2022), GADNR (Roy et al., 2024), and ADAGAD (He et al., 2024).
5.4. Baseline Analysis
CurvGAD consistently outperforms all baselines across all datasets, achieving the highest AUROC scores with an improvement of upto 6.5% over the second-best model (Table 1). Significant gains are observed on T-Social (+4.18%), Elliptic (+4.79%) and Tolokers (+4.611), all heterophilic networks, highlighting CurvGAD's ability to detect complex anomalies beyond homophily-bused as- sumptions (addresses limitation L3). Among specialized GAD models, ADAGAD and GADNR perform well on ho- mophilic datasets like Reddit and Weibo, ranking second third best in many cases. However, their performance deteriorates on heterophilic datasets, where curvature-
5.3. Experimental Results
7

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CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
AUROC
Reddit
Weibo
Amazon
YelpChi
T-Finance
Elliptic
Tolokers
Questions
DGraph
T-Social
OurvGADew.or
33.34::3.00
97.55
94.65
84.34
16.43--184
86.97-00.33
79.45-20.99
72.45-t0.35
79.94
33,54:3345
CurvGADequa
68.75:14.03
98.75:00.57
98.34:1:0.77
15.94
9..8711.23
87.4810066
1..5411001
71.32:10.66
81.37
83.56:12.88
CurvGAD (wer
62.03
93.45=00.47
92.76=t0.45
82.18+1224
96.75+11.24
84.65+0.755
81.244+0.77
67.55++0.67
78.76=2.77
84.75
CurvGAD flow
66.48.433.24
08.77=0.84
97.46:::0776
83.26++2344
34
1..46+1155
88.46::0.34
74.45-t0:46
71.74::0.55
80.45+3.01
86.34+3369
CurvGADunap
68.34.L.0.99
98.63.10.76
9..01
87.37
97.22.1.0.6
88.32.10.09
84.01 0..58
73.05.10.99
88.88 81.05_12.01
CurvCAD
70.42.11.03
99.04.10.34
99.62.1..11
89.33
98.13.11.34
90.13.10.99
85.22.1..11
74.45.10.04
83.77
89.23.13.89
Table 2. Ablation Study on CurvGAD Variants - AUCROC Score (Mean + 95% CT). Best performing scores per column are in Bold.
Cell shading reflects performance degradation relative to the best variant per dataset, with a darker Gray indicating higher degradation.
AUROC
Reddit
Weibo
Amazon
YelpChi
T-Finance
Elliptic
Tolokers
Questions
DGruph
T-Social
E224 g21
10,42:tt1.03
99.04=10.24
96.62=0.15
17.56
96.4441..45
85.45 #0.95
83,22=0017
67.56=0.45
75.34-2.3
88,45=3.05
(H#)² x (8 x giG
66.43-1-0.98
97.04-1046
9.662
85.66
1.08
9..56:11.334
88.65
84.56 18
71.86:00.17
82.46=1.87
84.97=345
E2 x S x B
68.
97.01
95.55
89.33 1.44
1.37
87.36::0.45
82.76:t0.18
70.56:00.07
80.35
37.56=33355
E/G x (3
1.1
96.11=00655
07.39=0019
1.56
9..76:21.31
86.75
76.65:20.19
72.76=00.19
81.76=2.01
89.23-3889
(2 x EiG
69.111
97.43-+0.33
96.01=0.188
87.35=1143
1..13=:1.34
87.65
78.67=:0.24
73.65=00.65
83.77-12.44
85.76=4.34
H24 x 1ga1
68.24:11.99
95.12:20.76
94. 11 0.65
1.11
97345
86.75
83.24110445
0.04
83.01=1.00
87.45=2.54
524 x g24
1.06
96.99:t0.65
84.35 97.62=0018
1.06
26.16:t22.38
87.6565:0.66
85,22:+0111
01.43:00.08
75.23=2.23
86.56=2666
1 326
x 1G
69.11++100
94.35-+0.31
95.61
88.65-+0.99
93.57:10076
90.1331:0.99
83.46:20.97
68.67=00.03
80.65=2.04
84.09=1199
Table 3. Ablation Study on Product Manifold Signatures AUCROC Score (Mean + 95% CI). Best performing signatures are in Bold.
Cell shading reflects performance degradation relative to the best signature per dataset, with darker Gray indicating higher degradation.
models like rGCN and QGCN perform better, suggesting that explicitly modeling curvature aids anomaly detection in such settings. Reconstruction-basse approaches like DOM- INANT' and AnomalyDAE, perform worse than dedicated GAD approaches, indicating that pure structure and attribute reconstruction alone may be insufficient for anomaly detec- tion. These models fail to capture deeper geometric irreg- ularities, reinforcing the necessity of integrating curvature information. HGCN and HGAT, despite leveraging hyper- bolic geometry, underperform compared to CurvGAD, as they operate on a constant curvature space, limiting their adaptability to datasets with mixed geometric properties.
drop is minimal (-0.27% which can be attributed to the low curvature variance in the ORC distribution of Weibo.
(c) Manifold signature analysis. The optimal curvature composition varies across datasets (Table 3). Reddit and Questions favor hyperbolic embeddings, while Amazon and YelpChi perform better with increased Euclidean capacity. This validates the necessity of nixed-curvature manifolds for adapting to diverse graph structures. To quantify the contribution of individual geometry types, we further con- duet ablations by removing each component in turn, e.g. H x $21 (E removed) and $24 122 (H removed).
(d) Curvature-invarian pipeline only (CurvGAD,nwr)) This variant omits curvature reconstruction, relying solely on structural and attribute reconstruction (removes Lc). While it still surpasses most baselines, performance declines significantly (-8.39% on Reddit, -6.86% on Amazon), confirming that structure and attributes alone are insufficient for robust anomaly detection (addresses L2).
5.5. Ablation Study
To assess the contribution of individual components in CurvGAD, we evaluate multiple ablations (Tables 2, 3).
(a) Euclidean-only variant (CurvGADeua).. This model learns purely in Euclidean space, discarding mixed- curvature embeddings. The performance drop (~ 5%) highlights the necessity of mixed-curvature modeling (addresses L1). Notably, degradation is more pronounced on curvature-sensitive datasets like Reddit ( (-7.0%% and T-Social (-5.6%%) but is minimal on T-Finance (-1.7%), where Ollivier- Ricci curvatures are predominantly zero.
(e) Curvature-equivariin pipeline only (GurvGADequi). This model detects only geometric anomalies, omitting curvature-invarian reconstruction (removes LA, Lx). It outperforms CurvGAD;ner (+6.72% on Reddit, +4.89% on Amazon), highlighting how effective the curvature- equivariant pipeline is in itself. Both pipelines compliment each other, and achieve the best results together.
(b) Without Ricci Flow (CurvGAD flow). Removing Ricci flow prevents curvature-invariant reconstruction, degrading performance on curvature-sensitive graphs (-10.77% on Tolokers, -3.94% on Reddit). This confirms Ricci flow's role stabilizing curvature variations and enhancing struc- tural and attribute reconstruction. Observe that in Weibo, the
(f) Unsupervised variant (CurvGADunsp). Removing the supervised classification loss Lats results in minor per- formance drop ( 1.5% avg.), demonstrating that CurvGAD remains effective even in an unsupervised setting, making it adaptable to woal-world scenarios with limited labels.
8

