Graph Neural Ricci Flow: Evolving Feature from a Curvature Perspective
Authors: Jialong Chen
Bowen Deng
Zhen WANG
Chuan Chen
Zibin Zheng
Venue: ICLR 2025 Poster (2025)
Keywords: Graph neural network
Differential equation
Curvature
Ricci flow
Primary area: learning on graphs and other geometries & topologies
TL;DR: Continuous-depth GNN based on attributed discrete Ricci flow

Abstract: Differential equations provide a dynamical perspective for understanding and designing graph neural networks (GNNs). By generalizing the discrete Ricci flow (DRF) to attributed graphs, we can leverage a new paradigm for the evolution of node features with the help of curvature. We show that in the attributed graphs, DRF guarantees a vital property: The curvature of each edge concentrates toward zero over time. This property leads to two interesting consequences: 1) graph Dirichlet energy with bilateral bounds and 2) data-independent curvature decay rate. Based on these theoretical results, we propose the Graph Neural Ricci Flow (GNRF), a novel curvature-aware continuous-depth GNN. Compared to traditional curvature-based graph learning methods, GNRF is not limited to a specific curvature definition. It computes and adjusts time-varying curvature efficiently in linear time. We also empirically illustrate the operating mechanism of GNRF and verify that it performs excellently on diverse datasets.

Published as a conference paper at ICLR 2025
GRAPH NEURAL RICCI FLOW: EVOLVING FEATURE FROM A CURVATURE PERSPECTIVE
Jialong Chen, Bowen Deng, Zhen Wang, Chuan Chen', Zibin Zheng Sun Yat-sen University
[chenjlong7, dengbw3) @erggbaww@aaaii @ma 112. sysu. edu. (wangzh665, chenchuan, zhzibin) @mail. sysu. edu. .n
ABSTRACT
Differential equations provide dynamical perspective for understanding and de- signing graph neural networks (GNNs). By generalizing the discrete Ricei flow (DRF) to attributed graphs, we can leverage new paradigm for the evolution of node features with the help of curvature. We show that in the attributed graphs, DRF guarantees vital property: The curvature of each edge concentrates to- ward zero over time. This property leads to two interesting consequences: I) graph Dirichlet energy with bilateral bounds and 2) data-indes eata-independent curvature decay rate. Based on these theoretical results, we propose the Graph Neural Ricci Flow (GNRF), novel curvature-awar continuous-depth GNN. Compared ditional curvature- based graph learning methods, GNRF is not limited to specific curvature definition. It computes and adjusts time-varying curvature efficiently in linear time. We illustrate that GNRF performs excellently on diverse datasets.
INTRODUCTION
Graph Neural Networks (GNNs) have currently achieved significant success in tasks such as com- munity detection (Liu al., 2020), product recommendation (Wu et al., 2022; Gao al., 2023) molecular design (Zhang et al., 2021; Wieder et al., 2020), and enhancing language models (Chen al., 2024; Jin al., 2023). One of the most successful ideas in designing GNNs is stack sev- eral cesssage-passing layers, allowing nodes to receive information and update their representations within multiple hops (Kipf & Welling, 2016; Veličković al., 2017; Wu et al., 2019).
Recent research has revealed a close connection between these layered GNNs and differential equa- tions (DEs). Oono & Suzuki (2019) first proposed the idea of viewing Graph Convolutional Net- works (GCN) (Kipf & Welling, 2016) as discrete dynamical systems. While Chamberlain et al. (2021), using the heat diffusion equation, derives the continuous-depth counterpart for Graph At- tention Networks (Veličković et al., 2017). By establishing DEs for node representations over time, more refined and theoretically sound evolution strategies can be designed, including energy con- servation (Rusch et al., 2022), anti-symmetry (Gravina et al., 2022), and repulsion (Wang et al., 2022).
Most DE-GNNs are based the heat equation and its variants (Chamberlain et al., 2021; Thorpe et al., 2022; Choi et al., 2023; Li et al., 2024; Bodnar et al., 2022). However, the classical heat equation forces the temperature in the system to become uniform over time, leading expressive node representations in GNNs inspired by this equation when reaching an equilibrium state. In this paper, we break away from this fixed mindset for the first time and turn to explore the benefits of another important differential equation- - the Ricci flow - for graph learning.
loss of
In differential geometry, Ricci flow is metaphorically described as a process where complex man- ifold gradually becomes "regular". This process is governed by the Ricci curvature, causing regions with larger absolute curvature values decay more significantly. When defining edge curvature based on node-attributed graphs, we find that the Ricci Flow works: it forces the edge curvature to concentrate towards zero quickly, thus efficiiffy yielding stable and non mon-smooth node repre- sentations.
"Corresoodiigg author.

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Published as a conference paper at ICLR 2025
Differential manifold
Curvature distribution
Attributed graph
F Feature space
0
73
Ricci Flow
Figure 1: Analogize the feature evolution process on the node-attributed graph to the Ricci flow in differential manifolds, where the curvature gradually concentrates zero.
Based on this observation, we design novel continuous-depth GNN called the Graph Neural Ricci Flow (GNRF). To our knowledge, GNRF is the first deep graph learning model based on time- varying edge Ricci curvature. (SelIRGNN (Sun et al., 2022) is a potentially confusing related work, but they focus on curvature in the embedded space rather than on the edges.) Previously, the main- stream paradigm for utilizing Ricci curvature in graph learning 2023; Fesser & Weber, 2024; Shen et al., 2024), where edge curvature was considered an intrin- sic property related only topology, precomputed, and stored. Additionally, they consider only specific curvature definition and require quadratic time complexity. GNRF, for the first time, de- fines time-varying edge curvature based on node attributes and introduces an auxiliary network to compute curvature for any given definition within linear time.
graph rewiring (Nguyen al.,
Our contributions can be summarized as follows:
We are the first to consider the evolution of node attributes from the perspective of attirbute diserete Ricci flow, providing theoretical guarantees decay rate and Dirichlet energy.
2. We propose GNRF, the first GNN, to apply time-varying edge curvature and introduce an auxiliary network for unified and efficient curvature computation.
3. We empirically test how well the GNRF fits the theory and describe its mechanism. We also verify its sigmificant validity on wide range of datasets.
2 PRELIMINARIES
Notations. We consider simple undirected graph G = (V, where V is a node set with size of |V] and edge & C V with size of E]. The i-th node is associated with feature vector æ; E RR, and the matrix form of all features is denoted as X [æ.,..., xx E R gNx" Similarly, let (t) = (t)........ fey (t)) Nxm be the node representations evolved to time according to differential equation. The initial representation H()) is obtained from X through a function F, i.e. HO)) F(X). We denote the time-varying attribute on edge ÿ as (iy(1). may be defined by H, in which case we have: (iy(() w(h,(t), (1)).. (ay(1) represents the edge curvature under any given definition.
Differential equation inspired GNNs (DE-GNNs). Continuous depth is the fundamental charac- teristic that distinguishes DE-GNN from other GNN architectures. It can unified as:
(H(t) f(g, (O),, H()), t E [0,
Øt
The update function f can be either pon-paramttric (Oono & Suzuki, 2019; Veličković et al., 2017; Wu et al., 2023) or parametric (Li et al., 2020; Xu et al., 2023). The heat diffusion equation is the most extensively discussed in DE-GNN methods (Chamberlain et al., 2021; Thorpe al., 2022; Li et al., 2024). A classic description is provided by Chamberlain et al. (2021):
(t) divig Vh;(t)
Σ (h,(tt),h (h₂ ht)))), jN
2

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Published as a conference paper at ICLR 2025
where g is the diffusion flux, and .). is the similarity function for representations which is usually assumed to be non non-negative (Chamberlain et al., 2021; Li et al., 2024).
Edge curvature weighted graph. For weighted graph G where each edge j corresponds to a weight Wij. the edge curvature Kij measures the tightness of the connection between the ego- graph of node and the ego-graph of node j. Kej has multiple definitions, with the earliest being the purely combinatorial one proposed by Forman (2003) via CW complex, known as the Forman-Rice curvature Kij In addition, Ollivier (2007) proposes Ollivier-Rieci curvature Ki which is based on the optimal transport distance between ego-graphs. Devriendt & Lambiotte (2022) proposes viewing weighted graph as a resistance network and establishing the resistance curvature Kij via effective resistance. See Appendix A.I for more detailed introduction.
Currently, curvature-based graph learning methods generally treat curvature as static property of edges, using it to perform graph rewiring (Nguyen et al., 2023; Topping et al., ;021; Fesser & Weber, 2024), edge sampling (Liu et al., 2023), or neighbor reweighting (Li et al., 2022). Their underlying idea is similar: to mitigate the influence of edges with extreme positive/negative curvature. owever, these methods often precompute Ricci curvature, resulting in a quadratic time complexity. Moreover, they focus on using a specific definition of curvature, which may lack sufficient robustness when dealing with different graph data (Southern et al., 2023; Attali et al., 2024). In contrast, Our method provides a way to evolve node features in linear time adjust curvature, which is applicable to any curvature definition.
Discrete Ricci flow (DRF). The Ricci flow was first introduced by (Hamilton, 1982) in differential geometry and was later extended to complex networks by (Ollivier, 2009; 2010) using the definition:
Ow., (t) Key (t) Ôt
Wey (t) 0,
which is referred to as the discrete Ricci flow. Recently, DRF has found its applications in network mining, such as in community detection (Ni et al., 2019; Lai al., 2022), network alignment (Flow, 2018), and biological structures prediction (Baptista et al., 2024). However, they all focus on graphs without node attributes, where edge weights are treated inherent properties, which is inconsistent with the majority of datasets used in modern graph deep learning.
3 APPLYING DRF ON ATTRIBUTED GRAPHS
We aim to establish a dynamical system on node-attributed graphs similar to Equation (3). The intuitive idea is to treat edge weights as function of the attributes of the two connected nodes, i.e., (iy(1) = w (1, (t)). where w(:,-) > 0 (1), h,,()))))) (1)) -K,5 (t), (t))..
We refer to this dynamical system as Attribute Discrete Ricci Flow (Attri-DRF). Note that cur- vature defined based on edge weights, which themselves are functions of node attributes. This implies that in Attri-DRE, curvature is also determined by the node attributes. Consequently, as node attributes evolve over time, the curvature changes accordingly. Attri-DRF also, for the first time, realizes the modeling of graphs curvature in continuous-time scenario.
