QDC: Quantum Diffusion Convolution Kernels on Graphs
Authors: Thomas Markovich
Venue: Accepted by TMLR ()

Abstract: Graph convolutional neural networks (GCNs) operate by aggregating messages over local neighborhoods given the prediction task under interest. Many GCNs can be understood as a form of generalized diffusion of input features on the graph, and significant work has been dedicated to improving predictive accuracy by altering the ways of message passing. In this work, we propose a new convolution kernel that effectively rewires the graph according to the occupation correlations of the vertices by trading on the generalized diffusion paradigm for the propagation of a quantum particle over the graph. We term this new convolution kernel the Quantum Diffusion Convolution (QDC) operator. In addition, we introduce a multiscale variant that combines messages from the QDC operator and the traditional combinatorial Laplacian. To understand our method, we explore the spectral dependence of homophily and the importance of quantum dynamics in the construction of a bandpass filter. Through these studies, as well as experiments on a range of datasets, we observe that QDC improves predictive performance on the widely used benchmark datasets when compared to similar methods.

Published in Transactions on Machine Learning Research (02/2024)
QDC: Quantum Diffusion Convolution Kernels on Graphs
Thomas Markovich
bmarkonich
CashApp
Cambridge, Massachusetts, USA
Reviewed on OpenReview: /ttps://openreen net/ forum? id= uXGUSX8GoY
Abstract
Graph convolutional neural networks (GCNs) operate by aggregating messages over local neighborhoods given the prediction task under interest. Many GCNs can be understood as a form of generalized diffusion of imput features on the graph, and significant work has been dedicated to improving predictive accuracy by altering the ways of message passing. In this work, we propose new convolution kernel that effectively rewires the graph according to the occupation correlations of the vertices by trading on the generalized diffusion paradigm for the propagation of a quantum particle over the graph. We term this new convolution kernel the Quantum Diffusion Convolution (QDC) operator. addition, we introduce a multiscalc variant. that combines messages from the QDC operator and the traditional combinatorial Laplacian. To understand our method, we explore the spectral dependence of homophily and the importance of quantum dynamics in the construction of bandpass filter. Through these studies, well as experiments range of datasets, observe that QDC improves predictive performance on the widely used benchmark datasets when compared to similar methods.
1 Introduction
Graphs are mathematical structures composed of vertices and edges, where the edges represent potentially complex relationships interactions between the vertices. Graph structured data involves complex patterns and relationships that can not be captured by the traditional deep learning methods that focus on tabular data. As result, graph structured data and graph machine learning models have become increasingly important in many fields, such as machine learning (Wu et al.] 2020 computer vision (Krzywda et al. 2022), and natural language processing (Wu et al. 2023). Indeed, graphs and graph structured data are ubiquitous in industrial applications ranging Trom Traud detection (Liu et al., 2020 Zhang el al.] 2022),
routing RRsek al. 2019 Chen al. 2022). weather predictions (Keisler 2022 Ma al. 2022), drug
discovery (Bongini et al. 2021 Han et al.
2021 Xiong et al. 2021), and personalized recommendations (Wu
al. 2022 Gao al.
2021
Graph neural networks (GNNs) are an increasingly popular modality for constructing graph machine learning
models [Zhou et al. 2020 Wu et al. 2020). There are many variations of architectures for GNNs, but most GNNs can be thought of as having a function that, for a given vertex, aggregates information from its neighbors, and a second function which maps this aggregated information the machine learning task under investigation, such as node classification, node regression, or link prediction. simple but powerful model is Graph Convolutional network (GCN), which extends the convolutional neural network (CNN) architecture to the graph domain by using localized filter that aggregates information from neighboring nodes (Zhang et al. 2019). By sharing weights across different nodes, GCNs can learn representations that capture both the local and global structure of the graph. These models have shown remarkable success in a variety of tasks such as node classification (Kipf & Welling 2016 Zhang et al. 2019), graph classification et al.) 2020). community detection (Jin et al. 2019 Wang al. 2021), ) and link prediction (Chen et al. 20200 Cai et al. 2019 Zeb et al.] 2022). GCNs learn filters on the graph structure be used inference time. Early GCN development learned these filters in the spectral domain (Bruna et al. 2013), but this requires

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the decomposition of large matrices. Due to the computational expense of these decompositions, spatial filters rose in popularity and have been the dominant paradigm. Significant effort has been dedicated to methodological improvements that make spatial convolutions more expressive (Bodnar et al. 2021 Bouritsas
ct al. 2022 and scalable (Hamilton et al. 2017 Ying et al. 2018
Further work has shown that is possible to unify many of these models based on spatial convolutions as generalized graph diffusson @Chamberlain et al.