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CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
These results validate the independent contributions of each component, showing that aurvature-awarent curvature- invariance, and Ricci flow collectively enhance anomaly detection in both homophilic and heterophilic graphs.
geometric anomalies, which is important in high-stakes applications where decision-making requires model expla- nations. In scientific domains including network science, biology, neurology, and transportation networks, where curvature-driven representations may aid in identifying anomalous patterns in actual systems, the knowledge gained via CurvGAD might be advantageous.
6. Conclusion
In this paper, we propose CurvGAD, the first approach to in- troduce the notion of curvature-based geometric anomalies and to model them in graph anomaly detection through mixed-curvature perspective. Our dual-pipeline architecture integrates curvature-equiivaiiat and curvature-invarianl re- construction, capturing diverse anomalies across homophilic and heterophilic networks. CurvGAD disentangles geomet- ric, structural, and attribute- anomalies, making in- terpretable. Extensive experiments across 10 benchmark- ing datasets demonstrate the superiority of CurvGAD over existing baselines. These results establish curvature as fundamental tool for GAD and pave the way for future advancements in geometry-aware graph learning while of- fering new avenues for exploring geometric anomalies in topologically complex networks.
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CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
A. Appendix
A.I. Notation Table
Notation
Reference
G= (V,E,A)
graph G with set of edges & vertices (nodes)
M:
A smooth manifold; p" component manifold in the product manifold (Mp E {E, H))
Kp. dp
Curvature and dimension of p" manifold component
expo R M
Exponential map, to map from tangent plane (Euclidean) to the product manifold Logarithmic map, to map from the manifold to the tangent plane Hyperbolic manifold; Spherical manifold; Euclidean manifold Product manifold of dimension dp
logø .M-- + Rd
II; S;
pdop
K;
Continous manifold curvature; Discrete Ollivier-Rice curvature (ORC)
Kry: Kay
ORC edge {a, w). ORC of edge (a. tth step of Ricci flow algorithm Probability mass assigned to node for ORC computation
m
8
Probability of the mass retained v/s distributed among neighbour nodes for ORC Computation
(xx)
Neighbourhood set of node æ This implics that x and y are adjacent nodes Wasserstein-1 I (Earth mover's) s) distance
x ~
Wi(.)
d)(x,y)
Shortest path (graph distance) between nodes and y on graph
Ma
Constant-curvature manifold with dimension dp and curvature Ky. Mp (H,8,E) Discrete Saplace-Beltrani operator
L
U. = [ud)o
Set of orthonormal eigenvectors of Lp Ordered real nonnegative eigenvalues associated with U+ Ollivier-Riie Ricci flow step size
[( R
Weight of the edge zy the th iteration Ricci flow algorithm.
dx
Input graph feature dimension
dp
Total dimension of the product manifold
X Rnxdx
Input feature matrix Mobius addition
Cx
8x
R
citt-marrrx-multipiiscee
n
Final (latent) node representation for node after encoder
A; A
Initial graph adjacency matrix, Reconstructed adjacency matrix Initial graph feature matrix, Reconstrueted feature matrix Initial edge curvature matrix, Reconstructed curvature matrix Curvature decoder reconstruction loss Structure decoder reconstruction loss Feature decoder reconstruction loss
X; X
C; Č
Le
La
Lx
La
Binary crossentropy supervised classification loss Weight of classification loss in total loss Geodesic distance over ph manifold component
dd
DM,
Du(ns ny)
Aggregated distance on the product manifold (i.e. sum over all component manifolds)
ng) ng)
Gaussian kernel used in curvature decoder over node embeddings n., ny
8,
Learnabee attentation weight for nh component manifold
as
Learnable attentation weight for f filter in the filterbank
Fif
Total number of filters the filterbank; Used to denote the 8th filter
P:P
Total number of components in the product manifold; Used to denote the th component Learnable trade-off parameters for loss components Sigmoid activation function Stopping criterion threshold for Ollivier-Ricai flow Graph filter operator
Ax. Ac. As. Act
(.)
db
So()
Neural network (MLP) that generates hidden state Euclidean features of dimension dp A graph signal V + IR
y
Kernel width hyperparameter for gaussian kernel (curvature decoder) Sensitivity parameter (Icarnable) for gaussian kernel (curvature decoder) Intermediate node representation from fth filter
(0
Ti(2)
Chebyshev polynomial order Diagonal matrix of cigenvalues Learnale weights for filters, for the th recursive level in Chebyshev encoder.
A
d1
12

---

CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
B. More on Preliminaries
B.1. Product Manifolds
Consider set of smooth manifolds M1, Mg...., Mp, each equipped with its own Riemannian structure. The product
manifold formed as their Cartesian product:
P Mi x M2 x x Mp.
(7)
This construction enables the representation of a point x E P as a tuple:
(01, T2, ................................ ,,.,,,,,,7 pp))
(8)
where each Tp corresponds coordinate in the individual component manifold MM.
Tangent Space and Geodesic Movement. For any given P, the associated tangent space is given by:
T P T Mi Mi x Tay M2
T Mp.Mp.
(9)
tangent vector IP is similarly represented as:
(11, 02 ..... Up),
(10)
where each Up T Mp represents the directional component along the corresponding manifold Mp-
Since optimization over Riemannian manifolds requires movement along the manifold surface rather than within the tangent space, we employ the exponential map, which maps tangent vector v at x back onto the manifold:
expx TxP
(11)
For product manifold, this mapping decomposes across individual components, leading to:
expx (v) = (exp,, (v1), expx, (o2),.... expxp (vp)).
(12)
This property allows optimization steps to be performed independently along each manifold component, significantly simplifying Riemannian optimization in mixed-curvature spaces. We use the Riemannian Adam optimizer from Geoopt pestps://githnn
www.opt Library for all our experimentation.
Curvature and Manifold Selection. The choice of component manifolds in I significantly influences model performance. Hyperbolic spaces (H) are well-suited for hierarchical structures, spherical spaces (3) capture cyclical relations, while Euclidean spaces (E) are ideal for line graphs (and provide standard representation). The flexibility of mixed-curvature embeddings allows for adaptive modeling across datasets with diverse underlying geometries.
Remark B. CurvGAD heuristically determines the optimal mixed-curvature manifold configuration (Algorithm 2, Appendix F) for given dataset, ensuring that embeddings conform to the dataset's intrinsic geometric properties. Remark B.2. The curvatures of the manifold components are dynamically learnt during the training process.
By leveraging product of Riemannian manifolds, our approach effectively models complex graphs where curvature varies locally, leading to superior anomaly detection across diverse graph topologies.
B.2. Stereographic Model
To effectively model mixed-curvature embeddings, we adopt the s-stereograpphic model (Bachmann et al., 2020) instead of adopting separate formulations for hyperbolic and spherical spaces. This unified framework provides consistent mathematical formulation across varying curvature regimes, enabling smooth interpolation between hyperbolic (x < 0), spherical (N > 0), and Euclidean (x 0) geometries. Efficient algebraic operations are essential for optimizing over non-Euclidean spaces, and the k-stereographic model offers closed-form expressions for key Riemannian operations.
Manifold Definition. The s-stereographie manifold of curvature x: and dimension is defined as:
kr.d_ = {z Rdd - =1|z|| <
(13)
13