Ricci flow is analogous to heat diffusion of the metric on a Riemannian manifold. A fundamental characteristic of standard heat diffusion is that the temperature distribution of any heat field gradually becomes smoother over time, ultimately reaching a steady state (thermal equilibrium). This implies that GNNs based on heat diffusion (such as GRAND Chamberlain al. (2021) and GRAND++ Thorpe et al. (2022)) will exhibit a similar steady state as time approaches infinity, where the node features gradually become consieet with the well-known over-smoothing phenomenon. By contrast, Ricci flow, as a 'diffusion on the metric', leads to a 'metric equilibrium', meaning that the curvature on the manifold gradually becomes uniform. Specifically, in the context of Attri- Attri-DRF, we can conclude the following:
Lemma 1 (informal), Consider the Auri-DRF on any edge i ~ j. if wi;(() is monotonic over a non-zero interval [t1. and Wer (t2) - (t)] is sufficiently close to 0, then the average is also sufficiently close O.
curvature over [t,, t2), i.e., Exc(tr
(l (Iss)(t))))
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Lemma I indicates that if the Attri-DRF edge approaches equilibrium state, the curvature of that edge must necessarily be close to zero. Since excessively large or small curvature can lead to over-smoothing over-squashin (Nguyen et al., 2023), Attri-DRF offers unified and novel solution to these issues: leveraging the evolution of the Ricci flow to drive the curvature towards O. It is worth noting that this process evolves automatically over time and is independent of any specific definition of curvature, which contrasts sharply with the currently popular graph rewiring paradigm (Nguyen al., 2023; Topping et al., 2021; Fesser & Weber, 2024; Shen et al., 2024).
To have a closer look, from now on, we let wiy(() = cos(h;(() (t) +1 + e be non-negative co- sine similarity, where is small positive number. Additionally, let [t)] to prevent numerical vanishing or explosion. At this point, w satisfies: w E (c.
3.1 DIRICHLET ENERGY
The Dirichlet energy E (t)) of a graph 9 is used to characterize the smoothness of attributes between nodes. As neighboring node representations become similar, E tends towards O. Proving that E' has a lower bound a common theoretical means to demonstrate the ability of GNN to resist over-smoothing ( (Wang al., 2022; Zhou et al., 2021a). Using Theorem 2, we demonstrate that when Autri-DRF stabilizes (i.e., k(() approaches 0), E has both upper and lower bounds. This means that the Attri-DRF avoids over-smoothing and prevents neighboring nodes from having excessive differences. See Appendix A.2 for more details about Dirichlet energy.
Theorem 2 (Informal). Consider the Auri-DRF with (t) = cos(h, (1,, + e and h. = 1. If within [t,, t2]. each edge of graph G reaches evolutionary equilibrium, then at this time G has both non-trivial upper and lower bounds on Dirichlet energy that are independent of the definition of curvature.
Here, we provide an intuitive explanation. Extreme smoothing results completely uniform node attributes, whereby the edge curvature is determined solely by the local topology. In most graphs without additional assumptions, differences in local topology lead to different curvatures, which contradicts Lemma 1. Therefore, for almost all graphs, can derive nonzero lower bound for the Dirichlet energy.
3.2 UNIFORM CURVATURE DECAY
We have characterized the asymptotic equilibrium of Attri-DRF and how it benefits graph learning. Now, we further discuss how this equilibrium is achieved, with particular attention to its practical significance agnificanee-ssecffccllll whether it can reach satisfactory state within a finite time.
Theorem (Informal). Consider the Attri-DRE with way (t) = cos(h, hy (t) + + e and h = 1 any edge ~ j. Assume that the curvature is bi- bi-Lipschitz continuous when considered as a function of the weights. For any arbitrarily small positive number 6, if (mir(0) > 6, then Raj will first decay to a value smaller than at t = O (In (6-1))
Theorem 3 implies that Attri-DRF requires about O 8-1) time to achieve a steady state. There are three points worth noting: (1) This bound is practically feasible: for 8 = 10 In 1n(8-) remains no greater than 10. (2) The result independent of (0) and (o): for any given 5, the result applies to every edge in any G. In contrast, Newton's 's law of cooling indicates that the time for heat diffusion to reach a steady state depends on the initial conditions (Winterton, 1999). (3) The decay rate is uniform: Theorem can be applied to all edges of the same graph, leading to synchronized decay of all curvatures. This eliminates the need to balance the differences in the evolution processes of various edges. In summary, the feature evolution of Attri-DRE is feasible, independent of the initial state, and uniform.
4 OUR MODEL: GRAPH NEURAL RICCI FLOW
4.1 INCORPORATING ATTRI-DRE INTO THE DE-GNN FRAMEWORK
Attri-DRE alleviates the issue of node representation quality degradation caused by excessively high or low curvature, and shows advantages in evolution time. To leverage these beneficial properties,
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next demonstrate how to derive the general form of the DE-inspired GNN (Equation (I)) from Attri-DRF (Equation (86)).
By expanding Equation (86) using the chain rule, obtain':
{ Tw(h, h2) (h,(t) Oh;
{ (wo(h,
h2)
@hz(t) - -Ki (f) Rtg).
+
(5)
aI
Ō
Equation (5) splits the effect of Attri-DRF on waj into two parts: the effect on h; and the ef- fect on hj- To weigh these two parts, we introduce a scaling function X(h (t), hy (t)), i.e.,
Ow (w.h.hh)) hn) (t)(t) (1ee(1)(1,(11) ( @w(h:s))) oh₂(0)) de
We can now focus solely on one side:
8h.
Dt
Sh;
@w((s @ht(t) )=
mey (t)(t) h2j)
1+ - (t), (t))
=
Oh;
Öt
Equation (6) provides the first constraint for (t, while [t)(t) = is the second one. Under the satisfaction constraint, we minimize [t),,, which means that h2(() always applies only the slightest change to satisfy the Autr-DRF, which guarantees that the evolution of h, (t) is numerically stable. Formally, this leads us to the following optimization objective:
(h,(t) Ot
Ow(h,, h2)
ah,(t)
Rey (ks(t))ohhnhh)) hz)
min
s.t. (I)
(II) h,,(t) = 1.
Oh,
Ōt
1+ h(s
Proposition 4. The optimization objective given by Equation (7) has closed-form solution with linear time and space complexity as follows:
- (ty(() he cos (hi 12 te
(8)
O
(h, where (t) (T+ (1),,, (A. h.)
Considering that the update of h2( is actually influenced by all the neighbors of the node denoted as (i), we let j in Equation (8) range over all elements in (V and take the sum. This leads to
deriving a Ricci flow-based node representation evolution, which forms the DE-GNN model. We refer to this as Graph Neural Ricci Flow (GNRF):
(h, Øt = -E Σ- (l (n,()) --
cos (l)((),).()) he 10
weight
damping factor
By comparing Equation (9) with Equation (2), one can get more insights between GNRF and heat diffusion. In early graph heat diffusion models like GRAND (Chamberlain al., 2021), the aggre- gation weight is always positive, which is considered the driving force behind the smoothing of node attributes (Wang et al., 2022). Negative weights, on the other hand, are considered repulsive forces, making node attributes dissimilar. In GNRF, the sign of the aggregation weights (-x') is opposite to that of the curvature (k), which results in nodes with positive curvature being pushed apart, while other nodes are pulled closer. As demonstrated by experiments, find that this helps in generating smoother decision boundaries.
Another distinction between GNRF and heat diffusion is the damping factor. It restricts the degree of the feature evolution. An intuitive explanation is that when [t((t]] = 1, then:
(t)l = cos? (h (1,) (t)). (t) cos (t), h.
(10)
so excessive similarity or dissimilarity between h; and h; (close to or -1) may weaken [[),),(t]]]-
4.2 UNIFYING CURVATURES VIA EDGENET
The key distinction of GNRF from other GNNs lies in its ability to perceive time-varying edge curvature. However, curvature is often computationally expensive. For instance, calculating the re- sistance curvature itC requires computing the matrix pseudo-inverse, while Ollivier-Rieci curvature
When unambiguous, we omit the independent variable for simplicity.
5

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ROR involves solving the optimal transport distance. These complexities make real-time curvature computation a major bottleneck, hindering the broader application of curvature.
GNRF addresses this challenge by introducing auxiliary network. Recent research has shown that the Wasserstein distance can be approximated in linear time using a simple network (Chen & Wang, 2024). Inspired by this, we aim leverage the universal approximation capability of neural networks to seek a general solution for approximating the curvature under any definition.
Theorem 5 (Informal), There exists a unified network structure called EdgeNet, which takes as imput the weights of all edges connected to nodes and j. and approximates Klg with arbitrary precision in linear time, i.e., (t))t) = EdgeNet (Wit (t) lk ~ it, { (usx(t)k k ~ j}). The definition of edge curvature can be Forman-Rice curvature Kx] (t), Ollivier-Ricci curvature mig approximate resistance curvature Kij (1).
OR (t) and
EdgeNet not only enables the computation of curvature in linear time, but more importantly, it over- comes the limitations of existing definitions by realizing adaptive curvature. Despite the variety of curvature definitions, to the best of our knowledge, there is no theoretical foundation that definitively identifies as superior. Moreover, based experimental results in Southern al. (2023) and At- tali et al. (2024), we observe that different curvature definitions have a significant impact on the effectiveness of graph learning, yet it remains challenging to establish clear empirical guidelines. Therefore, using adaptive curvature may be a more ideal approach, and our experimental results support this view.
Computational complexity. Following the calculation protocol in Blakely et al. (2021), the compu- tational complexity of GNRF is ((]VVnn2 + I[) where / is the number of iteration steps of ODE and n is the feature length. This complexity is linear with graph size ((E) and [V] and consistent with l-layer GCN. The method of pre-compuiing curvature usually requires square complexity, such as ((V))) (FOSR (Karhadkar et al., 2022)).
Differential Equation Solver. We use the Adams-Moulton method implemented by torchdi f feeq (Chen et al., 2018) as the default solver for GNRE Although GNRF performs well on most solvers, the adaams-Moulte method often achieves numerically stable solutions with larger fixed step sizes.
EXPERIMENT
Datasets. To evaluate the model fairly, we collect total of 14 datasets from commonly used node classification benchmarks. We report 12 of these datasets in the main experiment: Cornell, Wisconsin, and Texas from WebKB used in Pei et al. (2020); noman-Empire, Tolokers, Amazon- ratings, Minesweeper and Questions from Heterophilous Graph benchmark (Platonov et al., 2023); Full, Cora_ML, DBLP and Pubmed from CitationFull benchmark (Bojchevski & Günnemann, 2017). In addition, to verify the scalability of the model, we also introduce two larger-scale data sets: OBGN-Arxiv from Open Graph Benchmark (Hu et al., 2020) and OGBN- Year from Lim al. (2021). We report the performance and cost of models on these two datasets in scalability experiments. For all datasets, we uniformly adopted a random split strategy of 60%/20%/200 for the training, validation, and test sets. We report the mean and standard deviation of the experiments based on ten different splits.