2021a b Gasteiger al. 2019), with considerable attention
(Elhag et al. 2022 Di Giovanni et al., 2022). These gener- alized graph diffusion models can be understood variants of the heat diffusion equation. Heat diffusion graphs is well known to thermalize quickly, leading all vertices to average to the same temperature value. This physical phenomenon is similar to the oversmoothing behaviour that has plagued GCNs. There have been many approaches to combat this oversmoothing by altering the properties of the diffusion while keeping the underlying diffusion dynamics (Alon & Yahav 2020: Zhu et al. 2020 Topping et al. 2021
being focused on improving diffusion dynamics
This work is inspired by Gasteiger et al. (2019) and the recognition that the Graph Diffusion Convolution method (GDC) rewires the graph using a lowppass filter that is inspired by heat diffusion. We build upon this foundation by considering the question "Can we improve graph neural networks by rewiring the base graph with a learnable band-pass filter?". We find our answer in a rewiring method that is structured around a learnable band-pass filter and inspired by the Schrödinger equation. ion. Following in the steps of Gasteiger et al. (2019), we construct a graph Laplacian preprocessing framework that captures the dynamics of quantum diffusion through a system. We call this framework QDC, and find that this framework is very flexible and can be included into the architecture of many graph neural networks. In summary, this paper's 's core contributions are
infinite-time
We propose QDC, a quantum mechanically inspired diffusion kernel, a more powerful and general mothod for computing sparsified non-local transition matricos.
We propose a novel multi-scale message passing paradigm that performs message passing using QDC and the original combinatorial Laplacian in parallel.
We compare and evaluate QDC and MultiScaleQDC to a set of similar baselines on a range of node classification tasks.
We analyze the spectral dependence of homophily in graph datascts and show that many het- erophilous datasets are actually homophilous in filtered settings.
2
Related Work
Our method can be viewed as a technique for graph rewiring, because it changes the computation graph from the original adjacency matrix to a filtered one. Graph rewiring is a common technique in the literature for improving GNN performance by removing spurious connections. The Stochastic Discrete Ricci Flow (SDRF) method seeks to perform this rewiring by fixing instances of negative curvature the graph (Toping al, 2021 The Fully Adjacent layer method (+FA) attempts to improve performance by adding fully adjacent layer to message passing (Alon& & Yahav 2020). GDC is the most similar method to ours, but is based on the idea of heat, rather than quantum, diffusion which yields a low pass filter (Gasteiger et al. 2019).
Our method can also be understood within the context of physics inspired graph neural networks. Graph- HEAT proposes performing graph convolutions using a paramcterized version of the heat kernel (Xu et al., 2020). The Graph Neural Diffusson (GRAND) method recasts message passing as anisotropic diffussin on a graph, and provides framework with which to unify many popular GNN architectures (Chamberlain et all, 2021a: Thorpe et al. 2022). BLEND pushes this perspective further to explore diffusion in non-cuclidean domains (Chamberlain et al. 2021b). PDE-GCN looks further and seeks to combine diffusion with the wave equation to define new message passing frameworks [Eliaof et al.] 2021). To our knowledge, ours is the first. work that explores quantum dynamics as a message passing formalism.
Our method closely related the kernel signatures that have been explored in computer graphics. The Heat Kernel Signature was one of the first kernel signatures that was developed, and was developed by
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Quantum Diffusion
Sparsification
Figure 1: An illustration of the Quantum Diffusion Convolution Kernel. The graph is transformed through the propagation quantum particle as given by equation ??, and then sparsified the graph to account for the presence of rare events. The model is then trained on the post-sparsified rewired graph.
modeling the heat diffusion over a surface (Sun et al. 2009 Zobel et al., 2011]. It was observed that the Heat Kernel Signature was sensitive to motions that might introduce bottle neck, such as an articulation, which motivated the development of the Wave Kernel signature. The Wave Kernel Signature defines surface structural features by modeling the infinite time diffusion of quantum particles over the surface (Aubry et al., 2011). Building on the wave kernel signature, the average mixing kernel signature explores the finite time diffusion of a quantum particle over the surface (Cosmo el al. 2020). The wave kernel trace method instead explores solutions of the acoustic wave equation using the edge-Laplacian (Aziz al. 2014). These methods have all been used to develop graph features that would then be used to identify similar shapes, but we are instead using these kernels as convolution operators.
3 Background
Preliminaries Given an undirected graph, g = X), where V is the vertex set with cardinality [V] = N. and & is the edge set, and X E R.aad denote the matrix of vertex features, where d is the dimensionality of the feature set. admits an adjacency matrix, A E NNN, where Ay = 1 if and only if vertices and j are connected. Because we have restricted ourselves to undirected graphs, A.j = Ajr. While this & could have weighted edges, we focus on the unweighted case for simplicity. It is common to augment a graph with self loops, which is performed by A I A, to allow for message passing of depth I to include messages from all random walks of length < (+1. 1. With these definitions in hand, we define our combinatorial graph Laplacian as L = D- + AD 1, where D is the diagonal degree matrix of Á. Other normalizations are possible, but we restrict our discussion to symmetrically normalized graph Laplacians.
