1 Introduction↩︎

Message passing graph neural networks (GNNs) rely on aggregating signals via local averaging, which can be interpreted as convolving the node features with a simple, rectangular waveform that is non-zero only within one-hop neighborhoods of each vertex. It is known that this type of message-passing tends to suffer from over-smoothing if too many iterations are applied and from under-reaching if too few are applied [1]–[3]. One possible solution is to use multiscale message passing [4]. Another approach, [5]–[13] more directly related to our work, is to use graph wavelets [14], [15]. These wavelets can be viewed as convolving the input features with multiscale, oscillatory waveforms, in contrast to the simple, rectangular, one-hop waveforms used in message passing.

Here, we introduce DYMAG which uses dynamics on the graph to generate waveforms which we will convolve with the node features. We will use these waveforms as a form of multiscale message aggregation, which we show can effectively extract graph geometric and topological information and outperform baseline methods on graph-level tasks that rely on such graph properties.

We evaluate DYMAG on a broad spectrum of graph learning benchmarks spanning synthetic, citation, molecular, and materials science datasets. To assess its ability to recover generative and topological structure, we first test on synthetic graphs, including Erdős-Rényi and stochastic block models, where the task involves inferring graph parameters and persistent features. We then evaluate on citation networks, including homophilic datasets - Cora [16], Citeseer [17], and PubMed [18] - and heterophilic datasets - Texas, Wisconsin, and Cornell [19]. We further demonstrate DYMAG’s scalability on the largest dataset in the Open Graph Benchmark, ogbn-papers100M [20], demonstrating that it can recover topological properties of massive graphs. For molecular property prediction, we consider both protein graphs PROTEINS, ENZYMES, and MUTAG [21] and small-molecule graphs (DrugBank [22], Drug Therapeutics Program AIDS Antiviral Screen Data [23]). Finally, we test on the Materials Project dataset [24] to predict materials properties such as band gaps. Across these varied domains, DYMAG consistently outperforms standard GNNs and approaches the performance of pretrained, domain-specific models. Our main contributions are as follows:

2 Related work and Background↩︎

Previous work has either analyzed dynamics on graphs [25]–[31] or aimed to use dynamics as a framework for understanding GNNs. In the latter case [32]–[34] and [35] viewed message passing as a time-discretized diffusion PDE and used this insight to design novel GNNs. Unlike those methods, we view PDE solutions as waveforms and use convolution against these waveforms to define our aggregation rule. Additionally, existing work primarily focus on parabolic equations while we also consider hyperbolic and chaotic dynamics. We provide a further discussion of related work in . Now, we provide backgrounds on graph signal processing and dynamics. See for details.

2.1 Graph Signal Processing↩︎

In Graph Signal Processing, a node feature vector \(\mathbf{x}\in\mathbb{R}^n\) is viewed as a signal on the vertices of a weighted, undirected graph \(G=(V,E,w)\), \(|V|=n\) [36], [37]. Let \(L=U\Lambda U^\top\) be its Laplacian with eigendecomposition \(L\boldsymbol{\nu}_k=\lambda_k\boldsymbol{\nu}_k\), \(0=\lambda_1\le\cdots\le\lambda_n\). The graph Fourier transform is defined by projecting the singals onto these eigenvectors \(\widehat{\mathbf{x}} = U^\top\mathbf{x}.\) The projection onto the first several eigenvectors (small \(\lambda_k\)) captures the smooth portion of the signal and the projection onto the later eigenvectors (large \(\lambda_k\)) capture oscillatory ones. Classical message‑passing GNNs act as low‑pass filters [38], [39], effectively only keeping the projection onto the first several eigenvectors; DYMAG instead aggregates with waveforms spanning low, mid, and high bands. (Details in .)

2.2 Heat and Wave Dynamics on a Graph↩︎

For \(\alpha > 0\), we define the \(\alpha\)-fractional graph Laplacian by \(L^\alpha := U \Lambda ^\alpha U^\mathsf{T}\). For each \(i\in V\), we let \(\delta_i\) denote the Dirac signal at \(i\) given by: for \(\forall k \in V\), \(\delta_i(k)=1\) if \(i=k\), \(\delta_i(k)=0\) otherwise. We define \[\label{eqn:heat95eq95definition} \smash{-L^\alpha u^{(i)}_H(v,t) = \partial_t u^{(i)}_H(v,t),\quad u^{(i)}_H(v,0)=\delta_i(v), \quad \text{(Heat)}}\tag{1}\] \[\begin{align} \smash{-L^\alpha u^{(i)}_W(v,t) = \partial^2_t u^{(i)}_W(v,t),\quad u^{(i)}_W(v,0)=\delta_i(v), \quad \partial_t u^{(i)}_W(v,0)=c\delta_i(v),\quad \text{(Wave)}}\label{eqn:wave95eq95def} \end{align}\tag{2}\] as the heat and wave equations with a initial value \(\delta_i\) (and initial veclocity \(c\delta_i\) for wave). On a connected graph \(G\), they admit closed‐form solutions:

These expressions extend to disconnected graphs and, by Remark 1 of [40], are invariant to the choice of Laplacian eigenbasis. (See for details).

2.3 Chaotic Dynamics on a Graph↩︎

Chaotic dynamics, describing systems that have aperiodic behavior and sensitivity to initial conditions [41], can be modeled by the Sprott dynamics [42]:

Solutions remain bounded for \(b>0\). For \(b=0.25\), fully connected graphs with generic couplings or sparse graphs exhibit positive Lyapunov exponents (chaos).  [42], [43]. (Full details in .)

3 Methods↩︎

DYMAG is graph neural network consisting of two main parts.

1. Waveform Bank Creation: A diverse bank of multi-scale waveforms is constructed by solving the PDEs considered in Section 2. These waveforms define a set of basis functions that encode diverse patterns across spatial and temporal scales. (See .)

2. Multi-scale Aggregations: At each layer \(1\leq\ell\leq L\), node representations \(X^{(\ell-1)}\) are convolved with the waveform bank. The result is then passed through an MLP to produce an updated representation \(X^{(\ell)}\). This step replaces standard message passing mechanisms by aggregation via sophisticated, multiscale waveforms. (See .)