Comparison method. We compared GNRF with two categories of methods. The first category is discrete-depth GNNs, including two classic models: Graph Convolution Network (Kipf & Welling, 2016) and Graph Attention Network (Veličković et al., 2017), as well as two advanced state-of-the- art models: Feature Selection GNN (Maurya et al., 2022) and Directed GNN (Rossi et al., 2024). Recent studies show that simple modifications can significantly improve classic model performance (Luo et al., 2024; Platonov et al., 2023). Therefore, we add residual connections to GCN and GAT, resulting in enhanced versions: GCN+res and GAT+res. The second category is continuous-depth GNNs, including Graph Neural Diffusion (Chamberlain et al., 2021), GRAND++ (Thorpe et al., 2022), Allen-Cahn Message Passing (Wang et al., 2022) and High-order Graph Diffusion Network (Li et al., 2024). To evaluate the advantages of adaptive curvature.
also compared two variants: GNRF FRC and GNRF ARC- Instead of using EdgeNet, these variants directly use the definitions of Forman-Ra Ricci curvature and approximate resistance curvature to obtain K.

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Published as a conference paper at ICLR 2025
5.1
SEMI-SUPERVISED NODE CLASSIFICATION
Corn. 0.1227
Wise
Texas
R.
Tole.
Mine.
C.Fulll
PuhM.
DELP
C.MLL
Hom, level
0.1778
0.0609
0.0000
0,6344
0.6827
0.839
0.3803
0,5670
0.8024
0,8279
0,7885
4 Node
183
251
183
22,602
11,758
10,000
48,921
24,492
19,793
19,717
17,716
2,995
Discrte-depeh GNNs
GCN
(= 46)
55.14
(47700)
61,60
(1:445)
60.00
(#0.22)
71.23
79.61
74.79
50.21
(:0.61)
37,99
(+0.95)
68.06
(+0,47)
86.74
(H0.34)
83.93
(+1121)
87.07
70.11
69.50
71.65
73.91
83:44
90.13
75.45
18.17
69.53
86.91
R2.64
85.62
GCN+rs
(1:10.21)
(16.00)
(14113)
(10666)
(1061)
(10070)
(12.31)
(10.55)
(10.44)
(10.31)
(10.51)
(10.73)
GAT
(=11.1)
53.64
(+11.0)
60.00
(=8.17)
61.21
(+1.53)
77,40
(=0.2)
(+1.11)
(#0.88)
65.47
(=1.22
42.52
(+1.22)
67.55
(+00.55)
87.24
(+1:21)
80,61
(+0055
$4.12
65.42
72.20
73:45
81.55
83.91
92.45
76.95
50.00
67,33
87.50
83.51
85.11
GATeers
(L.7333)
(1:4000)
(1.6111)
(10.26)
(1003)
(10.77)
(L0.85)
(10043)
(10.68)
(10.00)
(10.72)
(10.19)
DirGNN
76.51
80.50
76.25
55.21
82.64
$1.52
59.95
46.66
67,80
86.94
81,22
K5.06
(-6:14)
(+5.50)
(+631)
(+0.44)
(-075)
(+0.4)
(+0.79)
(-061)
(+0.53)
(+0.55)
(+0.54)
(+0.31)
FSGNN
57:43
$7.60
85:15
83,31
(=3.65)
12:5.10)
(#:1911
(20.71)
(2006))
(30.4))
(20.32)
(=0.51)
(20065)
(#00.71)
(40055)
(4:0.4)
Continnous-eepti GNNs
$1.76
84.00
GRAND
(21399)
(17.50))
(18.42)
(10075)
(10.45)
(13.12)
(1:2.12)
(10.33)
(11.01)
(10.57)
(10099)
(10.81
GRAND++
(-7.13)
81.34
(+600)
81.50
79.34
(+0.51)
68.13
(-0.56
78.85
(+2.11)
78,55
(+0.88
60.14
(-0.50)
38.01
(+0.74)
67.53
(+0.35)
87.21
85:21
(+0.53)
88.44
ACMP
85.66
36.50
87.65
00M
00M
85.11
71.92
37.32
00M
88.01
82.31
76.11
(=5.10)
(=5.00)
(t3
(6.006)
(40.55
(=0.64)
(:1.44)
(+0.44)
(+2.11)
HID-Net
(=7.10)
83.53
(#6.60)
81.30
(=6.91)
77.11
(=0.73)
(=0.71)
79.50
(#0.65)
(=0.85)
63.77
(=1.03)
(#0.64)
68.11
(+:0.5)
88,60
(#0.33)
(=0.51)
89.00
GNRF
(-3.12)
87.28
38.00
(+4.13)
87.39
86.25
83.96
(+0.30 (0)
95.03
73.86
(+1 1.008)
46.89
72.12
90.37
(+-0.66)
85.73
[+0
89:18
GNRE
(=1
$5.59
(+10.55) 100
50.00
(+5,41)
$2.08
(40.68)
75.23
(0446)
76.17
(+1.07)
81.61
(+0,99)
61.78
(=0.43)
41.22
(:0.87)
67.51
(100.14
88.96
(10.32)
K2.55
(10.55)
87.29
GNREARC
$6.49
57.25
70.17
88.21
83,83
[=2 700
2-000
(=5.63)
(=0.33)
(=0.22
(=1.0))
(#1.33)
(=0.46)
(+0.61)
(+10.00)
(4:045)
(20.22)
Table I: We compare GNRF with two classes of methods on the node classification task. Highlighted are the top first, second, and third results. OOM means out of memory. Accuracy is the measure for the vast majority of datasets, and for Minesweeper, Tolokers, and Questions, we use ROC-AUC.
Main results (Table 1). GNRF consistently demonstrates outstanding performance across diverse datasets. Compared to Full, PubMed and Cora_ML, GNRF shows significant improvements of over 20% compared to GCN on the other 8 heterophilic datasets. This improvement is attributed to GNRF forcing positive curvature edges to repel each other, as ACMP also performs well with a similar repulsive bias. However, since ACMP by default uses an adaptive step solver (Dormand- Prince 5), excessively subdivides the step size when solving near-stiff equations, leading to an OOM. In contrast, GNRF remains stable on these datasets. Another observation is that G RC and GNRF A RC indeed show significant improvements over classic algorithms (such as GCN, GAT, GRAND, GRAND++), but they still fall short compared to GNRF using EdgeNet. This suggests that DRR itself is beneficial for graph learning, but adaptive curvature can maximize its utility.
Ablation study (Table 2). Two key modules: EdgeNet (c) and the damping factor (d), are used for ablation. When we remove EdgeNet from GNRF, we replace the aggregation weights with a train- able positive scalar. The results show that EdgeNet provides the primary improvement of GNRF, while the damping factor ensures numerical stability. Due to the inability to utilize negative weights, GRAND performs poorly on two heterogeneous datasets-Roman justing the damping factor does not lead to additional performance improvement. In contrast, ACMP performs much better, as it is similar to GRAND but can leverage negative weights. GNRF further improves upon ACMP, as it enables more fine- fine-graines grained control over the weight signs- signs-achieved achieved through Ricci flow. On the other hand, ACMP uses an adaptive step-size solver, making it diffi- cult to effectively handle near-stiff equations. However, when the damping factor is introduced, it produces stable and competitive numerical results.
and Tolokers- ad-
Resource consumption (Table 3). We validated the scalability of GNRF on the OGBN-Arxiv dataset. We fixed the depth of all compared methods to 3 and reported the model parameter count, runtime per epoch (averaged over 1000 iterations), and accuracy for hidden layer sizes of 16, 64, and 256. For GCN, we used the hyperparameters recommended by the official OGB guidelines (Hu et al., 2020); for GAT, we used the same hyperparameters as GCN and set the number of attention heads to 3. We performed full-batch training on OGBN-Arxiv. Our results show that, for the same model capacity (i.e., the same hidden layer size), GNRF outperforms the other three methods. The
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Published as a conference paper at ICLR 2025
Dormand-Prince
A Adams-Moulton
ODE Solver
(adaptive step)
(fix step)
Method
ACMP ACMP+d
ACMP
ACMP+d
GNRF
GNREd
GNRE/e
GRAND
GRAND+d
Roman-Emppe Empire
OOM
OOM
NC
72.44
86.25
NC
53.71
60.12
58.57
Tolokers
83,96
79.01
Cora_Full
OOM
OOM
NC
71.17
72.12
NC
71.55
67.66
67.31
Table 2: NC means Not Convergent. "d" and "e" represent the damping factor and EdgeNet respec- tively. We use "+" or "P" to denote adding or removing module.
#hidden=16 Time
#Hidden=64 Time
AHidden=256
#Param
Acc.
#Param
Acc.
#Param
Time
Acc.
GCN
3.15k
0.12s
60.95
15.2k
0.14s
68.55
110k
0.21s
71,65
Depth=3
GAT
15.0k
0.17s
59.52
86.8k
0.25s
64.39
788k
0OM
0OM
ACMP
4.18k
3.29%
61.03
32.0k
6.35s
68.89
374k
0OM
OOM
GNRF
5.50k
0.315
62.11
52.6k
0.78s
69.33
701k
0OM
00M
Table 3:
With a fixed depth of 3, we compare the performance and scalability GNRF with other models at hidden layer sizes of 16, 64 and 256 respectively.
number of learnable parameters in GNRF lies between those of GCN and GAT, and its training time is comparable to theirs, significantly faster than ACMP (the current state-of-the continuous deep GNN model). At hidden layer size of 256, GAT, ACMP, and GNRF experienced out-of- memory. We conclude that GNRF strikes meaningful trade-off between the high scalability but low accuracy of classical discrete deep GNNs and the low scalability but high accuracy of continuous- depth GNNs.
5.2
CURVATURE
We demonstrate whether GNRF faithfully adheres to the theoretical guidance provided in Chapter 3. We measure the curvature distribution of all edges in the graph dataset at different time points (Figure above) and record the variance of these curvatures over time (Figure 2 below). We perform a 0-1 normalization of the variance from = 0 to = 20 to scale multiple datasets to a unified scale.
As the results show, in homophilic graphs (Cora-Full and Pubmed), positive curvature edges dominate, while in heterophilic graphs (Roman-empire and Tolokers), negative cur- vature holds more weight. The reason is that in homophilic graphs, nodes of the same class tend to form tightly connected commu- nity structures, which lead to edges with pos- itive curvature (Topping et al., 2021).
Roman-empire
302
Pubmed
1.5
Densky
1.0
500
However, regardless of the curvature distribu- tion, as predicted by Lemma 1, the GNRF designed based Attir-DRF will force them to concentrate around zero. This avoids the over-smoothing information bottleneck is- sues caused by extreme curvature (Nguyen al., 2023). Another observation is that the speed of this "concentration towards zero" is roughly the same across different datasets, as their curvature variances follow a similar in- verse relationship, achieving relatively stable values around t 10. This means that re- gardless the dataset, the GNRF can always achieve stable curvature within feasible time, indicating that the model designed based Attri- DRF is consistently efficient in the evolution of node features and aligns with the expectations of Theorem 3.