Graph Signal Processing The central challenge of signal processing on graphs is to apply the intuition from signal processing on structured grids to nonuniform and unstructured graph domains. We do so by drawing an analogy between graph and rectangular domains. The Fourier basis in rectangular domains is given by ((x, k) eik= where k is the vector of frequencies, positions, which are the eigenfunctions of the Laplacian in rectangular coordinates. The definition of the Fourier basis allows us to define the Fourier transform and the convolution theorem:
wave numbers, and is the vector of
f(x) = (g h) = ( (F(g))F(h)))
(1)
where FF and F-1 1 is the Fourier transform and inverse Fourier transform respectively; is the spatial convolution; and is the point wise product. Convolutions are ubiquitous in signal processing, with spectral filtering being the application that we will focus on. From the perspective of spectral filtering, we have a signal, h(x), and filter given by g(.). We either apply this filter in the spectral conjugate spatial
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domain, but spectral filtering is preferred. If the optimal filter is unknown priori, we learn this filter through learning procedure. This can be done in variety of ways, such as learning the filter in the Fourier domain. This amounts to learning filter with infinite support and an infinite number of free parameters. While theorctically possible, this is a suboptimal technique. Instead, we frequently choose to expand into a convenient basis. In this way, we end up with:
- [(x) Σ (ckgk) =F=
E F((n) -FN))
where cı is the expansion coefficient for each basis function given by ga. Choices of basis function can include gaussians, wavelets, polynomials, or other functional forms depending on the desired properties. The implicit assumption with this filtering construction is that there are components of the signal which are spurions for a given application. Depending on the application and the signal, we might need to learn a filter that damps high or low frequency noise, or only selects narrow set of frequencies. These are called low-pass, high-pass, and band-pass filters respectively.
Analogously, we domain. In the case of an undirected graph, the Laplacian is a symmetric positive semidefinite matrix which admits an eigensystem with orthonormal eigenvectors and real, positive, eigenvalues. These eigenvectors form the graph Fourier basis and the cigenvectors form the squared frequencies. We can then write the eigendeconpooition as = UT AU, where U is the matrix of eigenvectors and is the diagonal matrix of eigenvalues.
the definition of the graph Laplacian to construct a Fourier basis in the graph
This definition allows begin to define filters. The original spectral GCN paper Bruna al. 20133 learned filters of the form g(O) diag(O), where diag(O) is learned set of parameters for each of the Fourier basis functions. This is then used for filtering as:
f(x) - (g(0) (UT x)
(3)
While effective and highly flexible, this filtering technique has O(n) learnable parameters, can be difficult learn, and requires decomposition of to construct eigenvectors U. Spectral convolutions are mathematically clegant, and the construction of the basis and learning of the filter computationally demanding, making them not ideal in many use cases. By contrast, convolutions can also be defined in the spatial domain. Spatial convolutions are spatially localized, so they have finite size, but they aren't 't guaranteed to be unique which makes them computationally difficult to learn and to apply. This mothod makes the assumption that we can approximate g as g(: {o}) Σx AxA which yields a filtered signal of the form:
K
f(z) = Σ CxC
(4)
This learned filter is more parameter efficient and spatially localized, because convolutions only extend to distance reachable by K applications of the Laplacian. Given that desire to limit our computational cost, we will restrict ourselves to learnable functional forms that result only in matrix products because for sparse matrices, these scale as O(EE). Chebyshev polynomials are one straightforward choice because they are the interpolating polynomial that provides the best function approximation under the maximum error norm. These polynomials yield a filter of the form g (A; [+}) = Σx 0gTx (A), where Ti is the 4th Chebyshev polynomial given by the recurrence relationship (x(x) 1x7)-1(x) - T Tn-2(x), and À is the rescaled eigenvalue matrix (Defferrard et al. 2016)
Diffusion as Convolution We can readily identify equation 4 as the Taylor expansions of the matrix exponential for particular choices of f which has a similar form to the Heat kernel. This amounts to solving the partial differential equation;
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t= 0
t=50
t= 250
t= 950
1.0
Heat
0.6
0.2
0.0
Quantum
0.8
0.4
0.0
Figure A comparison of both heat diffusion and quantum dynamics on barbell graph. The lop row corresponds diffusion according to the heat equation of the graph, and the bottom row corresponds to propagation of the Schrödinger equation. We simulated dynamics for 1000 unitless timesteps with the same initial distribution for both equations. We use the plasma colorbar, with blue and yellow corresponding to the minimum and maximum respectively. We observe the top row thermalizes within the cluster rapidly to the same temperature but encounters a bottleneck as the two ends the barbell thermalize. By contrast, quantum dynamics exhibits oscillatory behavior both within the clusters as well as across the clusters, and probability density migrates rapidly.