---

CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
Operation
Formalism in E
Unified formalism in r-stereographic model (H 8
Distance Metric
dx((,,) = ||x y122
tan,''
|| @D
yll
Vlll
Exp. Map
exp =
exp= tr
tank
|5
Viillvila
Log. Map
1og y
1og V ml12
Lam.
[c]
2nx y 1
Iyll
|
Addition
x x = = x
x Gx
1 2nxx
Table
Operations in Hyperbolic H Spherical s and Euclidean space B
where z E R represents point in the stereographic space, and the curvature parameter governs the underlying geometry: A < 0 corresponds to hyperbolic manifold (H > 0 corresponds to spherical manifold (8") and x = 0 reduces to Euclidean space (E*). The Riemannian metric tensor associated with M at point z is:
g = I,
(14)
where the conformal factor X is defined as:
X = 2 + x1|z
(15)
This ensures smooth transition between different curvature settings.
Algebraic Operations. To perform optimization and inference on the manifold, we require efficient algebraic formulations for fundamental operations such as vector addition, scalar multiplication, and distance computation. Table 4 summarizes the key operations under the x-stereographic model.
B.2.1. -RRRIITAAATRN---ULPPPIITTTT (BACHMANN AL., 2020)
For matrix E nx carrying x-stereographic embeddings across its rows, and weights represented by W E gaxe, the operation of Euclidean right multiplication be decomposed into individual rows (XW): X, W. Consequently, the m-rrigh-mmarrx- muliiiiimulilliiaaiionn is defined in the same row-wise fashion
(X X W): expo (llgg (X)
(16)
(XW): :
tan, (ax tan X(:ll)
(17)
[XW) and expă and logă denote the exponential and logarithmic map in the k-stereographic model. X.
where a;
B.3. Ollivier-Ricci Curvature (ORC)
In an unweighted graph, the set of nodes adjacent to given node x, denoted by (z), J is represented through a probability distribution following a lazy random walk model (Lin et al., 2011). This distribution is defined by the equation:
6,
if x,
(z) =
1-5
if e((r),
(18)
[ 0,
otherwise.
The parameter 8 determines the probability of staying the current node, with the rest of the probability (1 6) being evenly allocated among its neighbors. This approach links ORC with lazy random walks, affecting the interaction between local exploration and node revisitation. In our study, we set 8 = 0.5, resulting in equal division of probability between the node and its neighbors, creating a balance between breadth-first and depth-first search methodologies. The value of is significant and varies based on the graph's structure. A lower 5 value promotes more exploration within the local neighborhood, whereas a higher ó fosters node revisits, supporting the "lazy" aspect of the walk. For the experiments, 0.5 was chosen to ensure an equal division of probability between the node and its adjacent nodes.
14

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CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
C. Time Complexity Analysis
C.1. Preprocessing Complexity
C.I.1. COMPUTATIONAL CONSIDERATIONS FOR ORC
A crucial preprocessing step in CurvGAD is the computation of the Ollivier-Rieci Curvature (ORC) and subsequent Ricci Flow regularization. These computations serve as the foundation for both the curvature-invariat and curvature-equivarinn pipelines, making their efficiency paramount to overall scalability. We precompute both these to save computation time as they are used in both the CurvGAD pipelines.
Computing ORC involves determining (in this case, approximating) the Wasserstein- distance (W1) between the probability distributions of neighboring nodes. In discrete settings, this requires solving an optimal transport problem. The standard approach relies on the Hungarian algorithm (Kuhn, 1955), which operates in O(d)) time per node, making it computationally prohibitive for large graphs
To mitigate this cost, the Sinkhorn algorithm (Sinkhorn Knopp, 1967) is often used as a relaxation, reducing complexity (rd)) per node. However, this still remains impractical for million-scale graphs. In this work, we adopt linear-time approximation for ORC, allowing CurvGAD to scale efficiently while retaining interpretability.
C.1.2. APPROXIMATING ORC IN LINEAR TIME
Instead of directly solving the optimal transport problem, we employ combinatorial approximation for ORC, inspired by Tian et al. (2023). This approach estimates Wasserstein distances using local structural properties, significantly reducing computational overhead. The approximation is derived from the analytical curvature bounds proposed by Jost Liu (2014), which leverage local connectivity features such as node degrees and triangle counts. Let # (z, y) represent the number of triangles involving the edge (a, y), and define A b min(a, b), a. b max(a, b), where dz denotes the degree of node x. The Ollivier- Ricci curvature for edge (z. y) is then bounded as follows:
Theorem C.1 (Jost& & Liu (2014)). For unweighted graph, the Ollivier-Ricel curvature of edge (I, satisfies:
1. Lower bound:
# v)
K(x, v) 2 -
(19)
du M dg 11- 1 1 #(I, 2)
dz dy
+
#(2, v) de dg
d2 dy
de dy
2. Upper bound:
k(z, v) 2 # v) de dn
(20)
The curvature of an edge can then be efficiently approximated as:
R(x, v) = 2 (pe (x. y) + n lower (I. v))
(21)
Computational Complexity. This approximation scheme runs in linear time ((E)), where |E| is the number of edges. Since each edge's curvature estimated using only local information (node degrees and triangles), the process is inherently parallelizable across edges, making it well-suited for large-scale graphs
C.1.3. RICCI FLOW COMPLEXITY
Ricci Flow is an iterative process where edge weights are adjusted according to the ORC values, smoothing the curvature distribution over time. Each iteration requires one full computation of ORC, making the total complexity: ((1(E)), where I is the number of iterations required for convergence. Empirically, across the datasets used in this study, Ricci Flow converges in 12-13 iterations on average. Since I is much smaller than the total number of edges (I < {E), the process remains linear in complexity:, i.e. O(T)E)) ((E)) as 7 is fixed across datasets.
15