0.030
3,15
6:200
5.0
t=2.5
Hamatzeayars0
2.0
Wascarin Comcll Thons
Roman-empir lookes
Putmee Cora-ful
Figure 2: The distribution of curvature across dif- ferent datasets (above) and the variation of their variance over time (below).
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5.3 DIRICHLET ENERGY AND FEATURE VISUALIZATION
We present the evolution of the Dirichlet energy of GNRF with random parameters on synthetic graphs without any training. The synthetic graph consists of 10,000 nodes, where any pair of nodes has connection probability of 0.5, and the node features sampled from 10-dimensional standard Gaussian distribution. We measure the energy variations GNRF compared to GCN (Kipf & Welling, 2016), GAT (Veličković al., 2017), and GRAND (Chamberlain al., 2021) in the random graph. Specifically, to understand the impact of the damping factor (DF), two de- rived models-GNRR without DF and GRAND with DF-are are also used for comparison (Figure 3). The features of GCN, GAT, and GRAND be- come significantly smoother with the increase of time (for GCN and GAT, time corresponds to the number of layers). However, the damp- ing factor can reduce the length of the up- date gradient when features become similar, al- lowing GRAND to achieve non-zero energy lower bound. For GNRF, even without the DF, it will lead to decrease in (t)ll as ||EdgeNet,; (t)]l approaches O (due to curva- ture approaches O), which similarly causes the Dirichlet energy to converge to positive value. Under the combined influence of the damp- ing factor and curvature, the Dirichlet energy of GNRF exhibits near-steady-star behavior, which is also within the scenarios envisioned Theorem 2.
10910 E(t)
GEN
GRAND
CNRF
-2
0.0
2.5
5.0
7.5
10.0 12.5 15.0 17.5 20.0 Time (1)
Figure 3: The trend of Dirichlet energy changes over time across different models.
Wiscomen
Romse-impre
Polmed
Coraell
Tolokens
Tourr
109 (09.0E())
0.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
Time (1)
Figure 4: The evolution of Dirichlet energy of GNRF across different datasets (left); visualization of node representations for Roman-Enne Empire (right).
Next, we show the Dirichlet energy evolution of the well- trained GNRF under each dataset (Fig. 4, left). The re- sults illustrate that the GNRF guarantees that the Dirich- let energy is nearly constant, regardless of whether it is trained. However, constant energy does not mean stagna- tion of feature evolution. We use t-SNE to visualize the node features of the Roman-empire dataset (Fig. 4 right) and found that nodes of the same category do form larger and larger clusters, suggesting that the GNRF is beneficial for classification. We select two classes of node features in the dataset for visualization (Fig. 5) to further illus- trate the mechanism by which GNRF takes effect. When we look at the data at = O. we see that nodes of the same class form many very tight clusters. Moreover, the clus-
0.2
2
Figure 5: Visualization of node features for Roman-Empire's Class 11 and 13.

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Published as a conference paper at ICLR 2025
ters of the same class closer to each other than the clusters of different classes. As time evolves, small tight clusters gradually become loose and merge with other clusters the same class to form larger wholes. This, in turn, leads more regular decision boundaries. However, we also recog- nize that too long an evolutionary time can exacerbate category indistinguishability and, therefore, requires careful tradeoffs.
CONCLUSION
We propose a novel continuous-depth graph neural network called Graph Neural Ricci Flow (GNRF). Specifically,, we find that if the classical discrete Ricci flow (DRF) is generalized tributed graphs, this benefits the evolution of node features, including Dirichlet energy with bilateral bounds and data-independent evolution time. GNRF is built on the attribute DRF and is the first graph deep learning model to use vime-ngrying edge curvature. GNRF can fit edge curvature in linear time and adjust it to near zero, which prevents svor-smog and information bottlenecks.
ACKNOWLEDGMENTS
The National Natural Science Foundation of China under Grant 62176269 and The National Natural Science Foundation of China under Grant 62302537 support the research.
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A
RELATED WORK
A.1 RIEMANNIAN GRAPH LEARNING
Recent research has shown that studying GNNs from the perspective of Riemannian geometry can provide additional benefits. For instance, embedding graph data into a hyperbolic space can effec- tively address the neighborhood explosion problem that arises when performing message aggrega- tion on graphs with power-law distribution (Yang et al., 2022; Sun et al., 2021; Chami et al., 2019). The core idea of such work to map representations into a space of constant negative curvature (in contrast to the zero curvature of Euclidean space), thereby improving representation quality. An- other more direct approach to learn based on general Riemannian manifolds. They use graph curvature measurement for graph topology, and achieve structure-aware learning through graph rewiring (Nguyen et al., 2023; Topping et al., 2021; Fesser & Weber, 2024), edge sampling (Liu et al., 2023), or neighbor reweighting (Li et al., 2022).
A.2 EDGE CURVATURE ON WEIGHTED GRAPHS
Forman-Ricel curvature. The idea of transferring the concept of curvature from Riemannian geom- etry discrete graph-structured data was initially proposed by Forman (2003). This paper suggests constructing an analog of curvature in graph spaces using k-dimensional discrete CW complexes and
their weights. A commonly used definition considers I-dimensional and 2-dimensional complexes (Sreejith et al., 2016), which is:
FR Ki
W. w/
Wiy
Waj
(11)
Wia
Here, wy represents the weight on node i. However, in many cases, we only have edge weights without node weights. A simple approach in this situation is to consider the node weight as the sum of the weights of all its connected edges:
FR Pij =
Wij
Wij
(12)
Wiu
Wjo
Although there have been several improvements to Forman-Rice Ricii curvature, the most notable ones being balanced Forman curvature (Topping et al., 2021) and augmented Forman curvature (Iváñez, 2022). they cannot be defined on weighted graphs.
Resistance curvature. Resistance curvature is defined based on the equivalent resistance in a sistance network. We envision a network composed of multiple resistors connected, where each edge in the network has an associated resistor, and the positive edge weights represent the resistance values. Suppose we measure the voltage and current between any two vertices externally. In that case, the network resistor is referred to as the equivalent resistance. Equivalent resistance is a form of distance (De- vriendt & Lambiotte, 2022; Devriendt al., 2024), and compared other distance metrics defined on edge-weighted graphs, it effectively captures the connectivity information of the entire network. Specifically, if we define the following weighted Laplacian matrix Q:
be simplified to a single equivalent resistor, and the resistance value of this
Waj
if i
j j
(Q)
Σ Ejuitin if i=j
(13)
otherwise.
then the equivalent resistance between nodes and j can be defined as:
Tij = (e, 2T) ((ee-ee) -e).
(14)
where e; represents the i-th unit vector, and Q+ denotes the pscudo-inverse of Q. Resistance dis- tance helps understand the robustness of GNNs and addresses the overs-quashing problem (Shen et al., 2024; Black et al., 2023); it can also be used to define new discrete curvatures (Devriendt & Lambiotte, 2022; Devriendt et al., 2024). In this paper, we use the definition proposed by Devriendt & Lambiotte (2022), which is:
2 - Σ knw Tat Wie
Enng WikWiik
RC Kij
(15)
Tij
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Although this definition seems appealing, it is not computable in real-time. The reason is that calculating the pseudo-inverse matrix requires at least cubic time complexity. Fortunately, Radl et al. (2009) provides us with simple approximation:
Fig
(16)
Exrs Wui Leng Wej
Where N represents the number of nodes in the graph. This formula guarantees an error of O(1/N). The curvature obtained using this approximation is referred as the approximate resistance cur- vature:
--Eree Exws TIAWAI
Eknn FikWykk
RC Ke
(17)
Tij
Ollivier-Ricei curvature. Since Ollivier (2007) introduced this concept, ORC has become of the most commonly used mathematical tools for analyzing networks using geometric methods. has been widely applied in various areas, including complex network analysis (Paeng, 2012), community detection (Sia et al., 2019), hypergraph learning (Coupette et al., 2022), and understanding over- smoothing and over-squashing (Nguyen et al., 2023). A more accessible definition be found in Hehl (2024). Specifically, let us define the probability measure of random walk originating from any node u within its first-order neighbors as:
Way
if ~
Ma (v)
Σ
soa Was
(18)
otherwise.
.OR between any node pair (i, is defined as: Then, the Kij
W1 W1)
OR Kij
=
(19)
d(i,j)
where d(i,j) represents the shortest distance between nodes and j in graph G. For adjacent nodes and j. d(i.j) W1 denotes the I-Wasserstein distance between the two probability measures.
A.3 DIRICHLET ENERGY
Dirichlet energy is a commonly used measure of the smoothness of node attributes (Chen et al., 2020a; Zhao & Akoglu, 2019), and it is also employed regularization term mitigate GNN over-smoothing (Zhou et al., 2021b). First, we define the symmetrically normalized adjacency ma- trix for graph follows:
1
i~j j.
Ā (1++d)(1+d)
(20)
0,
otherwise.
Here, d, denotes the degree node i. When the node attribute matrix on G is H, the Dirichlet energy is defined as:
2
F(H) = TT (1A))) = Σ
h
hj
(21)
VI+d,
v + d
(6.))
B PROOF
[2), xy), the continuous function I satisfies (f(m)) [(2)) 2 L|x1 22), then f(x)) is monotonic in (1), xz).
Lemma B.1. If there exists a constant such that for any 31, 22
Proof. Consider proof by contradiction. Suppose f is not monotonic. Then critical point x" (I,, xr) must exist. Without loss of generality, assume f(x*) is a local maximum. Then there must
exist points x² and o. either side of x² such that f(x") > ==="==((z))
[f(z2) fx=2) L|x xi| 0.
(22)
This leads to a contradiction, so fz) cannot have any critical points x,]. which implics that f(x) is monotonic.
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Lemma B.2. (Marahashi el al., 2012) Let A denote the unweighted adjacency matrix of an undirected graph G, v, the r-th eigenvector of A, the vector composed of all eigenvalues of A. X implies that the k-th power is applied to each element of X. Then the shortest path distance between any two nodes and j is:
d(i.j) = min (v. X x v,) 1 0
(23)
Lemma B.3. (Chen & Wang, 2024) Let E > O, (St, da) be a compact metric space and let X be the space of weighted point sets equipped with p- w-Wassesstein, for any A, B X, there exist trainable networks @n, Ø2 and Øs with sufficiently large numbers of parameters such that:
wyds(t) )) <.
Wp W, - Φ1
@2
Σ (IA
wals(a))
22
Σ
(24)
(y.Rn)ER
Lemma B.4. For any symmetric adjacency matrix A E {0, 1jmxn, and its corresponding degree matrix D diag(d (d1, ... ,dn), the following inequality holds:
d2 E1da.
-
Ag
20
(25)
v(++d)
iE|n
RE|M]SE[M
The equality sign is taken when and only when A is regular graph.