3.1 Waveform Bank Creation Using PDEs↩︎

Let \(u^{(i)}(v,t)\) denote the solution to the chosen PDE dynamics (wave, heat, or Sprott equations) with initial condition \(u^{(i)}(\cdot,0) = \delta_i(\cdot)\), where \(\delta_i(j) = 1\) if \(j = i\) and \(0\) otherwise (a Dirac signal centered at node \(i\)). When applicable (for second-order dynamics), we also set \(\partial_t u(\cdot, 0) = c\,\delta_i(\cdot)\), with a fixed hyperparameter \(c\geq 0\). We choose \(K\) time points \(\mathcal{T}=(t_1,\dots, t_K)\) by fixing a maximal time \(T\) and then setting \(t_k=kT/K\). We then define \(\mathcal{U}=\left\{\mathbf{u}_{i,k}\right\}_{i \in V,1\leq k\leq K},\) where \(\mathbf{u}_{i,k}\) is the vector \(\mathbf{u}_{i,k}:=u^{(i)}(\cdot,t_k)\in\mathbb{R}^n\). We refer to \(\mathcal{U}\) as the PDE waveform bank (see Algorithm 4 and a). Each waveform \(\mathbf{u}_{i,k}\) is centered at node \(i\) and corresponds to a snapshot of the PDE dynamics at time scale \(t_k\). The bank \(\mathcal{U}\) collects such waveforms across all nodes and multiple time scales, similar to wavelets but more flexible thanks to the diverse dynamics. shows basic waveforms and more complex patterns created via combinations (from the MLP discussed below).

We note that \(\mathcal{U}\) can be computed offline prior to training for increased computational efficiency. Additionally, note that the waveforms can be computed efficiently via either Chebshev approximation or a Runge-Kutta scheme. We further discuss on complexity and scalability in .

3.2 Multi-scale Aggregation↩︎

In each layer, \(\ell\), we assume that we are given an \(n\times m_\ell\) feature matrix \(X^{(\ell)}\) (where \(X^{(0)}\) consists of the initial node features). We let \(\mathbf{x}_j\in\mathbb{R}^n\) denote the \(r\)-th column of \(X^{(\ell)}\), which we interpret as a signal defined on \(V\). For each waveform \(\mathbf{u}_{i,k}\) in the waveform bank \(\mathcal{U}\) (Section 3.1), we perform an inner product with the node features, thought of as a convolution: \[\begin{align} h_{j,k}^{(i)} &= \langle \mathbf{u}_{i,k},\mathbf{x}_j\rangle = \mathbf{u}_{i,k}^{\mathsf T}\mathbf{x}_j . \label{eq:inner-prod} \end{align}\tag{6}\] We then combine these convolved features by applying an MLP to the states \(h^{(i)}_{j,k}\) associated with each node \(v_i\), i.e., \(\mathbf{y}_i=\text{MLP}\left(\text{vec}\left(h^{(i)}_{j,k}\right)\right)\). We then reorganize the \(\mathbf{y}_i\) into a transformed feature matrix \(X^{(\ell+1)} =(\mathbf{y}_1^{\mathsf{T}},\ldots,\mathbf{y}^{\mathsf{T}}_n)^{\mathsf{T}}\) (so that \(\mathbf{y}^{\mathsf{T}}_i\) is the \(i\)-th row of \(X^{(\ell+1)}\)). See Algorithm [alg:mp] and b.

The inner product, Eqn. 6 , can also be interpreted as the feature \(\mathbf{x}_j\) being updated via a message from a source node \(v_i\) at scale \(k\). Indeed, message passing neural networks can be interpreted as performing such an inner product with a limited bandwidth rectangular waveform as shown in and then applying the MLP. We remark that since we use a waveforms based on PDE solutions of various time snapshots, we obtain multi‑scale embedding. As the time \(t_k\) increases, the waveform effectively dilates and spreads to a larger neighborhood of vertices. Furthermore, via the MLP, DYMAG is able to learn novel combinations of the waveforms, either from different source nodes or at different time scales. This includes the diffusion wavelets [15] which can be obtained by subtracting solutions to the heat equation at different time scales [40].

3.2.0.1 Downstream Readout

After \(L\) rounds of Multi-scale aggregation, the resulting node representations \(X^{(L)} = \{ \mathbf{x}_i^{(L)} \}_{i \in V}\) are used for prediction. For node-level tasks, a shared MLP is applied independently to each node feature vector. For graph-level tasks, node features are first aggregated using a permutation-invariant pooling operation (e.g., global mean or sum), followed by a task-specific MLP to produce the graph-level output. See c and Algorithm [alg:dymag].

3.3 Theoretical Properties Related to Dynamics-based Waveforms on the Graph↩︎

Below, we formulate properties of our waveforms and the information they are able to extract from the graph. These results serve as motivation for our method, which utilizes these dynamics as a novel aggregation paradigm for graph neural networks. Complete proofs are provided in .

3.3.1 Frequency Domain Characteristics of Waveform Based Message Aggregation↩︎

Standard message passing can be viewed as convolving the node features with a single, simple, rectangular waveform. From the perspective of graph signal processing, this corresponds to a low-pass filtering which only preserves the low-frequency (smooth) portion of the node features. In contrast, DYMAG employees a richer, more sophisticated bank of waveforms, which we will show allows DYMAG to extract a variety of different types of information. We first consider a function defined by \(u_{\mathrm{BP}}^{(i)}(v,t) \;=\; u_H^{(i)}(v,t_1)\;-\;u_H^{(i)}(v,t_2)\) for two fixed times, \(0<t_1<t_2\). In the graph-Fourier domain its response at frequency (eigenvalue) \(\lambda_k\) is \(e^{-t_1\lambda_k^\alpha}-e^{-t_2\lambda_k^\alpha}.\) This function (i) is \(0\) at \(\lambda_k=0\), (ii) tends to \(0\) as \(\lambda_k\to\infty\), and (iii) reaches a single maximum at \(\lambda^\star=\left(\frac{1}{t_2-t_1}\log\!\bigl(\tfrac{t_2}{t_1}\bigr)\right)^{1/\alpha}.\) Thus, \(u_{BP}^{(i)}\) suppresses both very low and very high frequencies, but keeps information in a moderate frequency band (which depends on \(t_1\) and \(t_2\)). Therefore, we call \(u^{(i)}_{BP}\) a band-pass function. Notably, DYMAG has the ability to learn this function via the use of the MLP which is applied after Eqn. 6 . We next consider the solution to the wave equation given by Eqn. 4 , for simplicity focusing on the case where \(c=0\). The frequency response at each \(\lambda_k\) is given by \(\cos(\sqrt{\lambda_k^{\alpha}} t)\). Since this function peaks and falls in multiple different “bands" we think of it as a multi-band-pass function. This leads us to the following proposition.