---

manifold for CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
C.1.4. SUMMARY OF PREPROCESSING COMPLEXITY
Ollivier-Ricci Curvature (ORC): O((ET) (linear time)
Ricci Flow Regularization: ({EE) (empirically bounded by constant number of iterations)
Total Preprocessing Complexity: O((E)) (parallelizable and scalable)
C.2. Runtime Comparison
Average Runtime per Epoch
Dataset
Reddit
Weibo
Tolokers
Questions
Time per epoci (6) [logrscale]
10a
10a
10-1
10
OCN
GraphSAGE
BenNeet
KGON
ChebNet
RGNN
BWGNN
PCGNN
Der
XGBOD
OuYGAD
Model
Figure 4. Runtime comparision of CurvGAD. Average runtime per epoch (training + inference) seconds (log scale) on Reddit (Liu et al., 2022), Weiho (Zhao al., 2020), Tolokers (Platonov al., 2023), likd Questions (Platonov al., 2023) datasets, for select baselines.
Dataset
GCN
GraphSAGE
GAT
BernNet
KGCN
ChebNet
BGNN
BWGNN
PCGNN
DCI
GHRN
XGBOD
CurvGAD
Reddit
34.37
10.19
17.55
11.27
158.45
9.52
507.41
10.52
12.50
150.61
11.70
3050.66
585.22
Weibo
6.95
8.58
10.63
7.65
99.34
5.13
1415.59
10.63
6.74
121.77
11.06
4889.16
276.70
Tolokers
10.12
9.02
12.17
10.80
105.23
9.16
295.89
11.94
10.37
149.58
14.08
1432.76
260.30
Questions
21.56
27.12
22.85
18.53
183.53
23.77
1972.41
25.41
21.44
408.37
26.55
188192.5
589.41
Table Runtime Comparison (milliseconds) of CurvGAD baseline models across multiple datasets. This table shows the exact numbers used in Figure 4. The results are shown for CurvGAD model with F = 8 filters and product manifold (HH*)² 816 x 1
C.2.1. RUNTIME ANALYSIS
The runtime comparison in Table C.2 provides key insights into the computational efficiency of CurvGAD relative to baseline methods. Our results report the execution time milliseconds for CurvGAD configured with J spectral filters and a mixed-curvature product manifold (HH522 x 1 16 Despite leveraging multiple curvature spaces and graph filters for representation learning, CurvGAD demonstrates reasonable computational efficiency. For reference, when analysing Figure 4, GCN and BerNet be thought of using just graph filter and KGCN can be considered learning representations (when compared to CurvGAD with 8 filters and 4 manifolds).
Scalability with mixed-curvature manifolds. Unlike KGCN, which operates on a single curvature space, CurvGAD effectively integrates four manifold components. Nevertheless, its runtime remains within an acceptable range, significantly outperforming KGCN across all datasets. For instance, on Reddit, KGCN requires 158,45ms, while CurvGAD completes execution in 585. 585.22ms, despite utilizing more expressive model.
16

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CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
Efficiency compared to BGNN and XGBOD. CurvGAD remains substantially faster than models like BGNN and XGBOD. On the Weiho dataset, BGNN exhibits extreme runtime of 1415.59ms, whereas CurvGAD operates only 276.70ms. Similarly, on Tolokers, BGNN requires 295.89ms, compared to 260.30ms for CurvGAD, making it significantly more efficient despite handling survature-sensiitve graph learning.
Reasonable overhead given model complexity. The additional computational cost incurred by CurvGAD is expected due its survature-based learning framework, which integrates Ricci flow, manifold embeddings, and spectral filtering. However, the runtime remains competitive, ensuring that the benefits of richer geometric representations do not come at impractical computational expense.
Overall, CurvGAD balances computational efficiency and model expressiveness, outperforming single-curvature models while maintaining significantly lower runtime than complex baselines like BGNN and XGBOD. Its efficiency relative to KGCN and its comparable performance to DCI suggest that the inclusion of curvature-based representations does not drastically hinder scalability, making it a viable choice for large-scale graph anomaly detection tasks.
D. Arcitectural Details
D.1. Definition 4.1: Intuition and Derivation
The curvature decoder in Section 4.1.2 reconstructs the Ollivier-Rict curvature (ORC) Cry between nodes æ and y using their latent embeddings ny; By E pdd, The key idea is define Gaussian kernel the geodesic distance within the product manifold capture the intrinsic curvature information. In this section, we derive the decoder formulation and provide mathematical intuition.
Distance-based curvature estimation. Curvature measures the deviation of a space from being flat. The Ollivier-Ricci curvature is based on the Wasserstein distance between local probability distributions around connected nodes and y. Intuitively, negatively curved spaces (e-g., hyperbolic manifolds) exhibit greater spread between transported probability masses, whereas positively curved spaces (e.g., spherical manifolds) result in concentrated transport.
Motivation:
Given that ORC is inherently tied to how distances between points behave in different curvature regimes, it is natural to model curvature reconstruction as a function of the geodesic distance DM (Na, ny). A widely adopted approach in Riemannian learning is to employ a Gaussian kernel, which serves smooth and flexible mechanism to relate distances to similarity metrics.
Geodesic Distance on the Product Manifold. For a product manifold IP composed of M individual manifold components, the squared geodesic distance between node embeddings ny and ny is computed as:
M DM ) DM, (n n.)
(22)
This formulation ensures that distances are computed per manifold component and then aggregated across all curvature spaces.
Gaussian Kernel Formulation. To reconstruct the curvature, we define a Gaussian kernel based on the computed geodesic distance:
K(nz, ny) exp (~n D DM ny)
(23)
x2
where y controls the width of the kernel and r is trainable sensitivity parameter that adapts to the curvature distribution.
Intuition. The kernel output is close to 1 for small distances (nodes close latent space) and approaches 0 for large distances (nodes far apart). This behavior aligns with the principle that lower curvature corresponds more spread-out distances in latent space.
Sigmoid transformation for curvature prediction. The predicted curvature Čity is derived from the kernel output via sigmoid transformation:
Cay = 2- sigmoid (1 K(nn, ny)) 1.
(24)
17