Proof. Notes that:
d
d
(26)
di
(1 + de) (1
d
kE|m]
iE(n]
and
ΣΣ
Ag = A D) d[
A(I
(27)
xE[m] JE[N]
Thus the inequality be proved can be transformed into:
1 [ D) - (D A) D) 1, 2..20 0.
(28)
Let 1, - and L D A, now need to prove:
x² La 2 0.
(29)
This is clearly valid according the semi-positive characterization of the Laplace matrix.
We now consider the conditions under which the equal sign holds. From linear algebra, we know that the quadratic form xx' Læ is equal to 0 if and only if æ lies in the null space of L, i.e., Læ 0. More specifically, (D- A) + D) 11, 0. Expanding into elemental form, we get that for every node i, it must satisfy:
de Σ
(30)
VI+ d,
I1+
Obviously, when A is regular graph, i.e., d; r, this condition always holds. However, if A is not a regular graph, then there must exist a node in A with the highest degree among all nodes, and it is connected to least one node with lower degree. Let us denote such node as i. In this case, we can conclude:
d;
Σ joi
>
> V
d.
(31)
V
VI+d
VI+d
jul
This leads contradiction, meaning that the equality can only hold when A is regular graph.
n
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B.1 PROOF OF LEMMA 1
Lemma B,5 (Formal version of lemma 1). Consider the Attri-DRE over the interval t2]. If the edge weight wt)(t) has a finite number of N extrema points within (t1, t2), denoted by Ti.... TN, and define To = I and TN+1 = l2, then:
A-1
1-
Beelt 12) (mngg(
(32)
t2 t
where Ç = minogigN (Ti+1 T:) represents the length of the most minor maximal monotonic interval within [ti, t2), and max. w(I.) denotes the ratio of the maximum to the minimum
miny wo(T)
value of w(t). Specifically, ws (L) is monotonic within (21,12), we have:
(w((2) t(ta)) > (ase
(33)
min(w(te w(t1)}
- t1
which (t((t)) - t(tz)) + 0 implies Exe(th ((s,) (t))) - O.
Since the edge i is arbitrary, omit the (t)(t) subscript in the proof. First, we integrate Equation (3) over an arbitrary time interval [t1, tz]:
w(t2) - w(11) = - /, k(t)w((tddt
(34)
Suppose that within [(,,ta] f2]. the function w(t) has N extremum points. These N points divide the interval [11, tz] into N + adjacent monotonic sub-intervals. Given that w(t) > 0, according to
Equation (I), it follows that the sign of (w(t) is determined by k(t). Moreover, the x(t) sign remains non-negative or non-positive within each monotonic sub-interval.
ăt
Let [2;, Ti+1 | denote any one of these N + monotonic sub-intervals. If ((t) 2 for E [i,,T++]]
then (t) is monotonically decreasing, which implies wT,)) > w(1) > w (T;+11) > 0. Therefore:
[
T++1 x(I)dt >
Pi+1
0 > (T++i)
x(()w(()dd > (T.) T;
x(I)dl.
(35)
JT;
If x()) < 0 for [T,, T.+1l, then w(t) is monotonically increasing, which implies w(T++1)
>
w(t) > (T,) > 0 and
Tii2 f x(t)dt >- -f
THi1
k(t)w(t)dt > (w()) /
T1i2 x(t)dt > 0.
-w(Ti+1)
(36)
Therefore, on any monotonic sub-interval, we have:
f k(()dt > w(T+11) - (T;) > -w(T)) ( x(l)dl.
Ti+1
-w((++1)
(37)
Tr
Take the absolute value of the above inequality:
Te+
( (fit) In(()]dl.
max max(w w w(i++))
[n([)]dt (u(T:+))-w(TT min((w(Ti), w(T++1))
(38)
Let Wmax (Tx)), w(T++1)) min(w(Ti), w(T+1))) > Wmin. have: -
[T(T++1) wT,))l
Titi
[s(t)ddt>
(39)
Wmin
Wmax
Summing over i from 0 to N and then dividing by 12 - t::
Eizo [wT(T++)) - w(TT)
1/ 11 1/
Ei (T wT(
In(()ddt Eserer (1ssg(()))))
Wmin (12 11)
Wmaxx (12 - 11)
(40)
Because (N + (Wmax - Wmin) > E.N (w((T++) - w(T:) > Wmax 2 Wmin- and
Wimax
then:
Wimin
(N + 1)(A 1) 1/ - 4
1-A-1
>
>
(41)
12 t
According to the definition of Ç, we have (t2 t1) > C(N + 1). Thus, the estimation of the upper
bound can be replaced by 111(2- 1) 42-
(N+1)0-11 thereby completing the proof.
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B.2 PROOF OF THEOREM 2
Theorem B.6. Consider the Atrri-DRF with edge weight wsy (t) cos h2) + 1 + and
h(() = 1. If for any j in 9, Wig(t) (1) has a finite number of monotonic intervals on [1, 12) then the average Dirichlet energy within [1,, 12) has following bound:
Bi 2 te (E(H(()) 2 B2.
(
(42)
where:
B1 = (12 - -11 Σ
(112 (112-11) (1 - Aij. (iJ)EE
Σ
1 1 d
VEV
(43)
d2
B2 E
Au
(t2- (1+e- e)
+ Σ 1+ d:
(6J)EE
di denotes the degree of node i, and À represents the symmetrically normalized adjacency matrix of graph G.
We begin the proof by expanding the definition of the Dirichlet energy. Noting that [[h,(t) 1 and (t)(t) (t)² (t) + 1 + we have:
((H(())
1
Σ
(t)
(t)
2
V1+d d
V1 d3 12
(HJ)EE
[t,(1)]1
(t)r
h,(t)Th,(t)
Σ
2
2
1 di
1
V(1 + d.) +
(KJ)EE
= Σ
146
Wip (t)
+
2
1 de
l+d,)
V(1 + d:)(1 + +dj)
v(1 di d:)(1 (1 dg)
(E,)EE
(44)
By defining cy A 2
1+d.
V(++))(+d) and (t) A cij
((++))(++d)
(+)
the
Dirichlet energy can also be simplified to:
(iy(1) V[I d:) (1))))
)
E(H())) = Σ
Caj
Σ Etj(1).
(45)
(1J)
(K,)EE
Take the partial derivative of the above equality to t:
(E(H(I)) (t))
owi; (t)
Σ
0L
V(1 +d)(1 d,)
Øl
(L,EEE
Riy (iy(t)wiy(t)
Σ
(46)
d1) + dj) kty(t)
V(1 d1)dd
(LEE
Σ
Jay Eig (t))
(E,))E
Noting that the above equation holds for any graph G, we can construct subgraph Gi such that it contains only two vertices (i, and a single edge ~ j. In this case, we have:
(ty(t) = (ty(t) Jay Et)(t)
(47)
Ōt
Integrate over arbitrary time interval (/1, 12):
Ety(t2) E.g(tr) = cij 1'h Ky (t) - 1 mi (t)
(48)
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We first prove the lower bound. Similar to the proof of Lemma 1, consider the N extremum points of wty(t) within (f1, t2): Ti.".. .Tv. For any i, Kiy(t) does not change its sign within
(T,, T.+1), and since Ety(t) 2 O. we have (.r (t) = -r. Eay (t) so:
Pit
Pit
(B:((T+)) Eiy(Ti)) = Cij
Key
- (e)(t)
Mis(()Ens(1)dd)
T1
J1,
T+t1
| Tits
Cij
Key (t)(t
Tr.
(i)(t))
T,
(49)
Ti+1
Tits
Cij
|nay(t)]dt
JT.
(rag(t)) Eis) (t)dt
T:
Ti+1
( Eiy(t)d(
Tits
[nsg(t)[tt Beele (a J.
2 Cij
T;
Sum over i from 0 to N and then divide by to t1:
N
Σ
Byg (Ti+1) - (T.( Fa (1t)((()) Ci Re
(50)
f/ h
Noting that (N + 1)Gis S - 1 and waj S + e, the left-hand side of the above inequality can be enlarged to:
1
N
Wi (T++1) - Wisg wig (T.)
-T.) E,G
1/2 4 i=0
12 1 1=
v(1 d;)(1 + dj)
(N + 1) (max Wij min way)
(t2 t1) (1+ d,)(1 d,)
(51)
47 max wiy (1 X7y) )
v1 + di)(1 + dz)
(2 + +)5T (1 (1- x711)
v (1+dd) (1)
Thus we have:
(2 12+0)5-(11220) 2 (e (tj (2el
(52)
(1 dj)
By applying Lemma B.I, we can obtain the lower bound for EsEte, Aa) (t))
(2+ (1 (1-x7²))
Ea (E((tt,))((EEtt))) 2a/
Ex (Ix(t)) (11 d:((1 + dz)
(2 + c)5 (1 X751)
1
1
1
1
+
+
+
Ere (k:j (1))
v(1+ d))(1 + d,)
(1 d;
1+
(2+ (2+c)55 (1 \551)
1
|1+-- -A.
+ "1 dd
Erelen (15:;(())) (t)l
2
1 di
1
1
2Ā (1 + - (2 - c) (12 - -11) +
+
2
1 d,
(53)
Take the sum over all edge:
Ā (1+c- (2+ (2+ (12 41) + Σ di
1. (E())) 2 Σ
(54)
1 + d;
(1.))
NEV
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Now, let's derive the upper bound. Recalling Eq. (49), for any maximal monotonic interval [T,, 1111 of (t)(t), we have:
T++1
[ |nay(t)ldt
Te+1
1N Esy css J
(Exy(T+a)) (Ty(()))
(55)
T
Sum over i from 0 to N:
- csj J. 1
N
Kiy
[kss(1][d -
Σ
Big (Ti(1) Eiy
wiy (Ti++) - wo
= Cij
(nis(t]lt
v(1 d:) dj)
1=0
(56)
max Wij min Waj
$ Cúj
[ks((t]lt
v(1 + d:)(1 + dg)
<(Aig -1)
< Cij
[NTg(1][t
V(I+ + d.) + dj)
Divide both sides by lo t1:
<(Ai - 1) (t2 - ti) [1) di) dr)
te (t)) S c.y (IK))(t)))
(57)
Applying Lemma 1, we obtain:
Xij t2 t1 Beel (Eij(1) (Ei)(()) S cooST (N: 1) (t2 (1 + di)(1 + dg)
<(Ay - 1)
(58)
Due to E(t) E(in)ee (t) we can derive the upper bound as:
E (E())) s (t2 - -1) Σ
GiddesCo
Σ
Exij (1+d)(++d)
(Bwj)EE
(iMj)EE
= =(2 1 -1)
Σ
AisGTJ
(bl +
(1 (1+()))))) I+
++
1
- Σ A
(RJ)EE
(WJJEE
Ejni NisSiaj
(12 4)
+ Σ (12- (1 ((to)-))-
1 di
NEV
(1.)EE
(59)
Combining the results of Eq. (53) and Eq. (59), we have:
Ejnr AusSi
(l2 - (1) Σ
+ ((12 - (1) (1 - <(ii))Ag
1 dj
1EV
(6,F)E
(60)
2 Eight E(t)) >
Σ (KS)E
Au
() (t2 - -t1) + Σ + d;
iEV
Theorem B.7 (Formal version of Theorem 2). Consider the Auri-DRF with edge weight wiy (t) =
cos(h; h2j) +1 + and [h(t)] = If G is a non-regular graph, and all Wij are monotonic on [1,, 12), let Amax = mij)eE Wiy,we have:
Xmnax Σ
d; E1dd sum(A) i+d
E (F(H())) > Σ d:
1+d, SA 0. (61)
Note that if x(()) is monotonically approaching 0 within (2, we have t2 - ti = = Then this conclusion is direct generalization of Theorem B.6 and Lemma B.4.