In addition to the above propositions, we note that DYMAG is also able to learn low-pass information, similar to standard message passing networks. This is a direct consequence of the fact that \(u_H^{(i)}\) has a decreasing frequency response \(e^{-t\lambda_k^\alpha}\). It can also learn high-pass information via function \(u^{(i)}_{\text{High}}=1-u^{(i)}_H\). We next discuss how the frequency-domain characteristics of DYMAG help alleviate the following limitations of standard GNNs:

Over-smoothing: Message passing networks utilize rectangular pulse waveforms, which act as low-pass filters, i.e., smoothing operators. With each layer, the features get smoother and smoother, eventually become nearly constant, which limits their usefulness. By contrast, DYMAG is able to learn band-pass, high-pass, and multi-band-pass information in addition to standard low-pass information. This allows it to avoid the oversmoothing problem. Under-reaching: Message passing networks only aggregate within local, one-hop neighborhoods. Thus their receptive field is equal to the number of layers, which must be kept small to avoid severe oversmoothing. This limits their ability to capture global structure or long range interactions. DYMAG, on the other hand performs aggregation via waveforms which are not confined to one-hop neighborhoods and is able to capture global structure. Heterophily: The local averaging operation in message passing networks, are particularly problematic on heterophilic graphs where many nodes have different labels than their neighbors. DYMAG’s diverse waveform banks are able to capture band-pass, multi-band-pass, and high-frequency information (in addition to low-pass). This makes them well-suited to heterophilic graphs. Additionally, we note that our experiment shows that the Sprott dynamics perform particularly well on node classificaiton on heterophilic graphs (see ), perhaps because of their ability to detect subtle changes in different portions of the network structure.

3.3.2 General Properties of Solutions↩︎

The following result shows that DYMAG is able to identify the connected components of \(G\).

Due to Proposition 2, we will assume that \(G\) is connected in the following sections. However, we note that many of the results still apply to disconnected graphs with suitable modifications.

3.3.3 Heat Dynamics↩︎

The continuous and global nature of DYMAG allows it to instantaneously have a receptive field over the entire graph. Intuitively, this corresponds to information spreading instantaneously over the graph (although for small values of \(t\) the energy \(u^{(i)}_H(v,\cdot)\) will be mostly concentrated near \(v_i\)). This is in contrast to message passing networks where the receptive field about each node is equal to the number of layers (which is usually must be kept small in order to avoid over-smoothing).

Our next two results analyze the energy decay of \(u_H^{(i)}\). They suggests that graphs with a larger \(\lambda_2\) will have a faster rate of energy decay. The second eigenvalue of a graph can be related to the isoperimetric ratio of a graph through Cheeger’s inequality, thereby revealing information on graph structure and how “bottlenecked" a particular graph is [44]. Additionally, they show that the properties of heat energy decay can distinguish between between graph structures. We note that although the assumptions for Proposition 5 represents a rather specific set of conditions, we expect that when two graphs have edges generated according to a similar rule or distribution, the more densely connected graph will have more rapidly decaying heat energy.

Finally, we restate some known results that provide additional foundation linking the behavior of the heat equation solutions to graph topology. Lemma 1 of [45] shows that the heat equation encodes shortest path distances \(d(v_i,v_j)\) between nodes on the graph:

We next consider, the Ollivier-Ricci curvature. This is a discrete notion of curvature, meant to parallel the tradition notion of Ricci curvature in Riemmanian geometry. It is defined by \(\kappa(v_i, v_j) = 1 - W_1(\mu_{v_i}, \mu_{v_j})/d(v_i ,v_j),\) where \(\mu_v\) is a probability measure centered around \(v\) (see [46] for details), \(W_1\) the 1-Wasserstein distance, and \(d(v_i,v_j)\) is the distance (shortest path length) from \(v_i\) to \(v_j\). The following result from [46] relates \(\kappa\) to the heat equation.

3.3.4 Wave Dynamics↩︎

The periodicity of the solution to the wave equation endows it with the ability to capture long range interactions. Central to this argument is the wave energy, analyzed in the following proposition which focuses on the case where the initial velocity \(c\) is equal to zero:

This shows that, unlike heat kernels (which decay over time), the wave energy oscillates, retaining signal over time, and thus can reflect non-local interactions such as those created by cycles in the graph. These oscillations allow the waveforms to “echo” through the graph and revisit distant parts of the structure - a behavior well-suited for recovery of topological features.

The graph eigenspectrum encodes a wide range of graph invariants and properties. Proposition 9 demonstrates that the solutions to the wave equation relate to graph spectral properties, and that this entire information is contained in the solutions of the wave equation at each node. For example, we have the following corollary:

More generally, when the graph is not a cycle but contains a cyclic subgraphs as a prominent topological feature, this proposition provides some intuition for why the wave-equation is well suited to pick up that a node belongs to a cycle and recover dimension 1 homology.

4 Empirical Results↩︎

We evaluate DYMAG across diverse tasks to assess its ability to recover geometric/topological structure and generalize to downstream biological, chemical, and materials applications. In this section, we set \(\alpha=1\) so that the fractional Laplacian coincides with the ordinary graph Laplacian. We conduct some experiments with other exponents \(\alpha\) in . Baselines include message-passing GNNs (MPNN [47], GAT [48], GIN [49]), diffusion-based methods (GRAND [33], GRAND++[35]), and GraphGPS [50], a state-of-the art graph transformer. We also compare against a GNN built with fixed-scale wavelets (GWT [15]) and a neural approximation algorithm of the extended persistence diagram (EPD) [51]. For molecular and materials prediction, we include pretrained models such as ProtBERT [52], MolBERT [53], and GeoCGNN [54].

In all tables, best and second best results are highlighted. Ties within one standard deviation are treated as equivalent. Unless noted otherwise, results reflect means over 10-fold cross-validation. Implementation details, including hyperparameter selection, Chebyshev approximations, and complexity analysis, are provided in . Full results with standard deviations appear in .

4.1 Geometric and Topological Properties↩︎

We first evaluated the expressivity of multiscale dynamics as a replacement for message passing by training DYMAG to recover geometric and topological features, including Ollivier–Ricci curvature and the persistence image representation [55] of the extended persistence diagram. Ollivier–Ricci curvature [56], a discrete analog of Ricci curvature from Riemannian geometry, captures local graph geometry. For persistent homology, diagrams are computed from ascending and descending filtrations of each node’s 5-hop neighborhood, using node degree as the filter [51], [57]. The resulting persistence images encode connected components (0 homology) and loops (1st homology).