0f 01
(5) where A is the Laplaco-Beltren Beltrami the graph, and admits solution f(t) cxp(-tt)f(0)) Viewed in this way, can draw the equivalence between solving the heat equation the graph and many message passing schemes. Building this observation, Gasteiger et al. (2019) observed that it is possible to define a generalized graph diffusion operator S as S = Dx Ox where T is transition matrix, such as the combinatorial Laplacian. As we can see from the definition of the heat kernel, exponentially squashes the contributions from eigenmodes with large sigenvalues, thereby providing us with a low-pass filter. Intuitively, this should help settings where averaging out the noise from neighbors can improve performance. Following this line of thought, many graph neural network architectures can be understood performing generalized diffusion over graph (Chamberlain et al. 2021b a..
-ff = -cCf,
From physical perspective, the heat equation is well known to drive towards an infinitely smooth equi- librium solution; by smoothing out the initial distribution exponentially quickly in acausal ways. The heat equation is also well known to exhibit thermal bottlenecks when the heat flux and temperature gradient vectors deviate (Bornoff et al. 20111 Grossmann Lohse (200)). This physical process analogous to the oversmoothing and oversquashing problems that have plagued graph neural networks respectively. We can observe both of these physical processes in Figure [ which presents comparison of heat and quantum diffusion at four different time steps. By t 50, observe that the top lobe of the barbell has completely thermalized, oversmoothed, and heat is slowly starting to leak into the center vertex, or oversquashed. In 200 timesteps, we observe very little in the way heat transfer. It is clear that there thermal bottle neck in our system, and this is hampering the flow of heat.
4 Methodology
Quantum Convolution Kernels Because the graph defines both the goometry of the system and its Laplacian, it is natural to ask if grounding our rewiring a different physical model would yield better results. There many possible options including the wave equation, which propagates the sharpness of
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solutions; the clastic wave equation which models both forces as displacements vector quantities; the convection equation which models the flux; the eikonal equation which simulates shortest path searches; or Schrödinger's equation which models quantum particles. Of these partial differential equations (PDEs), we turn our attention to Schrödinger's equation. While structurally similar to the heat equation, its dynamics are stable, do not lead to oversmoothing, natively capture both constructive and destructive interference, and are controllable through the engincering of potential energy surfacos (Chou et al., 2012 Jaffé 1987 Sakurai & Commins 1995). Indeed, Schrödinger's equation and the heat equation are related through Wick rotation by 2 (Popov) 2005). Qualitatively, we observe in Figure 2 that unlike thermal diffusion, quantum diffusion is able to quickly pass information through the bottleneck. Later timesteps show us constructive and destructive interference that provide structure to the propagations. Videos of the propagation show oscillations across the top lobe and along the vertical axis as a function of time. Qualitatively, these dynamiis do not seem to oversmooth and are less prone to oversquashing as a result. Motivated by these qualitative results, we bricfly introduce Schrödinger's equation, which are used to derive our quantum convolution kernel. The dynamics are governed by the time dependent Schrödinger equation, which, for a free partiele, is given by;
(xtttt) (ub(ar Ø1
(6)
On a graph, the eigenstates of A define a finite, complete, and orthogonal basis where the position arguments correspond to the vertex locations. Therefore, the expectation value (1b22, ()) computes the proba- bility that a particle on verlex i is found on vertex j at some time 1. In our setting, we wish to compute the long-term steady-state distribution, so we compute the energy-filteeed infinite time integral of the average overlap between any two vertices, which is given by;
00 L0 dt e e
222 + til (a, () = Ee Ce-(Ei-u) 1E:- (0 (z)))(((z)) = 2(z,,
where Q is our Quantum Diffusion Kernel (QDC), and both and are our two tunable parameters. Because GCNs learn spectral filters, we observe that this rewiring process is frequency dependent and has two learnable parameters in the diffusion kernel. Intuitively, we interpret Q(z;, as the time averaged probability of transition from vertex to vertex j. Analogously to GDC, we can use Q as our transition matrix, instead of combinatorial graph Laplacian. Doing so allows us to use QDC with any message passing neural network by simply replacing L with Q. We present schematic of our method in Figure | in which the first transformation corresponds to switching from the original graph laplacian to transition matrix.