---

CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
Justification: The sigmoid function normalizes the kernel output to the [0, 1] range, and the affine transformation maps the values into the [-1, range, consistent with the properties of Ollivier-Rice curvature.
Optimization via reconstruction loss. To ensure accurate curvature reconstruction, we optimize the decoder by minimizing the Frobenius norm loss:
Lc = Č C |
(25)
where Č denotes the predicted curvature matrix and C is the true ORC matrix.
The decoder exploits the structure of the product manifold by computing geodesic distances, mapping them through Gaussian kernel, and applying a transformation to ensure proper curvature scaling. This approach provides an effective means of capturing geometric distortions while maintaining computational efficiency.
Summary:
E. Experimental Details
E.1. Dataset Statistics
#Nodes
#Edges
#Feat.
Anomaly
Train
Relationship
Reddit (Kumar et al., 2019; et al., 2022)
10,984
168,016
3.3%
40%
Under Same Post
Weibo (Zhao et al., 2020; Liu et al., 2022)
8,405
407,963
400
10.3%
40%
Under Same Hashtag
Amazon (McAuley & Leskovec, 2013; Dou et al., 2020)
11,944
4,398,392
25
9.5%
70%
Review Correlation
YelpChi (Rayana Akoglu, 2015; Dou et al., 2020)
45,954
3,846,979
14.5%
70%
Reviewer Interaction
Tolokers (Platonov et al., 2023)
11,758
519,000
10
21.8%
40%
Work Collaboration
Questions (Platonov et al., 2023)
48,921
153,540
301
3.0%
52%
Question Answering
T-Finance (Tang al., 2022)
39,357
21,222,543
10
4.6%
50%
Transaction Record
Elliptic (Weher et al., 2019)
203,769
234,355
166
9.8%
50%
Payment Flow
DCiraph-Fin (Huang et al., 2022)
3, 3,700,550
4,300,999
17
1.3%
70%
Loan Guarantor
T-Social (Tang al., 2022)
5,781,065
73,105,508
10
3.0%
40%
Social Friendship
Table 6. Statistics of datasets used for evaluation. Percentage of anomalies and the respective train splits have been mentioned.
E.2. Hyperparameter Tuning
Hy Hyperparameter
Tuning Configurations
Description
F
(3,5,8, 10, 20, 25}
Total number of graph filters.
8
(0.2, 0.5, 0.7]
Neighbourhood weight distribution parameter for ORC
dr
(32, 48, 64, 128, 256)
Total dimension of the product manifold.
dropout
(0.2, 0.3, 0.5}
Dropout rate Number of training epuchs
epochs Ir
{100, 150, 1000)
(1e 4, 2e - 3, 0.001, 0.01}
Learning rate
weight_decay
{0, le 4, 5e 4)
Weight decay
Table 7. Hyperparameter configurations used in the experiments for CurvGADWe highlight the final configuration of CurvGAD in Red.
F. Signature Estimation
The mixed-curvature product manifold pdp is fundamental in capturing the geometric characteristics of graph-structurrd data. The optimal configuration of hyperbolic. spherical, and Euclidean components within this manifold is dataset-dependent, influenced by the underlying curvature distribution of the graph. To generalize across datasets, we estimate an appropriate manifold signature by leveraging Ollivier-Ree Ricci Curvature (ORC) as a guiding metric.
Our heuristic-driven approach assigns a spherical component to datasets exhibiting a predominance of positively curved edges, while hyperbolic component prioritized for datasets with a surplus of negatively curved edges. The stepwise procedure for this signature selection is formalized in Algorithm [eitationnwwitte wwithheld] By systematically analyzing the curvature distribution, our algorithm approximates a suitable manifold signature that aligns with the dataset's intrinsic geometric structure.
is used respect the double blind policy for one of our previous works, which is still under review the time submission. Appropriate citation shall be inserted later. This algorithm has been adopted from our previous work.
18

---

CurvGAD: Leveraging Curvature for Enhanced Graph Anomaly Detection
To ensure robustness, cluster the ORC distribution, identifying curvature centroids while preserving their sequence and frequency. This heuristic approach provides structured methodology inform curvature-aware representation learning. However, due to the inherent variability in optimal dimension allocation, treat the dimensions of component manifolds as tunable hyperparameter rather than an automatically inferred property. This strategy ensures fair and reproducible comparisons across datasets, as the ideal manifold configuration varies per dataset. While we do not claim that this heuristic yields the absolute optimal manifold decomposition, serves as an effective approximation, aligning curvature-sensttive embeddings with the dataset's topological properties to enhance graph anomaly detection.
Algorithm Product manifold signature estimation
Require:
Edge curvature histogram C {(Kss ft)
Threshold e' to distinguish between curved and flat regions
Maximum number of Hyperbolic (Hmax) and Spherical (Smax) components.
Total product manifold dimension de
(Optional) Preferred component manifold dimensions (h) des
(e)
Ensure: Product manifold signature póp P X My
1: Normalize frequencies: fi
15
2: Construct weighted curvature set: C' {(K,, f()
3: Determine optimal number of clusters K using methods like the elbow method, constrained by K S Hmax + Smax + 1b There
only Euclidean component
4: Cluster into K clusters using weighted clustering (e.g., weighted K-means)
5: Initialize empty lists H, S,
6: for each cluster in clusters do
Compute cluster centroid curvature Ric ofee 8C lc]
Compute total frequency weight uh = E ()m 9: if No < -é' and [7t] < Hmax then
b. Negative curvature b Curvature initialization
10:
Assign manifold component: MM = 11 Add (Mp; we) to H and [S] Smax then Assign manifold component: My = Sh Add (Mp, w.) S
11:
else if Ke
b Positive curvature P Curvature initialization
13:
14:
15:
else
Assign manifold component: Mp 17: Add (Mp, we) to
16:
Approximate curvature, i.e. K. [-c',e]]
18: end if
19: end for
20: Predefined dimensions (h) ( (ee provided then
21: Assign dimensions d, each component per predefined values
Dimension assignment
22:
else
23:
Set total number of components Q
H| + [ |E|
Dimension assignment
24: Allocate dimensions dp to each component p: dp = dr x
Proportional to weights
25:
Adjust dp to ensure E dp dp
26: end if
27: Formulate manifold signature:
H pdp (x [ (a)dew)) x (x118 (1 Rd0))
19