We evaluated the model on Erdős-Rényi graphs [58], \(G(n,p)\), with \(n\in \{100,200\}\) and \(p\in\{0.04,0.06,0.08\}\), stochastic block model (SBM) graphs, and several citation graphs. The results are shown in and (appendix). DYMAG significantly improves prediction accuracy on both Ollivier–Ricci curvature and persistent homology compared to standard GNNs and GRAND. For persistent homology, DYMAG performs on par with the purpose-built model of [51], a neural approximation of the Union-Find EPD algorithm, despite DYMAG not being designed specifically for this task. DYMAG generalizes well across both synthetic and real-world graphs, including Cora [16], Citeseer [17], and PubMed [18]. Full Results on ER graphs and additional experiments using alternative filtrations are provided in . We note that while persistent homology can be computed directly, it is computationally expensive with cost \(\mathcal{O}(g^3)\), \(g\) being the number of generators [59]. More importantly, these experiments highlight that DYMAG learns rich graph representations that capture topological features when useful for prediction, and it can adapt to the task to extract relevant information for regression or classification.

As shown in and (appendix), we evaluate expressivity via two tasks as proxies: recovering parameters of random graphs and node classification on homophilic (Pubmed, Citeseer, Cora) and heterophilic (Texas, Wisconsin, Cornell) [19] networks. We see that the heat and wave versions of DYMAG are the top performing models on most of the data sets whereas the Sprott version of DYMAG is the top performing model on the heterophilic data sets, possibly because these data sets need a model which is sensitive to small changes in local graph structure. presents results on node-level Ollivier–Ricci curvature, persistent homology with alternative filtrations, and an analysis of fractional heat and wave dynamics across different \(\alpha\) values.

4.2 Proteins, Molecules, and Materials↩︎

We evaluated DYMAG on graphs representing proteins, drug-like molecules, and materials shown in (appendix). We note that DYMAG’s strong performance on these data sets indicates its potential positively impact society by furthering the design of materials, drugs, or other healthcare treatments. More specifically, the tasks include predicting geometric and chemical properties such as dihedral angles, total polar surface area (TPSA), the number of aromatic rings, band gaps in materials, and anti-HIV activity. Datasets include PROTEINS [21], DrugBank [22], the Materials Project [24], and the AIDS Antiviral Screen [23]. Comparisons span standard message-passing GNNs (MPNN, GAT, GIN), diffusion-based models (GRAND, GRAND++), spectral methods (GWT), transformer-based models (GraphGPS), and domain-specific pre-trained models: ProtBERT [52] for proteins, MolBERT [53] for small molecules, and GeoCGNN [54] for materials. Across all datasets, DYMAG consistently outperforms standard GNNs, GRAND, GRAND++, GraphGPS, and GWT by a wide margin. It matches the performance of powerful, task-specific pretrained models within 1 standard deviation overall, and significantly surpasses them on the PROTEINS and AIDS datasets.

Overall, the heat and wave versions of DYMAG perform strongly across all molecular and material prediction tasks.The Sprott (chaotic) variant shows more variable performance, which may reflect its heightened sensitivity to local graph structure. This behavior appears beneficial in settings with recurring structural motifs, such as the Antiviral Screen dataset, and may be less advantageous in tasks where such sensitivity is less critical. Additional results, including accuracy and training time, are provided in .

5 Conclusion↩︎

We introduce DYMAG as a method for improving aggregation in GNNs. We use dynamics to generate a diverse bank of waveforms that span multiple frequency bands. Messages are aggregated by taking each node feature, interpreted as a graph signal, and projecting it onto this bank via inner products, producing a set of features that encode multi-scale information. The expressiveness of this representation, arising from the rich frequency structure of the waveforms, helps mitigate common GNN limitations such as oversmoothing, underreaching, and heterophily.

One limitation of our method is that the Sprott model is extremely sensitive to the graph structure. This is useful for some tasks which require detecting minute changes in structure. However, this be undesirable in other settings where one may want similar representations of nearly isomorphic. Another limitation is that our method is currently only applicable to supervised tasks. Extending DYMAG to unsupervised tasks, such as clustering, denoising, or signal reconstruction could could be in interesting avenue of future work. Overall, DYMAG establishes a theoretically grounded, empirically strong message aggregation paradigm; future work will broaden its application to diverse graph tasks and refine the accompanying speed-up techniques.

Acknowledgments↩︎

This research was partially funded and supported by CIFAR AI Chair [G.W.], NSERC Discovery grant 03267 [G.W.], NIH grants (1F30AI157270-01, R01HD100035, R01GM130847, R01GM135929) [G.W.,S.K.], NSF Career grant 2047856 [S.K.], NSF grant 2327211 [S.K., G.W., M.P.], NSF grant 2242769 [M.P.], NSF/NIH grant 1R01GM135929-01 [S.K.], NSF CISE grant 2403317 [S.K.], the Chan-Zuckerberg Initiative grants CZF2019182702 and CZF2019-002440 [S.K.], the Sloan Fellowship FG-2021-15883 [S.K.], the Novo Nordisk grant GR112933 [S.K.], Yale - Boehringer Ingelheim Biomedical Data Science Fellowship [D.B.], and Kavli Institute for Neuroscience Postdoctoral Fellowship [D.B.].

6 Related work↩︎

There is a long history of studying dynamics on graphs. For example, [25]–[28] analyze complex interactions such as brain processes, social networks, and spatial epidemics on graphs. The study of graph dynamics has recently crossed into the field of deep learning. with [29]–[31] using neural networks to simulate complex phenomena on irregularly structured domains.

Differing from the above mentioned works, there have also been several recent papers which aim to use dynamics as a framework for understanding GNNs. [32] and [33] take the perspective that message-passing neural networks can be interpreted as the discretizations of diffusion-type (parabolic) partial differential equations on graph domains where each layer corresponds to a discrete time step. They then use this insight to design GRAND, a novel GNN, based on encoding the input node features, running a diffusion process for \(T\) seconds and finally applying a decoder network. [35] builds on this work by extending it to diffusion equations with “sources” placed at the labeled nodes, leading to a new network GRAND++. They then provide an analysis of both GRAND and GRAND++ and show that they are related to different graph random walks. GRAND is related to a standard graph random walk, whereas GRAND++ is related to a dual random walk started at the the labeled data which can avoid the oversmoothing problem. We also note [60], which the graph heat equation to extract structural information around each node (although not in a neural network context) and [34] which used insights from hyperbolic and parabolic PDEs on manifolds to design a GNN that does not suffer from oversmoothing as well as [61] which uses convolution using unitary groups to improve GNNs ability to learn long-range dependencies.