Sparsification QDC defined as 2,),, is a matrix Qi, Q(z,,,)). where Qr is the probability of transition from vertex i to vertex j. Most graph diffusion results in a dense transition matrix, and QDC is no diffcrent. This happens because a quantum particle starting at site i will visit all vertices within its connected component given an infinite amount of time, yielding probabilities that can be small but non-zero. This is potentially major downfall of QDC when compared against spatial methods like Graph Diffusion Convolution Gaasteiger et al. 2019). This has the potential to introduce O(N2) storage costs. To address this issue, sparsily the resulting QDC matrix. We consider two different sparsification methods: simple threshold based approach, or an approach that only keeps the top-k highest weighted connections. We denote the sparsified QDC kernel as Q. While Q was both row and columm normalized, @ is not. Therefore, after sparsification normalize Q in the usual way, defining Qeym D0 2/2 1/2 We will drop the sym in
the following, such that all uses of Q are normalized.
Efficient Diagonalization QDC is a spectral method, and depends on the cigendecomposition of L. This is commonly viewed as loo computationally demanding of a procedure because the full eigendecompositiion of a matrix requires O(N3) time, and the storage costs of the resulting dense eigensystem are O(2)) where N the number of vertices. While this is generally true, we recognize from the form of our kernel in equation ??, we are constructing a band pass filter and are thus only interested in a subset of the eigensystem. As a result, are able to use approximate methods that are more computationally efficient.
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Due to the importance of cigendecomposition to the computational sciences, this problem has received considerable attention with algorithms such as power iteration Misse & Gollaczek-Geiringer 1929| divide
and conquer [Cuppen] 1980), Arnoldi iteration (Arnoldi] 1951), Lanczos iteration (Lanczos 1950 Ojalvo & Newman 1970 and LOBPCG Knyazov [2001]: Knyazey et al. (2007). In this work we used LOBPCG, or
Locally Optimal Block Preconditioned Conjugale Gradient, because it is provides a straightforrard method to compute a limited number of cigenvectors and eigenvalues while only depending on the computation of matrix veclor-products. In our applications, we use the folded spectrum method (MacDonald 1934) along with LOBPCG to compute eigenvalues centered around μ.. If the solver is unable to converge, we retry with n' = 14 + 6x, where 6Λ = le 6. In our settings, we compute min(512, N) eigenvalue, eigenvector pairs. In Table 2 we present average runtimes for training and testing for all GCN based methods. While we observe a significant increase in runtime, we attribute the majority of that cost lo the preprocessing of the Laplacian, which requires both an eigendecoopoosiionn and sparse matrix multiply. In applied settings, it may be possible to cache the Laplacian which would allow the amortizaiton of this cost. These techniques do not change the fact that diagonalization, even with matrix free methods, is expensive.
Multiscale GNN QDC can be used drop-in replacement for transition matrix for any message passing GNN. In section 5 we explore using QDC in place of & for both graph convolutional networks and graph attention networks. Because QDC provides a band pass filter, unlike GDC which provides a low-pass filter, it is interesting to explore the message passing across both L and Q in parallel. In this setting, we pass messages in parallel using L on one side and Q on the other. We then combined messages from each tower by either adding or concatenating them together. Finally, we feed the resulting messages into a readout function for predictions. We term this method MultiScaleQDCC, because we are able to pass messages across multiple length scales of the graph.
5 Experiments
We tested our method on the task of semi-supervised node classification in both homophilic and heterophilic settings to understand the performance of our method, Broadly, we observed that QDC and MultiScaleQDDC provided significant, performance improvements upon the baseline methods that they were incorporated into. In some cases, the improvements can be as large as 20%. To understand these results, we analyze the spectral dependence of homophily and observe that QDC and MultiSealeQDDC are able to rewire the graph in a way that makes it more homophilie. In the remainder of this section we present the details for these experiments as well as a more detailed discussion of the results.
Experiment Setup In an effort to ensure a fair comparison, we optimized the hyper-parameters of all models presented in Table ! on all data sets, We performed 250 steps of hyper-parameter optimization for each method, including basclines, and the hyper-parameter search was performed using OPTUNA, a popular hyper-parameter optimization framework. All tuning was performed on the validation set, and we report the test-results associated with the hyper-parameter settings that maximize the validation accuracy. The parameters, and the distributions from which they were drawn, are reported in Appendix AA.4 All experiments were run using PYTORCH GEOMETRIC 2.3.1 and PYTORCH 1.13, and all computations were run on an Nvidia DGX A100 machine with 128 AMD Rome 7742 cores and 8 Nvidia A100 GPUs.