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B.3 PROOF OF THEOREM: 3
Theorem B.8 (Formal version of Theorem 3). Consider the Attri-DRF with Wiy(t) =
cos(h;, h2) + + e and [h(tt] = and h = on any edge j. Assume that the curva- wrre is bi- bi-lipschitz continuous when considered as a function of the weights, iee, there exist I, K such that [w((2) w((1)] 2 [x((t) - x<t1)] | 2 [.(w((2) w(t1)]- )) Let [1ss(0)] O. For any
arbitrarity small positive number 8, it holds that:
2L +
min (1s((1)) = S Le 5 In t00)
(62)
6(2 +
Similarly the proof of Lemma 1, we omit the edge i j subscript. Since the proof process for x(O) < O is entirely analogous to that for x(0) > 0, we can assume k(0) > 0 without loss of generality in the following proof.
Denote min (15s) (4) = 8) = T Because x()) > for all / E [0. T*), then awi()) and w()) w(). In this case, the condition (x((2) k(t)) 2 L|w((z) w(t1)) will lead to the following two cases:
at
1. <tt) 2 x(0) + ((w(0) m(t)
2. (() < =(0) - (() (()
We first discuss the case 1, we will demonstrate that this case is impossible. For any E 00,T",, we can fix the values of w for all edges except the one currently under consideration. At this point, k(t) can be expressed as function of wt()) i.e., x(() x(w()). According to x(() > k(0) and Lemma @x(w(t))
B.I, we know that x(w(t)) increases monotonically as w(() decreases, i.e., dw(() I - 0.
Please note that this holds for all t E [0, +o0). not just within [0,T*). So for any t E [0, +o0), we have:
dx(t) @n(w(tt) @w(() -x(w(t))w(() [k((w(t)) 0. Owl(t)
(63)
Ot Sw(()
Therefore:
@w(() = =x(())=(t) S -x(0 < 0, Vt E [0, (00+
(64)
However, this is impossible because it would lead to w() decreasing without bound, while wc() 2 6. This results in a contradiction, so this situation cannot occur.
Next, we consider Case 2. Using similar proof method as in Case I, we can show that for x(t) > 0,
On(w(()) > L >> 0 holds. Reviewing w() over T*), it is monotonically decreasing. To maximize (w(()
T*, we need to: 1) maximize (0)0- (T*), where lead to w(0) = 2+ and w(T*) = c. 2) minimize @w Dt
Since ((T") = 8, for any t E [0, T",,, the minimum value of xct) satisfies k(() - x(T") =
[(w(t) - w(T')) )), so: R(t) = ((w(t) <) + 8. Therefore we have: @w(t)
= - (I(((()) - c) + wl)(() = - Lww(t)² - (Te - 5)v(())
(65)
Øt
This is first-order nonlinear differential equation of the form y' = -ay² + by. whose general solution is -a + where L and b
By manipulating the general solution,
we obtain:
bt aw(t)
(66)
aw(l) -
When f = 7", w()) e, and when 0, w(t) -2 Substituting these values, T can be solved as follows:
aw(( (T)
aw)(
T =
(T((T) -
aw(()
(67)
1
21. + 5
In
Le - 8
5(2 +
Treating I. and c as non-zero constants, we have:
T= 0 (In (8-1))
(68)
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B.4 DERIVATION OF PROPOSITION
From the condition [h,(()] know that 0, (() 0, which expands using the chain rule
to:
(t, 0.
(t), Ōt
(69)
Moreover, due to (t) - h,,t) (t) + + e, we know that 8w((s, notation, we omit the independent variable t and denote @h,(t) this point, the original problem takes the following quadratic programming form:
ht(t). To simplify
5h;
as y. Let Ny (0,0h,
At
at
(0,,0,,00),,;00))
min yd y s.t. h y , h y 0.
(70)
(71)
Consider the method of Lagrange multipliers:
vyL(y)=
V. (y y 1(h y- (ub Th =
2y + Mh, + =.
(72)
Left-multiplinng Eq. (72) by h. and combining it with Eq. (71) and h h; 1, we get:
T T= -= hj-
(73)
Similarly, left-multilling Eq. (72) by h
2h y + -h h:
2/1 + x A(h3'h;)²
À = 2 2/ (1 hi)²)
(74)
Thus we can obtain the solution to this optimization problem:
3" Ahj The Xh.
A h.h?) h
2
2
1 (I h,h?) hj
11
h2 cos (h:: 12,)
(1 te: = -m/5(0) no cos (1a, 11,,,1)00 12
1 h.)² h.)²
1
(he h2)
)
(75)
Note that the constraint condition does not provide an upper bound for || but || 2 always holds. Therefore, y" corresponds to the point of minimum y.
B.5 PROOF OF THEOREM 5
We first define EdgeNet layer follow:
(k+1) (LP(R) e
Σ MLP(A (R)
0, (c(h)
(k)
(1)) Gig
MLP(1) Σ
(M) 01 (c
(R)
(76)
where, Giy represents the attributes edges i j. which can be scalar vector. || indicates concat operation. EdgeNet can be viewed as a natural generalization of DeepSet (Zaheer et al., 2017) graphs, which performs permutationninvariint mapping of the neighhorhood of an edge.
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Theorem B.9 (Formal version of Theorem 5). Let When Forman- Ricci curvature FF,, Ollivier-Ricte curvature xO or approximate resistance curvature RRC are used as the definition of edge curvature in GNRF, there exists an EdgeNet that can approximate the aggregation weight of GNRF k' with arbitrarily high precision. Respectively, have:
(0) Iʃ = x FR then a I-layer EdgeNet with inputs Ejj = ha(t))lby() approximate w', i.e.
(1) = Egg
(0) IfN RRC, then a 2-layer EdgeNet with imputs egg
h. (1 approximate K', i.e.
(1) = (2))
If k. OR, then I-layer EdgeNer with inputs Cij (ttt))lh w,, approximate k',
(0)
i.e., (t) cij
Here, vi is the i-th eigen-vector of the adjacent matrix A.
According to the standard Universal Approximation Theorem (Hornik et al., 1989), MLP can approximate continuous function to any desired accuracy. Therefore, to prove Theorem 5, we only need to assign each learnable MLP in the EdgeNet a continuous function. By approximating eachMLP to its corresponding continuous function, EdgeNet can ultimately approximate any defined curvature.
1, then there exist a continuous function f such that:
K/j 2 2(wir
Wa
(77)
For simplicity, also denote MLP(()
8n as pn let vi (h,(t))lh((t) (i)(t)-Ou 0..5 and
2 (x]|v||=111) f(2 (v²x + ex +)) + 1 + e) then approximate Forman-Rico curvature:
Σ vi
(h (t)l())),
h (1)00)))))))))))))
E Φ1 x1 (h)(())))()))
=P2
umi
mmy
= P2
Σ wi
(ttt))l),((t)
Σ w.y
umi
anj
FR Kij
let 1100 (ha(1)]()((() = wij (1), (0) (<||v|=]||r) = ( + *r) 1|10=+++ +1 + (x]l)) = xy and
22 (2) (x|[||/=1]] = = - 2) then we approximate RCC.
(0)
Σ 10) (h. (t)lla)) uni
( x10 (1)((())))))()))
14 =42
(0)
Σ win A,(0)11,(0)
Σ Waj
=42
and
-PogllWis
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(2)
=
(1)
Σ
(1)
(Fuil Suillwns)
Pis
wij
Σ vnd
xp
(Fngllwos)
um0s
=P2
(1)
Σ
Tui Wui
Pij
Wij
Σ vnjj
Tei Waj
Wej
un0
RCC
For Rig the situation becomes a bit more complex. Noting KO = W1 (P:: Mý), our objective transforms into approximating the 1-Wasserstein distance on the graph using neural networks. Due to the result of Lamma B.3, we need find a representation for each node such that there exists distance function on this representation to form compact metric space. For Ollivier-Ricci Curva- ture, this distance must be the shortest path distance. To achieve this, we can to perform spectral decomposition on the unweighted adjacency matrix of graph G to obtain set of eigenvalues and eigenvectors:
(78) Then, according to Lamma B.2, there exists function f such that f(v;, 12) d(i.j). where d(i,,) is the shortest path distance between nodes and j. Therefore, {u,) forms a reasonable set points in the graph shortest distance space. In this time, there exist trainable functions @1, 22, and da such that:
{u:): A SVD(A).
Wi(ui, = $3 @22
Emmnon aurti (vw)) + Φ2 amm
Σ onjj
Woy (onndi())))) (v.))
(79)
Let 11
(0) (t)
(t)lle;) (t,
wir (t) and 5 (=]||||=][||]] ((1-))))))(((-(lleell (011)))))))))00000000000000)))))
1 + é) then we have:
(0)
Σ (0) (hu(t)) (1((t)))) umi
Σ p10 (m..t)llb,(ttllle))) enoy
2
(0)
Σ Wiidi (ww) (h. (h.(0))),,
Σ Ees (ww) Dnd
=P2
uni
OR
Rij
C EXPERIMENTS
C. IMPLEMENT DETAILS
Code. An implementation is available at:
https//gittab.com/LLong hat.a//ccoooooooooccC/////EEEEE new
Models. In the experiments, EdgeNet to be single-layer because found that this performed well enough. Specifically, we perform experiments using the following formula:
(t,
Σ
EdgeNNel, (1) [2,(0) - cos (1,(1),,,)))
(80)
jnv
where
EdgeNett, (1) = MLP
Σ MLP (ewi) ai!