Our network method differs from these previous works in several important ways. Most importantly, whereas [33] and [32] primarily focused on PDEs as a framework for understanding the behavior of message passing operations, here we propose to use the dynamics associated to the heat equation as a new form feature aggregation, replacing traditional message passing operations. Additionally, we consider both the heat equation (the prototypical parabolic PDE) and the wave equation (the prototypical hyperbolic PDE) as well as chaotic dynamics, whereas previous work [32], [33], [35] has primarily focused on parabolic equations. Notably, similar to GRAND++, the wave-equation and the Sprott versions of DYMAG do not suffer from oversmoothing. However, the long-term behavior of these equations differs from the diffusion-with-a-source equation used in GRAND++ in that they only depend on the geometry of the network and not on locations of the labeled data. (Additionally, since we does not require labeled data as source locations, our method can be easily adapted to unsupervised problems by removing the MLPs.)

7 Detailed Background↩︎

This section is a more detailed version of the background on graph signal processing and dynamics provided in Section 2.

7.1 Graph Signal Processing↩︎

In graph signal processing, node features are interpreted as signals (functions) defined on the nodes of a graph [36], [37]. Each signal can then be decomposed into different frequencies defined in terms of the eigendecomposition of the graph Laplacian and an associated Fourier transform.

Formally, we let \(\mathbf{x}:V\rightarrow\mathbb{R}\), denote a function (signal) defined on the vertices of a weighted, undirected graph \(G=(V,E,w)\) with vertices \(V=\{v_1,\ldots, v_n\}\). For convenience, we will identify \(\mathbf{x}\) with the vector whose \(k\)-th entry is \(\mathbf{x}(v_k)\). Thus, we will write either \(\mathbf{x}(v_k)\) or \(\mathbf{x}(k)\), depending on context. We may also write \(\mathbf{x}(v)\) if we do not wish to emphasize the ordering of the vertices. We let \(L\) denote a graph Laplacian with eigenvectors \(\boldsymbol{\nu}_1,\ldots,\boldsymbol{\nu}_n\) and eigenvalues \(0=\lambda_1\leq\lambda_2\leq \ldots \leq \lambda_n\), \(L\boldsymbol{\nu}_k=\lambda_k\boldsymbol{\nu}_k\). Unless otherwise specified, we will assume that \(L\) is either the unnormalized Laplacian \(L_{U}=D-A\) or the symmetric normalized Laplacian \(L_{\text{sym}}=D^{-1/2}L_\text{U}D^{-1/2}\), where \(D\) and \(A\) are the weighted degree and adjacency matrices. In these cases, we may write \(L=U\Lambda U^\mathsf{T}\), where \(U\) is a matrix with columns \(\boldsymbol{\nu}_1,\ldots,\boldsymbol{\nu}_n\) and \(\Lambda\) is a diagonal matrix, \(\Lambda_{k,k}=\lambda_k\).¹ The graph Fourier transform can be defined as \(\widehat{\mathbf{x}} = U^{\mathsf{T}} \mathbf{x}\) so that \(\widehat{\mathbf{x}}(k)=\langle\boldsymbol{\nu}_k,\mathbf{x}\rangle\). Since the \(\boldsymbol{\nu}_k\) form an orthonormal basis, we obtain the Fourier inversion formula \(\mathbf{x}=\sum_{k=1}^n \widehat{\mathbf{x}}(k)\boldsymbol{\nu}_k\). The eigenvectors \(\boldsymbol{\nu}_k\) are referred to as Fourier modes and the eigenvalues \(\lambda_k\) are interpreted as frequencies. Therefore, the Fourier inversion formula can be thought of as decomposing a signal \(\mathbf{x}\) into different the superposition of Fourier modes at different frequencies.

Standard message passing neural networks are known to essentially perform low-pass filtering [38], [39]; i.e., they preserve the portion of the signal corresponding to the first one or two eigenvectors, while suppressing the rest of the signal. As we discussed in Section 3.3.1, the waveforms utilized in DYMAG may span a broader range of frequency behavior and can act highlight different aspects of the frequency spectrum by acting as either as low-pass, high-pass, or band-pass filters.

7.2 Heat and Wave Dynamics on a Graph↩︎

For \(\alpha > 0\), we define the \(\alpha\)-fractional graph Laplacian by \(L^\alpha := U \Lambda ^\alpha U^\mathsf{T}\). We note that \(L^\alpha\) has the same eigenvectors as \(L\) and the eigenvalues of \(L^\alpha\) are given by \(\lambda_k^\alpha\), i.e., \(L^\alpha\boldsymbol{\nu}_k=\lambda_k^\alpha\boldsymbol{\nu}_k\). Additionally, we see that when \(\alpha=1/m\) for some \(m\in\mathbb{N}\), we have \((L^{1/m})^m=L.\) For each \(i\), we let \(\delta_i\) denote the Dirac signal at \(i\) given by \(\delta_i(k)=1\) if \(i=k\), and \(\delta_i(k)=0\) otherwise. We say that a function \(u^{(i)}_H(v,t)\) solves the \(\alpha\)-fractional heat equation with a initial value \(\delta_i\) if \[\label{appx:eqn:heat95eq95definition} -L^\alpha u^{(i)}_H(v,t) = \partial_t u^{(i)}_H(v,t),\quad u^{(i)}_H(v,0)=\delta_i(v).\tag{7}\]

We say that \(u^{(i)}_W\) solves the \(\alpha\)-fractional wave equation with initial Dirac data \(\delta_i\) and an initial velocity \(c\delta_i\) (where \(c\) is a constant) if \[\begin{align} -L^\alpha u^{(i)}_W(v,t) = \partial^2_t u^{(i)}_W(v,t),\quad u^{(i)}_W(v,0)=\delta_i(v), \quad \partial_t u^{(i)}_W(v,0)=c\delta_i(v).\label{appx:eqn:wave95eq95def} \end{align}\tag{8}\]