Reviews:
[{"review_id": "gpCPhaimFd", "reviewer": "Reviewer_Vzbi", "summary": "This paper proposed a mixed-curvature graph autoencoder that detects curvature-based geometric anomalies. It combines geometry reconstruction, using a Riemannian encoder and Gaussian kernel-based decoder, with structure and attribute reconstruction, regularizing graph curvature through discrete Ollivier-Ricci flow to isolate non-geometric anomalies. This approach refines anomaly classifications and uncovers new curvature-driven anomalies.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their positive assessment and thoughtful questions. Please find our detailed responses below:\n\n> [Q1] Could the authors provide visualizations or quantitative results showing the curvature of the regularized graph?\n\nWe thank the reviewer for pointing this out. We would like to clarify that **Figure 3** in the main paper visualizes the Karate Club graph before and after Ricci Flow regularization. The edge curvatures are color-coded—red/blue indicating positive/negative curvature before the flow, and grey edges (≈0 curvature) after the flow. This demonstrates that Ricci Flow effectively neutralizes geometric irregularities, enabling subsequent modules to focus on non-geometric anomalies. We shall add similar visualisations (and their corresponding curvature values) for other example graphs in the camera ready version.\n \n> [Q2]  Which specific components of the model contribute to its effectiveness in such cases? (Homophily)\n\nWe appreciate this insightful question. CurvGAD is equipped to handle both homophilic and heterophilic settings through a learned spectral **filter bank**, as described in **Section 4.1.1**. This bank consists of:\n- Low-pass filters to capture smooth, homophilic signals,\n- High-pass filters for heterophilic patterns, and\n- An identity (unfiltered) pathway to retain raw structural information.\n\nThese filters are assigned learnable weights, enabling the model to adaptively emphasize the appropriate frequency bands depending on the graph type. As a result, CurvGAD is able to generalize across both homophilic and heterophilic graphs by **selectively attending** to different parts of the **eigenspectrum**, which is reflected in its strong performance on diverse benchmarks.\n\n> [Q3] Does this curvature difference extend to non-graph data?\n\nThe idea of using curvature as a measure of structural irregularity indeed extends beyond graphs, particularly in data types that can be embedded into metric or manifold spaces, such as text, images, or time series. While such extensions are beyond the scope of the current paper, we are actively exploring them in future work.\n\n> [Q4] The proposed method includes two scores for anomaly detection.\n\nWe appreciate the opportunity to clarify this. Yes, all the three reconstruction-based anomaly scores (curvature, features, adjacency matrix) are normalised using min-max scaling. We shall clarify and highlight this more explicitly in the camera-ready draft and code repository. We also use learnable trade-off weights $(\\lambda_{\\mathbf{C}}, \\lambda_{\\mathbf{A}}, \\lambda_{\\mathbf{X}}, \\lambda_{\\text{cls}})$ to adaptively balance the contributions of all losses. This ensures that no single score dominates (**Section 4.3**). We further perform extensive ablation studies to isolate the impact of different scores (Table 2). Section 5.5 entails the different CurvGAD variants which *omit specific losses* out of these four scores.\n\nWe sincerely thank the reviewer again for their thoughtful and constructive feedback, which will help us further improve the clarity and completeness of our work.", "claims_and_evidence": "The authors claim that curvature plays a crucial role in graph anomaly detection, supported by evidence in Figure 1. The motivations outline three key challenges, each backed by citations to the corresponding supporting evidence.", "methods_and_evaluation_criteria": "The experimental settings, including datasets and evaluation criteria, make sense.", "theoretical_claims": "Not related.", "experimental_designs_or_analyses": "The experimental settings, including datasets and evaluation criteria, align with existing works.", "relation_to_broader_scientific_literature": "The contributions and motivations  related references are all listed in this manuscript intuitively.", "essential_references_not_discussed": "The references are comprehensive and well-discussed.", "other_strengths_and_weaknesses": "**Strength:**\n\n1)The paper is well-written, and well-organized.\n\n2)The experimental results are comprehensive.\n\n3)The motivation is justified and clear.\n\n**Weaknesses:**\n\nI still have some questions:\n\n1)The authors claim that \"the input graph needs to be regularized to standardize the curvature of the graph, which allows the subsequent structure and attribute reconstructions to focus solely on non-geometric anomalies.\" Could the authors provide visualizations or quantitative results showing the curvature of the regularized graph to support that this operation effectively removes geometric information?\n\n2)The authors state that CurvGAD can handle both heterophilic and homophilic networks, with the primary contribution focusing on addressing heterophilic problems. How does the proposed model perform on homophilic anomalies? Which specific components of the model contribute to its effectiveness in such cases?\n\n3)Does this curvature difference extend to non-graph data? If so, could the approach be generalized to other types of data structures?\n\n4)The proposed method includes two scores for anomaly detection. Are these scores normalized to the same scale? What happens if the scores differ significantly, potentially causing one of them to fail? Have the authors considered such a scenario, and if so, how is it addressed?", "other_comments_or_suggestions": "No more suggestions. The paper is well-written.", "questions_for_authors": "Please refer to Weaknesses.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "owteOZ88su", "reviewer": "Reviewer_BDH1", "summary": "The authors propose a method for graph anomaly detection that directly takes into account the curvature of the graph, thus capturing the structure of the graph in an arguably more appropriate way than what can be done with Euclidean representations (i.e., the common approach of embedding nodes/graphs via GNNs). The proposed approach, CurvGAD uses 1) a mixed-curvature encoder that captures irregularities in curvature and 2) an encoder-decoder module based on Ricci Flow to capture the structure of the adjacency matrix and the node features.\n\nExperimental results show a substantial improvement in graph anomaly detection (posed as a classification task) compared to several strong competitors.\n\n## update after rebuttal\nAfter going through the other reviews and the response from the authors, I will maintain my original score. I was initially (and still am) overall positive on this paper, and I have not found any major counterpoints in the other reviews.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their insightful feedback and constructive suggestions. We hereby address the raised concerns.\n\n### **Computational Considerations for ORC**\n\n> Computing ORC is computationally expensive\n\nWe appreciate your concern and apologize for not sufficiently emphasizing this aspect in the main paper. To clarify, we employ a **linear-time combinatorial approximation** for computing Ollivier-Ricci Curvature (ORC), which avoids the expensive optimal transport computations typically involved. A detailed analysis of the computational considerations for ORC is provided in **Appendix C.1**, particularly in **Section C.1.2**.\n\nAs outlined in **Theorem C.1** and Appendix C.1, our approximation computes curvature using only local structural properties—specifically node degrees and triangle counts—allowing each edge's ORC to be estimated as the average of analytically derived upper and lower bounds. This approach results in a **linear-time** computational complexity $(\\mathcal{O}(|\\mathcal{E}|))$. Furthermore, because the computation for each edge is independent, the process is naturally parallelizable, making it scalable and efficient even for large graphs.