Because our method can be viewed as a Laplacian preprocessing technique, we use QDC in place of the traditional Laplacian in both graph convolution networks (GCN) (Kipf & Welling) 2016 Zhang et al. 2019) graph attention networks (GAT) (Veličković et al. 2018), and H₂GCN, a GNN designed to specifically handle hetcrophilic datasots (Zhu et al. 2020). QDC is similar in structure to graph diffusion convolution (GDC) Gasteiger et al. (209)) and SDRF (Topping et al., 2021), ), so we have chosen to compare QDC to both GDC and SDRF in addition to an umprocessed Laplacian in a GCN, GAT, and a H₂GCN. Our GCN and GAT models are implemented using the relevant layers from PYTORCH GEOMETRIC, and H₂GCN was run using an open source reimplementation (Gitvventhandler 2022). Similarly, our GDC implementation uses the publicly available version in PYTORCH GEOMETRIC, and we used the reference SDRF implementation from the authors [Topping )022).
Datasets We evaluated our method on 9 data sets: CORNELL, TEXAS, and WISCONSIN from the WebKB dataset; CHAMELEON and SQUIRREL from the Wiki dataset; Actor from the film dataset; and citation graphs
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Table 1: Dataset statistics and experimental results on common node classification benchmarks. H. 1V |8| denote degree of homophily, number of vertices and number of odges, respoctively. Top results for cach of the GCN and GAT families are highlighted in bold.
Cornell
Testas
Wasconein
Chamelcon
Squireel
Actor
Cona
Citesect
Pichmeed
0U06
0.30
11:25
0.22
0.24
0.51
0..73
[V]
183
183
251
2,277
5,201
7,000
2,708
3,327
18.717
[E]
280
255
456
31,421
198,433
26,702
5,278
4,076
44,327
GCN
05.08
63.51
30:22
27. 79
29.
87.
76.61
1:25
66.47
GCN+GDC
47.03 5.09
63.51
6.07
57.25
2.83
40.03 42
2.93
27.97
0.93
20. 10.14
0.91
S7.63
0.91
76.58
1.21
88
1. 0.55
GCN+SDRF
45.14 5.20
62.97
5.55
57.64
1.52
40:35
1.52
28.17 1 0.97
20.07
1.00
$7.44
1.10
76.85
1:47
88.47
1 0.34
GCD+
61.81 5.95
08.32
63.73
50.44
01.37
1.04
1:17
74.70
1.34
84.33
0.56
GCN+QIX
63.78 9.68
72.70
6.67
65.29
0.80
68.22
1.56
00.02
1.94
36.08
0.64
56.00
1.56
75.10
1.52
84.65
0.44
CCN MalliSeskDD
66.22 6.44
71.77
4.63
61.71
4.47
54.71
= 2.79
42.24 #: 1.77
30.55 # 1.46
87.85
0.85
76.72
= 1.49
M8.32
0.47
GAT
60.81 + 5.40
68.11
5:24
63.14
7.58
44.55 88
1.64
31.47
1.44
30.45
1.17
86.68
1.64
75.64
1.55
$4.11 + 0.70
GAT+GDC
63.33
31.65
31.18
75.92
0.55
GAT+
50.19 6:33
67.30
63.02
5:20
45. 58
1.05
31.76
1.00
31.15
0.78
85.20
1.34
75.90
1: 1.27
87.47 0.48
EXT+QDCC
77.57 1: 3.83
87.57
5.56 86.22
85.04
3.33
30.50
1: 2.16
35.33
1.81
35.57 1: 1.05
S1.08
1.54
75.21
1.30
H7.55 1 0.31
17.03
4.06
2.60
1.11
76.39
87.59
0.38
H&GCN
74.05 4.22
87.84
5:30
55.69
9.51
58.00
2.08
28
7.14
39.54
1.11
88.15
1.22
06.82
3.54
80.44
0.48
GDC
74.50 +.95
67.03
5:24
K3.92
+ 1.71
00.73
1.27
26.76
+ 2.87
33.45
+ 1.26
88.19
1.00
75:57
1.45
80.42 + 0.55
HDGU
74:05 3.24
71.35
4.71
S1.11
9.56
58.60
1,78
96.08
1.24
33.52
0.78
58.11
1.49
76.85
1.47
60.44 1. 0.51
HCC
77.84
57.45
$5.10
4.13
61.52
1.91
36.30
0.96
34.35
0.59
74.76
2.29
75.63
1.53
T3.34 3,10
76.01 3.72
88.38
4.84
86.86
3.51
59.14
1.61
36.64
1.88
34.70
0.94
$7.87
1.43
76.13
1.79
89.00 0.38
CORA, CITESEER, and PUBMED. Where applicable, we use the same data splits as Pei et al. 20200 Results are then averaged over all splits, and the average and standard deviation are reported. These datasets represent a mix of standard heterophilic and homophilic graph datasets. The statistics for the datasets are presented in the first three rows of Table ! where we have used the definition proposed by Pei et al. 22020) for homophily, given by:
1
ΣΣ [(())
H(G)
(8)
M
NEN
where N' is the neighborhood operator and 1 is the operator that returns the label of the vertex.