If \(G\) is connected, solutions to the heat and wave equations are given explicitly by \[\begin{align} \tag{9} u^{(i)}_H(v,t) &= \sum_{k=1}^n e^{-t\lambda_k^\alpha}\langle \boldsymbol{\nu}_k, \delta_i \rangle \boldsymbol{\nu}_k(v), \quad\text{and}\\ u^{(i)}_W(v,t) &=\sum_{k=1}^n \cos(\sqrt{\lambda_k^\alpha} t)\langle \boldsymbol{\nu}_k,\delta_i \rangle \boldsymbol{\nu}_k(v) + t\langle \boldsymbol{\nu}_1,c\delta_i \rangle\boldsymbol{\nu}_1 +\sum_{k=2}^n\frac{1}{\sqrt{\lambda_k^\alpha}}\sin(\sqrt{\lambda_k^\alpha} t)\langle \boldsymbol{\nu}_k,c\delta_i\rangle\boldsymbol{\nu}_k(v). \tag{10} \end{align}\] We note that Eqns 3 and 4 can also be adapted to disconnected graphs with simple modifications. Furthermore, following Remark 1 in [40], we note that the solutions \(u^{(i)}_H(v,t)\) and \(u^{(i)}_W(v,t)\) defined in Eqn. 3 and 4 do not depend on the choice of orthonormal basis for the graph Laplacian, see Section [appendix:joyce95statement] for details.

7.3 Chaotic Dynamics on a Graph↩︎

We next consider dynamics exhibiting chaos, a behavior that may be informally summarized as “aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions" [41]. As a prototypical example of chaos on graphs, we consider the general, complex, and nonlinear graph dynamics on graphs described in [42]:

where \(b\) is a damping coefficient, and the \(c_{k,j}\) represent interactions. We refer to Eq. 5 as the Sprott equation and denote its solutions by \(u_S(v,t)\). When \(b>0\), solutions \(u_S(v,t)\) remain bounded. In the case of fully connected graphs, with \(b = 0.25\), chaotic dynamics (corresponding to positive Lyapunov exponents, see, e.g., [43]) were observed when a sufficiently large fraction of interactions were neither symmetric (\(c_{j,k} = c_{k,j}\)) nor anti-symmetric (\(c_{j,k} = -c_{k,j}\)). Sparsely connected networks also exhibited positive Lyapunov exponents with a value of \(b = 0.25\) [42].

8 Computational Complexity↩︎

Here, we discuss the computational complexity of DYMAG and show that it may be scaled to large graphs.

We utilize Chebyshev polynomials to approximate \(u^{(i)}_H(v, t)\) and \(u^{(i)}_W(v,t)\) as defined in Eqn. 3 and 4 motivated by the success of Chebyshev polynomials in the approximation of spectral graph wavelets [14] and GNNs [62]. This removes the need for eigendecomposition (which can have \(\mathcal{O}(n^3)\) computational complexity and \(\mathcal{O}(n^2)\) memory). The polynomial approximation has linear complexity for sparse graphs [62]. For example, if \(G\) is a \(k\)-nearest neighbor graph and the order of the polynomial is \(m\), then the time complexity for solving the heat/wave equation is \(O(kmn)\).

DYMAG’s runtime complexity is \(\mathcal{O}(r|E|FT)\), where \(|E|\) is the number of edges, \(F\) is the number of features, and \(r\) is the degree of the Chebyshev polynomial. No closed form solution is available for the Sprott dynamics, so we instead use a fourth-order Runge-Kutta method, which has been successfully applied to chaotic systems [63], [64]. This approach has complexity \(\mathcal{O}(g|E|FT)\), where \(g\) is the number of solver steps between sampled time steps.

For each of the underlying dynamics, DYMAG exhibits linear complexity, with respect to the number of vertices for sparse graphs (setting \(|E|=n\bar{d}\), where \(\bar{d}\) is the average degree). This makes it efficient and scalable to large graphs, where the focus is often on local or node-level properties rather than global topological properties (although predicting global properties is the primary focus of our work). In such cases, local properties can be examined with a smaller feature set of size \(F' \ll n\), by concentrating on subgraphs around nodes of interest. Parallelization is feasible for large sparse graphs since the \(r\)-th order Chebyshev polynomials act over \(r\)-hop neighborhoods, allowing DYMAG to scale with standard MPNN techniques. Furthermore, the feature space \(F\) can be selected or adjusted to be small by utilizing random features, Diracs on a subset of nodes, or natural graph signals.

9 Proofs of Theoretical Results↩︎

10 Implementation Details↩︎

10.1 Experimental Computation Resources↩︎

All experiments were conducted on a high-performance computing server equipped with an Intel Xeon Gold 6240 CPU (18 cores, 36 threads, 2.60 GHz base frequency) and 730 GB of system RAM. The server is configured with 4 NVIDIA A100 GPUs, each with 40 GB of VRAM, enabling efficient parallel training of deep learning models. The system runs on Red Hat Enterprise Linux 8.8 with CUDA version 11.8 and cuDNN 8.5.0. Experiments were executed using PyTorch 2.0.0 and Python 3.10. Unless otherwise stated, all models were trained using mixed precision to optimize memory usage and throughput.

10.2 DYMAG Parameters↩︎

Here we describe the architecture and training setup of DYMAG used to generate the experimental results presented in this paper. DYMAG supports both node-level and graph-level tasks, and is evaluated on datasets comprising either a single large graph (e.g., citation networks such as PubMed) or collections of small graphs (e.g., synthetic graphs, molecular graphs, and protein structures).

We use a stacked architecture consisting of \(L = 3\) DYMAG layers, each simulating \(K\) discrete time steps of heat or wave dynamics, where \(K\) is selected via grid search specific to each dataset. Between layers, we apply a 3-layer MLP with LeakyReLU activations for node-wise transformation. For all downstream tasks, we use a 5-layer MLP with LeakyReLU activations as the task-specific head.

For node-level tasks such as node classification in citation graphs, we directly use the hidden node embeddings produced by DYMAG and feed them into the 5-layer MLP to perform classification or regression. For datasets composed of multiple small graphs (e.g., synthetic Erdos–Renyi graphs, molecular graphs, etc.), we apply mean pooling across the node dimension to obtain a graph-level embedding. This pooled embedding is passed to the same 5-layer MLP for classification or regression.