\n\n**Summary of Preprocessing Complexity (Appendix C.1.4)**\n- Ollivier-Ricci Curvature (ORC): $\\mathcal{O}(|\\mathcal{E}|)$ (linear time)\n- Ricci Flow Regularization: $\\mathcal{O}(|\\mathcal{E}|)$ (empirically bounded by a constant number of iterations)\n- Total Preprocessing Complexity: $\\mathcal{O}(|\\mathcal{E}|)$ (**parallelizable** and scalable)\n\n\n> I don't think \"my model wins\" is appropriate in a scientific paper.\n\nWe thank the reviewer for pointing this out. The caption could be more formal and we will revise it to adopt a more appropriate and professional tone in the camera-ready version.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Not much to check. The model leverages existing theory.", "experimental_designs_or_analyses": "Yes. No issues.", "relation_to_broader_scientific_literature": "This paper contributes to the graph anomaly detection literature. This is a very active area of research. However, methods that consider the curvature are still not very commonly used.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Weaknesses:\n- Computing ORC is computationally expensive", "other_comments_or_suggestions": "Call me a killjoy, but I don't think \"my model wins\" is appropriate in a scientific paper. I would suggest choosing a different caption for Table 1.", "questions_for_authors": "NA", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "IW8HA8x75H", "reviewer": "Reviewer_PTSu", "summary": "The paper proposes CurvGAD, a novel graph anomaly detection framework that incorporates curvature information through a mixed-curvature graph autoencoder consisting of two parallel pipelines: curvature-equivariant geometry reconstruction and curvature-invariant structure and attribute reconstruction.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their detailed and thoughtful comments. We deeply value the opportunity to clarify our contributions and improve our work based on your feedback. \n\n### **[Q1] Why curvature?**\n\nThe core aim of our work is to learn node representations on a **non-Euclidean manifold** and study how the underlying geometry—characterized by **curvature**—reveals anomalies.  While several geometric metrics exist, curvature helps in quantifying the intrinsic structural irregularities in graphs (our hypothesis for this paper). \n\nUnlike static scalar descriptors (e.g., clustering coefficients or local assortativity), curvature allows us to reason about how node-pair relations behave in a global geometric context, making it suitable for tasks like anomaly detection that are sensitive to subtle perturbations in local structure. Furthermore, graph regularization via Ricci Flow allows us to isolate non-geometric anomalies for reconstruction. If there are specific alternative geometric properties you had in mind, we would be eager to understand and consider them in future work.\n\n### **[Q2, Q4] Homophiliy (Heterophily) and Low-pass (High-pass) Filtering**\n\nWe appreciate this clarification and fully agree that homophily and low-pass filtering are not the same concept. We apologize if this was inadvertently conflated in the phrasing. Our intention was not to equate them, but to note that low-pass filters are particularly effective in capturing homophilic signals, which tend to lie in the low-frequency **eigenspectrum** of the graph Laplacian (and high-pass for heterophilic).\n\nTo support our statement, we sincerely refer to the source cited by the reviewer [1], which states:\n\n- *“High-frequency signal, which can catch the neighborhood dissimilarity, is found to be helpful for heterophily.”*\n- *“It is commonly stated that low-pass filter mainly captures the smooth components of the signals (neighborhood similarity), while high-pass filter mainly extracts the non-smooth ones (neighborhood dissimilarity).”*\n\nThus, we only argue that low-pass filters assist with homophilic patterns and high-pass filters help capture heterophilic patterns, which aligns with the aforementioned literature. We will revise our phrasing in the CR version to avoid ambiguity.\n\n[1] The heterophilic graph learning handbook. 2024 Jul 12.\n\n### **[Q3] Novelty of Equations 2-4**\n\nYes, Equations (2–5) are indeed novel contributions of our work. They extend Chebyshev spectral filtering to the mixed-curvature Riemannian setting, and define the filterbank using the $\\kappa$-stereographic model. Specifically:\n- Eq. (2): Projects Euclidean features to the manifold.\n- Eq. (3–4): Define manifold-aware Chebyshev filtering and aggregation. They extend the typical Chebyshev recurrence to the mixed-curvature spaces.\n\nThese gyrovector operations (Mobius addition, $\\kappa-$matrix multiplication) are standard in non-Euclidean geometry and are described in Appendix B.2. They help us extend the Euclidean operations to the product manifold. \n\n### **[Q5] Heterogeneous or heterophilic?**\n\nThis is correct and we apologize for the typo (and therefore, the unintended abuse of terminology). We indeed meant \"heterophilic\" and \"homophilic\", not \"homogeneous\" or \"heterogeneous\" in the sentence mentioned. We will rectify in the CR version.\n\n### **[Q6] Additional Experimentation on Heterophilic Networks**\n                    \nThank you for this valuable suggestion. We initially evaluated CurvGAD on a diverse mix of homophilic and heterophilic benchmarks:\n- Homophilic: Reddit, Weibo, Amazon, YelpChi, Tolokers, Questions\n- Heterophilic: T-Finance, Elliptic, DGraph, T-Social\n\nBased on the feedback we conducted additional experiments on heterophilic datasets from [1]:\n- Malignant: Cornell, Texas\n- Ambiguous: snap-patents, twitch-gamers\n\nAs these datasets lack organic anomaly labels, we follow a common setup by treating the minority class as anomalous. We compare CurvGAD against strong (select) baselines (due to spatial and time constraints): GCN, ADAGAD, KGCN, and BWGNN.\n\n#### **AUROC (%) Results on more Heterophilic Datasets**\n\n| Dataset             | GCN                | ADAGAD         | KGCN           | BWGNN          | **CurvGAD (Ours)** |\n|-------------------  |----------------|----------------|----------------|----------------|---------------------|\n| **Cornell**        | 72.3 ± 1.8       | 73.5 ± 2.1     | 78.8 ± 1.9     | 75.2 ± 2.9     | **84.6 ± 1.7**      |\n| **Texas**          | 74.5 ± 0.3        | 78.2 ± 0.2     | 79.9 ± 2.0     | 81.3 ± 1.2     | **85.7 ± 0.6**      |\n| **snap-patents**   | 41.2 ± 1.1     | 49.5 ± 1.5     | 48.5 ± 0.4     | 49.1 ± 1.2     | **52.03 ± 0.9**      |\n| **twitch-gamers**  | 66.8 ± 1.9        | 71.1 ± 2.4     | 73.3 ± 2.1     | 75.2 ± 2.1     | **78.34 ± 1.1**      |\n\nThese results demonstrate CurvGAD’s generalization to heterophilic graphs. We will include full analysis (all baselines) in the CR version.", "claims_and_evidence": "yes", "methods_and_evaluation_criteria": "yes", "theoretical_claims": "no", "experimental_designs_or_analyses": "yes", "relation_to_broader_scientific_literature": "graph anomaly detection", "essential_references_not_discussed": "no", "other_strengths_and_weaknesses": "see below", "other_comments_or_suggestions": "1. What is the special property of curvature over other metrics on graphs that is required for anomaly detection. In other words, there are lots of metrics that can capture geometric information, why do you choose curvature instead of others for anomaly detection, what makes it better than others?\n\n2. \"...under the homophily assumption (i.e., low-pass filters) and fail in heterophilic graphs where...\" Homophily and low-pass filter are two different concepts. I think you've mixed them up. See [1] for the definition and explanation.\n\n3. Is there anything new about equation (1)? Why do you use \"mobius addition, mobius subtraction, κ-right-matrix-multiplication and κ-left-matrix multiplication\" in equation (2-4), what is your motivation and what is their relations with graph anomaly detection? Are these your novel contributions or you borrow from other papers?\n\n4. \"filter bank captures both high-frequency (heterophilic) and low-frequency (homophilic) signals\" It is not true that high-frequency signals can capture heterophilic information and low-frequency signals capture homophilic information. See the analysis and results in [2]\n\n5. \"enabling CurvGAD to generalize effectively across heterogeneous and homogeneous graphs.\" Heterogeneous/homogeneous or heterophilic/homophilic? See their differences in [3].