Eig
Σ Dno
MLPo, (evz)
(81)
and cij (t) (,, Both MLP,, andMLP,, 02 are 2 layers. Among them, the number of their hidden neurons and the number of output neurons of MI.Po, 0o are consistent with the dimension of
25

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Published as a conference paper at ICLR 2025
Algorithm Solve GNRF with Forward difference method Geometric style)
1: Input: Features X, labels y, edge index E
2: H -ree-teeeeeeee pre.trans pre.trranformm )X);
3: sour -our,ddr,a desl_idz E:
4: for t= = 0, ... T do
5: H Hourr (t) H [s)rr idz); (t) Httsdesst idz];
6: if use Net then
7:
0(0) H sour (t) H dest (t);
8:
for = 0, 1, ................................
do
9:
E scatter(MLP ())
E
sour -id);;
10:
E(i)
dest scatter (MI.P
(1)
(E()
dest_idx);
B1
11:
E(++1) MLP!!
(0) sour
E
Best
82
12:
end for
K =E(L;
14:
else
15:
Compute curvatures k(t) by (t) and E;
K ko() coo( H zour(() (U)) +1+e),
16:
2-2 cos" sour (d,
17:
end if
18:
(t) - cos (oouur (f), (dest(t)) sourr (t): (t)
19:
H( H(1) 11 scatter (-K o H(), sour idx)
end for
21:
Z trasstrranfommm H (T));
22: Compute loss and back propagation via Z and y.
h. The output dimension of MI.Pa, is 1. One can also extend the output dimension of MLP to
/hl. which call channel-wist curvature.
An explicit scheme GNRF can be given using forward time difference:
h4 = h; 7 Σ
EdgeNett, jmm
hn [hy- - cos (hg h1) h.)
(82)
where 7/ is step size. When people set termination time T, this update process will be executed multiple times within [o, 7). eventually producing output h(T)). The division of time slices is auto- matically performed by the ODE solver and is highly related the solution algorithm.
Experimental Platform. Our code is implemented in Python 3.11.5, with the primary libraries be- ing PyTorch 1. PyTorch Geometric 2.4.0, and Torchdiffeq 0.2.4. All experiments are conducted on a single NVIDIA 4090 GPU with with 40GB of VRAM.
Hyperparameters, We fine-tune GNRF within the hyperparameter search space, performing up to 100 trials on each dataset. The hyperparameter search space is as follows:
Hyperparameters
Search Space
Distribution
Remark
learning rate
[10-5 10-2
log-uniform
N/A
weight decay
[10 110-6:10 10
log-uniform
N/A
dropout
[0.01,0.99] 164,128,256)
uniform
N/A
hidden dim
categorical log-uniform
For Ogbn-arxiv. it is fixed at 64. N/A
time
[0.1,10]
Table 4: Hyperparameter Search Space
26

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Published as a conference paper at ICLR 2025
Compute curvature via EdgeNet. In GNRF, we use the output of EdgeNet the value of Kiy (t):
Kig (t)(t) heg)
EdgeNet,,, (t) (t) (1 + ((h,(1) ((h((0,))()))
(83)
h2)))
Where À is a scaling factor used adjust the influence ratio of Attir-DRE on h; and hj for edge i ~ j. Currently, we set À = 1 by default, and the calculation formula for the learnable time-varying
edge curvature is follows:
(t) 2 122h"1 h h h2 +1
EdgeNet,y (1).
(84)
This equation is applied in Section 5.1.
Homophily. To comprehensively evaluate the model's performance, we pay particular attention to the diversity of the datasets when making our selections. As a result, we chose three homophilic graphs and five heterophilic graphs. The homophilic graphs exhibit higher level of homophily in comparison, with homophily defined as follows:
K(u, v) (u, v) EE = yoHl
H
(85)
[3]
This is also referred to edge homophily ratio in the literature.
C.2 MORE EXPERIMENTS
Graph rewiring (Table 5). Graph rewiring typically has complexity of O(V)?) (FOSR (Karhad- kar et al., 2022)) or ((VIE (SDRF (Topping et al., 2021)), which restricts its application to smaller datasets. In contrast, GNRF operates with O(V) + {E|) complexity while still offering similar curvature adjustments as graph rewiring. Moreover, graph rewiring overlooks label informa- tion, potentially losing valuable structural priors that are useful for downstream tasks, which hinder consistent improvements. The end-to-end GNRF effectively addresses this issue.
Backbone
Rewiring
Cornell
Wisconsin
Texas
GCN
FOSR
57.75(1 2.61]
62. 02.50( 0.900
57,33(| 2.67) 65.11() 4.89)
SDRF
56.63(1 1.49)
61.60(" 0.00)
FOSR
86.19(] 0.79)
88.67(- 0.67)
83.38(1 1.01)
GNRF
SIRF
87.39(1 0.09)
87.33(. 0.67)
84.98(1 2.41)
Table 5: GNRF as the backbone model for graph rewiring
Graph classification (Table 6). We report graph classification results on three commonly used molecular graph or protein graph datasets (NCI1, DD. PROTEINS). We use the widely adopted 80%/10% 10% train/validation/test ratio for random splitting (Ma et al., 2019; Zhang et al., 2019; Ying et al., 2018). And report the mean and variance over 10 different divisions. We perform both Sum and Mean pooling for all methods. The parameter search range remains consistent with the main experiment (Table 1). We found that continuous-depth GNNs generally perform better. We speculate that this may be because the graph-level task requires fusing information from all node information in the entire graph, which is a challenge for discrete GNNs, but is easier for continuous GNNs. This is because in order to achieve sufficiently high accuracy, the ODE solver often needs to perform many time step within (0, 7]. and it is usually much more than the common layer setting of discrete GNN (for example, within 5). GNRF performs better than the current advanced continuous depth GNN, namely ACMP.
Long range graph learning (Table 7). We verify the ability of GNRF to combat the over-squashing problem on two graph classification datasets with more than million node reviews- Peptides-func and Peptides-struct. We use the same partitioning and verification methods as in Dwivedi et al. (2022). We use two commonly used position/structure encodings: LapPE and RWSE enhance model performance Rampášek et al. (2022). GatedGCN Li et al. (2015) and SAN Kreuzer et al. (2021) were added for comparison. Based on our results, we find that GNRF shows significant
27

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Published as a conference paper at ICLR 2025
NC11
DD
PROTEINS
Pooling
Sum
Mean
Sum
Mean
Sum
Mean
GCN+res
75.28 + 1.33
76.26 = 1.05
74.81 = 0.96
76.12 0.57
75.42 = 1.30
75.82 + 0.35
GAT+res
73.25+
73.65
1.35
76.68 0.88
77.26 2.01
74.44 1.35
74.51 0.96
GRAND
76.54 1.51
77.82
0,68
76.56 + 0.55
78.51 1 0.87
77.12 = 0.53
78.25 + 1.14
ACMP
77.42 0.60
79.09 - 0.77
75.82 = 1.83
78.44 0.53
78.88 = 0.33
78.34 = 0.66
GNRF
79.59 + 0.09
81.67 + 0.51
78.52 0.64
79.08 + 0.88
78.59 + 2.12
80.12 0.54
Table 6: We compare GNRF with other models on graph classification datasets.
improvements over classic message-passing-basee GNNs. Without additional encoding, GNRF im- proves performance by at least 3% over GCN on both Peptides func and Peptides-struct. When additional encodings are used, GNRF's performance can rival that of SAN (a Transformer-base ar- chitecture). However, we acknowledge that GNRF still struggles to match the state-of-the-ar Graph Transformer methods on the LRGB dataset. But it is forgivable because GNRF remains a fully message- paassaggg architecture, where first-order neighbors are the only direct source of information for feature updates. Compared to Graph Transformer methods, GNRF has much lower computa- tional complexity and is more suitable for large-scale single-graph scenarios.
OCN
SAN--LipE
SAN+RWSE
GNRI
L
GNRE-RWSE
Peptides
0.9930e11.0233
0.634460:0221
0,6439b0 0075
0.623360.0000
0.64
0640000.0056
Pepndes MAI()
0.5496811.003
033357
0,265250
0.25
0.3166
0,2675
02811=00031
Table 7: We verify GNRF on Long Range Graph Benchmark.
Model depth (Table 8). We measure the resource consumption and performance of the models dur- ing training on the OGBN-Arxiv and OGBN- datasets. Since these two datasets share the same graph structure and attributes (differing only in labels), we only present the resource consumption results for AGBN-Arxit The length of the hidden representation for all models was fixed at 64. Two classic deep GNNs: APPNP (Gasteiger et al., 2018) and GCNHI (Chen et al., 2020b), are used for comparison. In GCNII, each layer of discrete-depth GNNs uses different learnable parameters, resulting in parameter count, memory. and time consumption increasing linearly with the number of layers. contrast, continuous-depth GNNs have parameter count and memory usage indepen- dent of depth, giving them significant advantage at greater depths. Compared with ACMP, the training time of GNRF does not increase as the depth increases. This is an advantage brought by the fixed-step ODE solver. In terms of performance, whether it is the homophilious graph (Arxiv) or the hetrophilious graph (Year), coutinious-depth GNNs (In particular, GNRF) shows significant advantages when the depth is 4 or 16. When the depth becomes deeper, the performance of both ACMP and GNRF decreases due to the accumulation of errors in the ODE solver.
GCNI
APPNP
ACMP
GNRF
#Parum
Mem.
Time
Acc.
#Param
Mem.
Time
Acc.
#Turam
Mem.
Time
Acc.
APrrum
Mem.
Time
Acc.
Arxiv
4.
27.2k
222G 0.14s
63.55
33.94
15.0k
1.6GG
0.16s
39.58
64.52
19.5k
7.15G
8.43s
67. 16
47.55
35.9%
11.5 0.79:
48.55
69.25
& Year
16
76.0k
4.06G 0,24s
35.01
64.37
15.0k
1.64G
0.22s
39.03
64.51
19.5%
7.15G
13.7%
43.53
65.72
35.9k
11.5G 0.79:
44.13
65.14
0.66s
67.55
0.55
64.66
39.08
51.72
42.31
1150
55.25
40.15
Table 8: We verify the performance of GNRF at different depths and report the model overhead.
C.3 FUTURE DIRECTION
In this paper, we focus the application of Attribute Discrete Ricci flow in message passing- based discrete/contimuuu--ephhh GNNs. However, in view of the excellent performance of Graph Transformers (GTs), especially graph-level tasks, we believe that it is also meaningful to consider the application of Attri-DRE in this type of method. We believe that this generalization may be feasible, based on two observations:
28

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Published as a conference paper at ICLR 2025
(1) In theory, there certain curvatures that can be defined any node pair (i,j) without requiring i and j to be adjacent (For example, Ollivier Ricci curvature). This in GTs is very useful because GTs directly aggregates information from the entire graph.
(2) In practice, a significant difference between our model GNRF and GARND that the ag- gregation weight replaces the attention coefficient with curvature-aware coefficient. Given the widespread reference of attention coefficients in GTss, this replacement is likely natural.
We also provide a possible promotion here. Let PE(-) be some position encoding function and -im(::-) be some similarity function. We can let wiy()) (1) = sim(PE(i,,) (), PE(j, ()) to get the general- ization of Attri-DRF:
Ssim (E((,)))PP((()) (Rij())sim(PE(( 1), PE(J, t)).