Node Classification We present the results from our experiments in Table I We observe that QDC provides improvements in accuracy across the heterophilic datasets, but seems to provide mixed results for Cora, Citescer, and Pubmed. By using MultiScaleQDCC provide improvements across all datasets. This validates our hypothesis that QDC can provide a viable step forward to improving GNN performance. These results are consistent for both QDC and MultiScaleQQDC modifications to our three base models GCN, GAT, and H₂GCN.
see that multi-scale modeling appears to
A Analysis of Hyper-parameterss (e.g. number of layers or number of hidden units), QDC has multiple hyperparameters, which are unique to the rewiring process that we tuned as part of our experiments. These hyperparameters correspond to μ,, the mean of the gaussian, 0, the standard deviation of the gaussian; and k. our cutoff parameter. This is not dissimilar from hods like GDC or SDRF. GDC includes hyperparameter for a and k which correspond to the diffusion strength and the cutoff parameter, respectively. SDRF has hyperparameters that correspond to the maximum number of iterations; the temperature, т; and Ricci-curvature upper-bound, C+. QDC only introduces one additional hyperparameter when compared with GDC, and has the same number of hyperparameters as SDRF. To understand the sensitivity of our method to these hyperparameters, we first present a violin plot in Figure 3 which plots a kernel density estimate of the model performances from the experiments on GCN, GCN+QDC, and MultiScaleQDD. In the case of the Cornell dataset, we clearly observe that MultiScaleQUDC has two humps, which correspond to the GCN and QDC distributions. We see similar patterns in the Texas, Wisconsin, Squirrel, and Actor datasots as well. This robust behaviour also holds for GAT based models, as can be seen from Figures b Furthermore, we clearly see that there are many experimental settings that out-perform the baseline model. While there are many effective experimental settings, we highlight that our method still quite sensitive to choice of hyperparameter. Furthermore, cach step of hyperparameter tuning is more expensive than in the case of GCN or GAT because we have compute new sets of approximate eigenvectors.
addition to the hyperparameters associated with the underlying model
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0.9
MultiscaleQOC
Mean Test Accuracy
0.8
GCN
GCN+QDC
0.7
0.5
0.4
0.2
0.1
Comell
Wisconsin
Chameleon
Squirrel
Actor
Cora
Citeseer
Pubmed
Figure Violin plots of our experiments +NN+MultiSallQDC (groen), GCN (yellow), and GCN+QDC (blue), where these plots generated by aggregating over all experiments associated with each model. We observe that both GCN+QDC and configurations.
generally have high density of near-optimal
Comell
Texas
Wisconsin
Chameleon
Squirrel
Actor
Cora
Citeseer
Pubmed
0.75
0.50
0.25
Figure 4: Scatter plots of mean test accuracy plotted against hyperparameters μ and In(a) in the first and second rows respectively for GCN+QDC (blue) and CCN+MultiScaleQDCC (green). We observe that each of QDC and MultiScaleQUIC are robust with respect to deviations in each of the hyperparameters.
We next turn our attention to the sensitivity of our model to and a for both QDC and MultiScaleQUDC models by plotting mean test accuracy against μ and or the first and second rows of Figure respectively. We have plotted both GCN+QDC (blue) and our MultiScaleeDC (green) on the same plot. We observe that in general, there are many settings of μ and that provide near equivalent performance which indicates find
that method is robust to potentially suboptimal choice of hyperparameters. Interestingly, that the optimal μs for GCN+QDC and our MultiScaleQDDC model are quite different. This is because in the MultiScaleQDCC case, are looking for eigenvectors that correct for any deficiencies in the original combinatorial Laplacian. In Figure g we present a 3D triangulated surface generated from the same data used to generate the scatter plots Figure 4 so that we could better understand the correlations between both sets of hyperparameters. consistent with the findings in Figure although these surface plots are somewhat difficult to interpret without the aid of the 2d projections presented in Figure We observe similar robust behaviour for GAT-
this figure we find relatively wide plateaus of high performance that are
based models as well,
be seen from Figures 6 Figure and Figure
Importance of Gaussian Filter At the core of QDC is the choice of filter. In our development of our method we have chosen to use Gaussian filter because it models inhomogeneous broadening, which is a physical effect that is observed as a transition frequencies taking on Gaussian profile due to microscopic details of the system such as atomic motion. This physical model intuitively sensible if we imagine that our vertices are analogous to the atoms, and the latent embeddings are the atomic positions. While we can provide physical arguments from analogy for this choice, citation networks not molecular systems. This

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Cornall
Squiral
0.5
0.50
025
0
tigenvalue
Eigenvalue
Eigervaluee
Eigenvalue
Egenvalue
Egenvalue
Eigenvalue
Bgenvalus
Bicenvauue
Figure 5: Plots of the homophily function of the eigenvalues. We observe that homophily has strong spectral dependence, and that the mid-band peaks in homophily agree with recovered optimal jas.