\n\n6. The heterophilic graphs used in experiments are not good enough as shown in [1]. Tolokers and Questions are actually homophilic graphs and both lead to beneficial effects to message passing instead of negative effects. Therefore, some analyses and conclusions in this paper need to be carefully re-examined based on correct evaluation and evidence. You need to use the real challenging heterophilic graphs, especially the malignant and ambiguous heterophily datasets identified in [1].\n\n\n\n\n[1] The heterophilic graph learning handbook: Benchmarks, models, theoretical analysis, applications and challenges. arXiv preprint arXiv:2407.09618. 2024 Jul 12.\n\n[2] When do graph neural networks help with node classification? investigating the homophily principle on node distinguishability. Advances in Neural Information Processing Systems. 2024 Feb 13;36.\n\n[3] Beyond homophily in graph neural networks: Current limitations and effective designs. Advances in neural information processing systems. 2020;33:7793-804.", "questions_for_authors": "see above", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "O0p0sKtaBo", "reviewer": "Reviewer_24eB", "summary": "The paper introduces CurvGAD, a novel graph anomaly detection method that leverages mixed-curvature geometry to detect anomalies overlooked by conventional structural and attribute-based approaches. CurvGAD employs two parallel reconstruction pipelines: a curvature-equivariant pipeline that captures geometric anomalies using mixed-curvature embeddings, and a curvature-invariant pipeline utilizing Ricci flow for structural and attribute anomalies. The approach demonstrates superior anomaly detection, achieving up to 6.5% improvement in AUROC scores over state-of-the-art methods across ten diverse datasets.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer for their insightful feedback, valuable assessment and constructive suggestions. \n\n### **[Q1] Scalability and Complexity**\n\nThank you for this important point. We provide a detailed time complexity (and scalability) analysis in **Appendix C**. Specifically:\n1.  **Appendix C.1.2** explains our linear-time approximation of Ollivier-Ricci Curvature (ORC), which scales as $\\mathcal{O}(|\\mathcal{E}|)$ and is fully parallelizable.\n\n**Summary of Preprocessing Complexity (Appendix C.1.4)**\n- Ollivier-Ricci Curvature (ORC): $\\mathcal{O}(|\\mathcal{E}|)$ (linear time)\n- Ricci Flow Regularization: $\\mathcal{O}(|\\mathcal{E}|)$ (empirically bounded by a constant number of iterations)\n- Total Preprocessing Complexity: $\\mathcal{O}(|\\mathcal{E}|)$ (**parallelizable** and scalable)\n\n\n2. **Appendix C.2** presents runtime comparisons (Table 5, Figure 4), showing CurvGAD’s efficiency across diverse datasets. Figure 4 lays down the average runtime per epoch comparision of CurvGAD with other baselines.\n\n\nThe linear time approximation of ORC ensures that our model is scalable to million-scale graphs. Despite using multiple curvature manifolds and filters, CurvGAD achieves runtime comparable to or better than baselines like BGNN, XGBOD, and KGCN. For example, on Reddit, CurvGAD is faster than KGCN and significantly more efficient than XGBOD, while being far more expressive. Thus, our design ensures both scalability and geometric fidelity for large graphs.\n\n\n### **[Q2] Robustness to adversarial perturbations?**\n\nWe appreciate this insightful suggestion. Robustness to adversarial attacks is a valuable direction—especially for anomaly detection—but it lies beyond the current scope of this work. CurvGAD focuses on the role of mixed-curvature geometry in identifying anomalies. Our results establish curvature as a fundamental tool for GAD and pave the way for future advancements in geometry-aware graph learning while of- fering new avenues for exploring geometric anomalies in topologically complex networks. We are actively exploring robustness extensions in future work.\n\n### **Other suggestions**\n\n- **Typos & Visuals**: Thank you for pointing these out. We will fix the typo (“compliment” → “complement” in Section 5.5(e)) and enhance Figures 1 & 3 with clearer legends and annotations in the camera-ready version.\n- **Baseline Hyperparameters**: We acknowledge this concern and will include detailed hyperparameter settings for all baselines in the camera-ready version and code repository. Due to the rebuttal length constraint, these could not be included here.", "claims_and_evidence": "The claims regarding the superiority of CurvGAD in detecting anomalies are well-supported by extensive experimentation and comparative analyses (Table 1). However, the claim of interpretability through curvature-based anomaly detection is partially supported only qualitatively (Figure 1); a quantitative interpretability evaluation is absent, weakening this claim.", "methods_and_evaluation_criteria": "The proposed methods, including curvature-equivariant and curvature-invariant reconstructions, effectively address the stated limitations. The evaluation using AUROC across ten datasets (Table 1) is appropriate and thorough, adequately capturing the method's efficacy in various contexts. The chosen baseline comparisons are relevant and robust, allowing for a clear assessment of CurvGAD’s relative performance.", "theoretical_claims": "The paper does not present new theoretical proofs, but it effectively builds upon existing theoretical foundations of Riemannian geometry and Ricci curvature. The correctness and definitions in sections 3 and 4, particularly related to curvature and Ricci flow, appear mathematically sound and well-explained.", "experimental_designs_or_analyses": "The experimental design, including the methodology for dataset splits, anomaly scoring, and evaluation metrics (Sections 5.3 and 5.4), is sound. One minor weakness is the absence of a detailed discussion regarding the computational overhead introduced by Ricci flow regularization across large datasets. More transparency regarding runtime comparisons with baselines would enhance confidence in practical applicability.", "relation_to_broader_scientific_literature": "CurvGAD is well-positioned within the context of existing literature on graph anomaly detection, clearly distinguishing itself by incorporating mixed-curvature and geometric perspectives (Section 2). While the paper cites recent works effectively, an expanded discussion on how CurvGAD’s geometric perspective fundamentally differs or extends beyond existing curvature-based embedding methods could further clarify its novelty", "essential_references_not_discussed": "The paper extensively cites the relevant literature, including critical references in mixed-curvature graph embeddings and anomaly detection. However, discussion on recent works involving adversarial robustness in geometric and spectral graph methods, such as those addressing structural robustness or attacks on anomaly detection systems, would strengthen the context.", "other_strengths_and_weaknesses": "The paper exhibits clear originality and significance in utilizing curvature for anomaly detection, offering substantial practical implications and interpretability. However, a major weakness is the lack of detailed hyperparameter settings for baseline methods, which limits reproducibility.", "other_comments_or_suggestions": "The writing is clear, though a few typos are present (e.g., \"compliment\" instead of \"complement\" in Section 5.5 (e)). Also, improving readability in Figures 1 and 3 through clearer legends or annotations would benefit interpretation.", "questions_for_authors": "How does computational complexity scale with large graphs, and is CurvGAD scalable enough for real-world networks containing millions of nodes?\n\nHave you considered robustness analyses against adversarial perturbations, given the relevance to anomaly detection tasks?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: Four reviewers have evaluated this paper and scored it 2, 3, 3, 4. Overall, the reviewers had a number of concerns ranging from the issues with computational efficiency (some remained unconvinced), whether the method is purely applicable to graphs only and issues with the abuse of terminology (e.g., confusion between homophily and low-pass filtering). On balance, while the paper is borderline, the rebuttal is mostly sound and the AC recommends a weak accept.
Decision: Accept (poster)