10.3 Parameters for Sprott Dynamics↩︎

For Sprott dynamics (), in our experiments, we set \(b=0.25\) and the coupling coefficients as \(\smash{c_{k,j} \sim \frac{1}{\sqrt{n-1}} \left(2 \cdot \mathrm{Bernoulli}(0.5) - 1\right)}\), which assigns each \(c_{k,j}\) a value of \(\pm \frac{1}{\sqrt{n-1}}\) with equal probability.

11 Additional Experiments↩︎

We present further experimental validation of DYMAG, focusing on its ability to predict both biomolecular properties and topological descriptors such as curvature.

In , we present the complete results for predicting Ollivier–Ricci curvature and extended persistence images using synthetic and real-world datasets comprising of Erdős–Rényi graphs and citation networks. For all experiments, DYMAG is trained using a uniform signal as input. Ground truth Ollivier–Ricci curvature values are computed directly from the adjacency matrix, and persistence images are generated using node degree as the filtration function. Results are reported as mean with standard deviation in parentheses. Across all settings, DYMAG variants with heat or wave dynamics consistently outperform baseline methods. Given the high computational cost of computing Ollivier–Ricci curvature on large graphs, we restricted the PubMed evaluation to a subgraph comprising the 2,000 most highly cited papers and omitted this evaluation for OGBN-MAG and OGBN-Papers100M.

In Table 4, we evaluated the performance of DYMAG on three publicly available datasets for biomolecular graph classification from the TUDatasets benchmark [67]. The ENZYMES dataset [68] consists of protein secondary structures with ground truth annotations of catalytic activity. In the PROTEINS dataset [21], the task is to classify whether a protein functions as an enzyme. The MUTAG dataset [69], contains small nitroaromatic compounds, and the task is to classify their mutagenicity on the S.typhimurium bacterium. DYMAG achieves strong validation accuracy across all datasets, with the wave equation variant performing best on PROTEINS and MUTAG.

In , we present results corresponding to Figure 5, where the task is to recover generating parameters of random graphs. On both Erdős–Rényi and stochastic block model (SBM) datasets, DYMAG achieves the best overall performance, further validating its capacity to capture latent structural properties.

11.1 Ollivier-Ricci Curvature↩︎

In Riemannian geometry, Ricci curvature quantifies how a space deviates from being locally Euclidean by measuring volume distortion along geodesics. The discrete Ollivier-Ricci formulation extends this notion to graphs by capturing how neighborhood structures contract or expand under local optimal transport, thereby providing a principled measure of local geometric distortion.

To evaluate the capacity of DYMAG to capture such local geometric properties, we consider the task of predicting node-level Ollivier-Ricci curvature [56]. Ground truth curvature values are computed using the GraphRicciCurvature package (v0.5.3.1). The method first calculates edge-level curvature scores via optimal transport between neighborhood distributions, and then aggregates these values to the node level by averaging over all incident edges.

Because Ollivier-Ricci curvature is determined entirely by the graph topology - specifically, the adjacency structure and any edge weights - it does not require node features. We therefore compute curvature values directly from the graph’s adjacency matrix. The resulting node-level values are used as regression targets in a node-level prediction task.

We report results on both real-world and synthetic graphs in . Due to the computational cost of Ricci curvature estimation on large graphs, we restrict evaluation on PubMed (19,717 nodes) to a subgraph comprising the 2,000 most highly cited nodes.

11.2 Extended Persistence Image↩︎

To compute extended persistence diagrams for graphs, we define a scalar filtration function \(f: V \rightarrow \mathbb{R}\) that assigns a real value to each node. By default, we use node degree, but the framework supports any scalar-valued function (e.g., centrality, clustering coefficient, or domain-specific metadata).

From this node-level function, edge values are induced by setting \(f(u, v) = \max\{f(u), f(v)\}\). We then construct a filtration over the graph using both sublevel and superlevel sets: in the sublevel filtration, nodes and edges are added in order of increasing \(f\), capturing the evolution of connected components and cycles; in the superlevel filtration, nodes and edges are included in decreasing order, allowing for the identification of global topological features that persist across the entire graph. The extended persistence diagram combines information from ordinary, relative, and extended homology classes to characterize these multiscale topological changes.

Each extended persistence diagram contains a collection of birth–death pairs for two types of features: dimension 0 features correspond to connected components, while dimension 1 features correspond to cycles. To convert these diagrams into a vector representation, we map each birth–death pair \((b, d)\) to a birth–persistence pair \((b, p)\) where \(p = d - b\). We then place a Gaussian kernel (with bandwidth \(\sigma = 0.005\)) centered at each \((b, p)\) coordinate and discretize the resulting function onto a grid to obtain persistence images [55]. To reflect the distinct statistical profiles of the two types of features, we use different grid resolutions: for dimension 0 features, which are typically short-lived, we use a compact \(25 \times 1\) grid; for dimension 1 features, which exhibit greater variability in both birth and persistence, we use a full \(25 \times 25\) grid. These two images are flattened and concatenated into a 650-dimensional vector.

We treat persistence image prediction as a graph-level regression task. The pooled graph embedding from DYMAG is passed through a 5-layer MLP to predict the flattened persistence image vector.

Results using node degree as the filtration function are reported in (bottom). In , we present results obtained using clustering coefficient as the filtration function. The clustering coefficient of a node \(v\) is defined as \[c(v) = \frac{2T(v)}{\text{deg}(v)(\text{deg}(v)-1)}\] where \(T(v)\) is the number of triangles through node \(v\) and \(\text{deg}(v)\) is the degree of node \(v\). Compared to degree-based filtrations, the higher MSE values in this setting suggest that persistence images generated using clustering coefficients are more challenging to predict.

11.3 Node Classification Accuracy on Homophilic and Heterophilic Datasets↩︎

In Table 8, we also consider node classification on homophilic (Pubmed, Citeseer, and Cora) and heterophilic (Texas, Wisconsin, and Cornell) networks from [19]. On the homophilic real-world graph datasets, we see that the wave version of DYMAG outperforms the heat/Sprott versions, achieving performance that is roughly comparable with the more standard GNNs. However, on the heterophilic datasets, we see that DYMAG with chaotic Sprott dynamics outperforms other models.