(86)
We leave application on GTs of this definition to future work.
29


Reviews:
[{"review_id": "UFjaTsDjtf", "reviewer": "Reviewer_tn6E", "summary": "The paper introduces the dynamical system Attribute Discrete Ricci Flow (Attri-DRF) and incorporates this to propose the Graph Neural Ricci Flow (GNRF), a curvature-aware continuous GNN. This ensures that the graph Dirichlet energy can be bilaterally bounded and that the curvature decay to 0 independent of data. Using an auxiliary network (EdgeNet), the model can theoretically incorporate different types of curvature definition. GNRF has excellent performance on many data sets against a variety of discrete and continuous GNNs.", "questions": "1) Why does having a data-independent curvature decay rate a good thing?\n2) Where does equation (3) come from? I checked the Ollivier paper and I can't find this equation there.\n3) Does GNRF satisfy the theoretical results in Section 3. If it does, it would be great if the authors can clarify this a bit more within the paper.\n4) Over-smoothing and over-squashing are problems caused by the message-passing design in GNNs. Considering that GNRF does not strictly adhere to the message passing design, is it appropriate to mention these problems in this work?\n5) Does GNRF work on larger datasets?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Theoretically, the paper provide several interesting results. \n1. Section 3 provides guarantees on the curvature decay rate and the stable curvature limit of Attri-DRF when certain conditions are met, along with providing a bound on the Dirichlet energy when the curvature stabilizes. This indicates it may be able to avoid over-smoothing/over-squashing. \n2. Incorporating recent results, the paper uses an auxiliary network (EdgeNet), which is capable of approximating arbitrary edge curvature with high precision.\n\nExperimentally, the paper performs well on a variety of popular node classification tasks against a number of old and new discrete and continuous GNN architectures. Section 5.2 and 5.1 provides good evidence that theoretical guarantees hold in reality.", "weaknesses": "1. It is not clear to the reviewer how the theoretical results tie together/what assumptions are made at each step of the way.\n2. The design of EdgeNet is glossed over within the paper, with only a few formulas mentioned within either the main paper or the appendix to explain it. There's also no comparison between EdgeNet's curvature values compared against any other type of curvature that it supposedly can approximate.\n3. The datasets used in the experiments are relatively small datasets.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "6P1ZSqJETu", "reviewer": "Reviewer_tPnw", "summary": "his paper introduces Graph Neural Ricci Flow (GNRF), designed to model a dynamic system called Attribute Discrete Ricci Flow (Attri-DRF) on graph.\nUnlike traditional GNNs, which use multiple layers and pass outputs from one layer as inputs to the next, the model in this work employs only a single layer.  Instead, it iteratively updates node features and curvatures—treated similarly to edge weights—over discrete time steps according to the Attri-DRF ODE.", "questions": "1. Could you share your thoughts on the relationship and differences between the curvature predicted in this architecture and edge attention mechanisms? And do you think it is possible to apply attention mechanism in the framework?\n\n2. In EdgeNet, are the edge features from the previous time step used as input for the current time step? Equation 13 suggests that the previous edge features should be used, but the code appears to rely only on the previous node features without incorporating the prior edge features.\n\n3. What is the formulation for updating node features at each subsequent time step? As a suggestion, including an illustrative figure or a pseudo-algorithm would help readers gain a clearer understanding of the overall framework.\n\n4. While there is only one layer in the implementation, is it possible to apply a multiple layer GNN? And is it possible to connect layers with time steps in this case?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The model dynamically learns the curvature instead of relying on precomputed values, enabling it to adapt in response to both internal hidden features and the graph’s topology. \n2. This approach is closely aligned with the heat flow equation.\n3. As shown in Figure 3, the proposed framework achieves stable curvature over sufficient time steps. At the same time, the curvature concentrates around zero that can facilitate smoother information flow across the graph.", "weaknesses": "1. The network architecture used in this framework is relatively general. And it would be valuable to discuss the potential benefits of using more complex GNN architectures. Moreover, exploring the motivation behind the proposed framework with other related works [1, 2, 3] that focus on graph curvatures could provide additional insights.\n\n2. The experiment primarily focuses on node classification tasks on small-scale graphs. To better validate the effectiveness of the proposed framework, applying it to larger graphs would be beneficial. For example, the ogbn-arxiv dataset could serve as a graph classification dataset with GNNs as baselines. Additionally, for non-homophilous graph datasets, larger datasets and relevant baselines are available in [4].\n\n3. An efficiency study would be helpful. The computational cost of applying the ODE method should be explicitly discussed so readers can better understand its applicability. For instance, comparing parameters, training time, and GPU memory usage between this approach and other GNNs, such as GCN and GAT, would clarify its potential advantages and trade-offs.\n\n[1] Curvdrop: A ricci curvature based approach to prevent graph neural networks from over-smoothing and over-squashing.\n[2] Curvature Graph Neural Network.\n[3] Hyperbolic variational graph neural network for modeling dynamic graphs\n[4] Large Scale Learning on Non-Homophilous Graphs: New Benchmarks and Strong Simple Methods", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "kwM2qZC4PU", "reviewer": "Reviewer_QrP8", "summary": "The paper proposes a continuous GNN dynamics, namely GNRF by incorporating curvature based on the Ricci flow. In particular, by expressing edge weights as a function of node features, GNRF propagate features following a discrete (graph) Ricci flow. In order to avoid the costly computation of Ricci curvature on graphs, the paper proposes an auxiliary network for modeling curvature and learned end-to-end.  The paper provides several theoretical guarantees in terms of bounded Dirichlet energy and fast curvature decay. The experiments support the effectiveness of the method.", "questions": "1. In Line 285, the paper claims the sign of EdgeNet_ij aligns with the sign of k_ij. I am not sure how this is achieved without the supervision from the actual curvature. \n\n2. In Line 175 of Theorem 2, w_ij should be changed to k_ij?\n\n3. In Section 5.1, the curvature seems to be computed from the EdgeNet? What about the actual curvature?\n\n4. I am also curious whether there could be improvements when EdgeNet is replaced with actual curvature? This could be part of ablation study.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Compared to curvature-based graph rewiring, it is interesting and natural to incorporate Ricci curvature into the propagation of node features.\n\n2. Theoretical developments are supportive of the claims.", "weaknesses": "1. It is unclear how EdgeNet approximates edge curvatures? In particular, given there are trainable parameters and in the experiments, EdgeNet is trained end-to-end with supervision only from the task, instead of actual curvature. How to ensure EdgeNet approximates the curvature in this case? \n\n2. Theorem 5 is unclear. Does this mean there exists some network \\phi_1, \\phi_2 such that the network can approximate any curvature? Please give more explanations.\n\n3. Even though the theory is well-developed, the main GNN algorithm in (14) seems to resemble GRAND, especially the EdgeNet seems to act like a re-weighting term as in graph attention. How does EdgeNet differ to the graph attention module? Can you add experiments to verify the difference?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "vw1bWn0nIW", "reviewer": "Reviewer_1v62", "summary": "The paper introduces Graph Neural Ricci Flow (GNRF), a novel method for evolving node features in Graph Neural Networks (GNNs) using a differential equation-inspired approach based on the Ricci flow. The authors generalize the Discrete Ricci Flow to attributed graphs, where each edge's curvature converges toward zero over time. This has two key consequences: it bounds the graph's Dirichlet energy and provides a data-independent curvature decay rate. GNRF is unique because it computes time-varying curvature efficiently in linear time, unlike traditional curvature-based methods, which are typically precomputed and limited to specific curvature definitions. \n\nThe motivation behind the GNRF stems from the limitations of existing GNNs that rely on heat diffusion equations, which often lead to over-smoothing. Instead, the paper explores an alternative differential equation—Ricci flow—to mitigate over-smoothing and create more stable, non-smooth node representations. This innovative approach contrasts with traditional methods that view curvature as a static, precomputed property tied only to graph topology. By allowing curvature to evolve with node features, GNRF enables more dynamic and flexible graph learning.", "questions": "Could the authors explain why they are only using the Tolokers and Roman Empire datasets from “A Critical Look at the Evaluation of GNNs under Heterophily” and not the three other node classification datasets?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- Novelty: The paper introduces an interesting and innovative approach with GNRF, applying Discrete Ricci Flow to attributed graphs in a novel way. By allowing edge curvature to evolve dynamically with node attributes, GNRF moves beyond the limitations of traditional methods that rely on static, precomputed curvature. Its ability to work with any curvature definition and to compute curvature in linear time addresses key concerns around scalability and efficiency. This flexibility offers a practical and effective way to handle common challenges such as over-smoothing and over-squashing in graph neural networks. Overall, GNRF presents a meaningful advancement in curvature-aware graph learning.\n- Theoretical results: The paper offers solid theoretical contributions that help establish the soundness of the Attribute Discrete Ricci Flow framework. One key result is the demonstration that edge curvature naturally converges toward zero, ensuring a stable evolution of node features and addressing potential issues like over-smoothing and over-squashing. The bounding of the Dirichlet energy provides additional assurance that node representations maintain a balance between being too homogeneous or too distinct.", "weaknesses": "- Limited experimental results: The paper's experimental evaluation has some limitations. The focus is exclusively on node classification tasks, raising the question of why the method wasn't tested on other common tasks like graph classification or regression, which would provide a broader view of its applicability. Additionally, of the seven node classification datasets used, three (Cornell, Wisconsin, and Texas) are notably small, making it difficult to draw definitive conclusions about the method’s performance on more challenging or larger-scale data. Furthermore, on two of the larger datasets (Cora_Full and PubMed), the proposed method performs only within the statistical margin of error compared to the baselines, which limits its ability to demonstrate a clear and significant improvement over existing methods. Overall, I believe the paper would benefit significantly if the authors added some experimental results on graph classification/ regressions tasks, for example the LRGB datasets [1].\n- Baseline comparisons: The paper’s comparison with baseline methods raises some concerns regarding its evaluation methodology. Specifically, on the Tolokers and Roman Empire datasets, the authors use a 60/20/20 train/validation/test split, but the results reported for baseline models like GCN are significantly lower than what is found in the original work, which used a 50/25/25 split. When compared with the results in “A Critical Look at the Evaluation of GNNs under Heterophily” (2023), the proposed method (82.55 on Tolokers) does not seem to outperform a simple baseline like GCN on Tolokers (83.64), raising questions about whether the method truly offers improvements in these settings.\n\nOverall, I would be happy to increase my score to a 5 or 6 if the authors can convincingly address the above two points and show the practical usefulness of their method.\n\n[1] Dwivedi, Vijay Prakash, et al. \"Long range graph benchmark.\" Advances in Neural Information Processing Systems 35 (2022): 22326-22340.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)