11.4 Fractional Heat Equation Dynamics↩︎

In Tables 5 and 6, we conduct experiments investigating the role of \(\alpha\) in the fractional Laplacian \(L^\alpha\). We highlight the case \(\alpha=1\) in gray to emphasize that when \(\alpha=1\), the fractional Laplacian reduces to the standard (non-fractional) graph Laplacian. The values of \(\alpha\) that achieve the best performance are highlighted in blue. In cases where there is a tie and \(\alpha=1\) is one of the co-best methods (e.g., in the MP dataset), we highlight the \(\alpha=1\) case in gray and the other top-performing method in blue.

Table 5, reports the mean squared error (MSE) for predicting extended persistence images. We observe that varying \(\alpha\) significantly impacts the model’s performance. For most Erdős–Rényi (ER) graphs with \(n=100\) nodes, lower values of \(\alpha\) yield better performance than the standard Laplacian (\(\alpha=1\)), suggesting that fractional heat diffusion processes capture relevant graph features more effectively in this context. For larger graphs with \(n=200\) nodes, the optimal \(\alpha\) is still lower than 1, though not as low as 0.25.

In Table 6, which reports the \(R^2\) scores for predicting various geometric and graph topological properties of molecules, we observe that the performance across different \(\alpha\) values is relatively similar, indicating robustness to the choice of \(\alpha\). For example, on the PROTEINS dataset for predicting dihedral angles, the highest \(R^2\) score is \(0.89\) at both \(\alpha=0.25\) and \(\alpha=0.50\), while at \(\alpha=1\), the score is \(0.87\). In the case of the Materials Project (MP) dataset for predicting band gap, there is a tie in performance between \(\alpha=0.50\) and \(\alpha=1.00\), both achieving an \(R^2\) score of \(0.59\).

Overall, these results demonstrate that non-local smoothing achieved with the fractional Laplacian featuring various \(\alpha\) parameters allows the model to perform better on certain tasks. Specifically, fractional Laplacians with \(\alpha < 1\) can enhance performance in recovering the topology of randomly generated graphs, while different values of \(\alpha\) do not significantly impact DYMAG’s performance on molecular and material science datasets.

12 Dataset Description↩︎

Cora [16] is a citation network comprising 2,708 scientific publications classified into one of seven categories. Each node represents a publication and is associated with a 1,433-dimensional binary feature vector indicating the presence or absence of specific words from a predefined dictionary. Edges represent the 5,429 citation links between documents.

Citeseer [17] is a citation network containing 3,312 scientific publications categorized into six classes. Each node corresponds to a publication and is described by a 3,703-dimensional binary feature vector based on the presence or absence of specific dictionary words. The graph includes 4,732 citation links, forming edges between related documents.

PubMed [18] is a citation network of 19,717 biomedical research articles from the PubMed database, all related to diabetes, and categorized into three classes. Each node represents a publication and is associated with a 500-dimensional feature vector based on TF-IDF weighted word frequencies. The graph contains 44,338 citation edges.

Cornell, Texas, and Wisconsin [19] are subgraphs extracted from the WebKB dataset, comprising webpages from the computer science departments of the respective universities. Each node represents a webpage, described by a bag-of-words feature vector derived from its textual content. Edges correspond to hyperlinks between pages. The classification task involves predicting the type of webpage (e.g., student, faculty, course, project, staff). These graphs are relatively small, with 183-251 nodes and 295-499 edges. Notably, all three datasets exhibit strong heterophily, where connected nodes often belong to different classes, posing a challenge for traditional homophily-based graph learning methods.

ogbn-papers100M and ogbn-mag are large-scale academic graphs from the Open Graph Benchmark (OGB) collection. ogbn-papers100M is a directed citation network comprising over 111 million papers indexed in the Microsoft Academic Graph (MAG) [70], where each node represents a paper with a 128-dimensional word2vec feature vector, and edges denote citation links. ogbn-mag is a heterogeneous graph also derived from MAG, containing four node types - papers (736K), authors (1.1M), institutions (8.7K), and fields of study (60K) - and four directed edge types: authorship, citation, affiliation, and topic assignment. Only paper nodes have input features (128-dimensional word2vec embeddings), while the other node types are featureless.

PROTEINS [21], part of the TUDataset benchmark suite [67], is a graph classification dataset consisting of 1,113 protein structures, each labeled as either an enzyme or a non-enzyme. In each graph, nodes represent amino acids, and edges are formed between pairs of amino acids that are within 6 Ångströms of each other in 3D space.

ENZYMES [68], part of the TUDataset benchmark suite [67], contains 600 protein tertiary structures categorized into six enzyme classes, as defined by the BRENDA enzyme database. Each protein is represented as a graph, where nodes correspond to amino acids and edges capture spatial or sequential proximity.

MUTAG [71], part of the TUDataset benchmark suite [67], is a graph classification dataset consisting of 188 chemical compounds labeled according to their mutagenic effect on Salmonella typhimurium. Each compound is represented as a graph, where nodes correspond to atoms (with one-hot encoded atom types as features) and edges represent chemical bonds. There are 7 discrete node labels. The task is to predict the binary mutagenicity label based on molecular structure.

DrugBank [22] is a publicly available resource that integrates detailed information about drugs and their molecular targets. We use version 5.0 of the database, released in 2018, which contains 6,712 drug entries, including 1,448 FDA-approved small-molecule drugs. While the database includes a wide range of chemical, pharmacological, and structural properties, we focus on predicting two geometry- and topology-related molecular attributes: total polar surface area (TPSA) and the number of aromatic rings. Each molecule is represented as a graph, with atoms as nodes and bonds as edges.

The Materials Project (MP) dataset [24] consists of a large collection of inorganic compounds labeled with physical and chemical properties computed using density functional theory (DFT). We use version 2018.6.1, which includes 69,239 materials and a range of properties such as formation energy, bulk and shear moduli, and electronic band gap. In our experiments, we focus on predicting the band gap (e.g., in eV), a key electronic property available for 45,901 compounds. Each material is represented as a graph, with atoms as nodes and edges defined by interatomic bonds or distances derived from crystal structures.

The Antiviral Screen Dataset [23] originates from the Drug Therapeutics Program (DTP) AIDS Antiviral Screen, which evaluated the anti-HIV activity of 43,850 chemical compounds based on their ability to inhibit HIV replication. Each compound is represented as a molecular graph, with atoms as nodes and bonds as edges. Screening outcomes were originally categorized into three groups: confirmed active (CA), confirmed moderately active (CM), and confirmed inactive (CI). As part of the MoleculeNet benchmark [72], the CA and CM categories are merged, resulting in a binary classification task: predicting whether a compound is active (CA/CM) or inactive (CI).

References↩︎