raises the question of whether the Gaussian form of our filter important, or whether any band-pass filter would be sufficient. To answer this question we implemented a variant of QDC given by
B(x,, (Fa
- 1 +)
(u +7 7- Fa) ob (x:)
where o-) is the logistic sigmoid function, f is the center of our bandpass filter, y is the width of our band-pass filter, and B is the band-pass version of QDC which we term the Band Pass Diffusion Convolution(PPDCC Using this filter, we performed experiments on a range of data sets using BPDC as our transition matrix with a GCN and have presented those results below in Table [ significant lift across the heterophilic datasets, but that lift is in general smaller than that observed with QDC.
We observe that BPDC able to provide
Spectral Dependence of Homophily It has previously been observed that the performance of Graph Convolution models correlates with the homophily of the graph, which motivates us to ask whether homophily is spectrally dependent. To answer this question, we constructed adjacency matricies from subsets of the eigenvectors that corresponded to each unique eigenvalue. In the case where the eigenvalues were degenerale, we computed the mean homophily. We then sparsified the resulting adjacency matrix by removing all entries smaller than - 7, and plotted the results in Figure 5 We observe that the homophily highly spectrally dependent. Actor appears to be outlier in this regard, with the optimal μ near 1 and the homophilic peaks existing in the range [.1. We attribute this to the generally poor performance on the Actor dataset, with a wide but flat performance envelope. In the case of the Cornell dataset, we observe that the datasot is generally quite heterophilic but becomes more homophilic in higher portions of the spectrum; and observe that the cluster for GCN+QDC in Figure | corresponds to this highly homophilic region. Similar trends are found for both Texas and Wisconsin. We observe spectral variations of homophily for Chameleon and Cora well; and note the same agreement. between the optimal μ and this observed spectral peaks in the homophily curves.
6 Conclusion
In this work we have introduced quantum diffusion kernel that we have termed QDC, and a multiscale model that we have termed MultiScaleQDC. We have motivated this convolution kernel through a deep connection with quantum dynamics on graphs. In experiments we have shown that QDC generally helps in cases of heterophilic node classification, and MultiSealeQDCC scems improve in both homophilic and heterophilic node classifications settings when included in three different base GNN models GCN, GAT, robust across configurations and that Quantum
and H,GCN. Additionally, find that the improvements inspired filters provide better results than more traditional square band-limited filters. We have additionally explored the spectral dependence of homophily, and found that not only is homophily spectrally dependent, but that this can explain the efficacy of our band pass filters.
Limitations While we are able use iterative, matrix-free, cigensolvers, our method still more expensive than spatial convolutions and limits the applicability of our method large graphs. Additionally, propa- gating gradients through approximate eigensolvers is quite challenging, making it difficult to optimize the
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parameters of the diffusion kernel during training time. Finally, because our method is spectral, we are only able to use this method in transductive settings. We believe that quantum convolution in the spatial domain will open up avenues to address these issues, and are excited to explore this approach in followup work.
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A
Appendix
A.1
GAT Hyperparameter plots
0.9
0.8
Mean Test Accuracy
0.6
0.5
0.3
MultiscaleQOC
0.2
GAT+QDC
Corell
Texas
Wisconsin
Chameleon
Squirrel
Actor
Cora
Citeseer
Pubmed
Figure 6: Violin plots of experiments GAT GAT (green), GAT (ycllow), and GAT+QDC (blue), where these plots are generated by aggregaling over all experiments associated with each model. We observe that both GAT+QDC and GAT+MultSSealeDCC generally have high density of near-optimal configurations.
Cornell
Texas
Wisconsin
Chameleon
Squirel
Actor
Cora
Citeseer
Pubmed
0.6
0.2
0.4
Figure 7: Scatter plots mean test accuracy plotted against hyperparameters p. and In(or in the first and second rows respectively for GAT+QDC (blue) and CAT+MMltiScaleQDCC (green). We observe that cach of QDC and MultiScaleQDC are robust with respect. to deviations each of the hyperparameters.
A.2
Runtime Costs
A.3
Hyperparameter 3d Plots
A.4 Model Details
We performed 250 steps of hyper-parameter optimization for cach of the models presented in Table L All training runs were run with maximum of 1000 steps for each split, with early stopping turned after 50 steps. In the interest of reproducibility, we outline the parameters and ranges that we're optimized for each model below.
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Table 2: Runtime costs for the QDC and MultiScaleQDCC compared to a variety of baselines. We find that QDC requires a nontrivial increase in compute costs, but that this expense often carries with it a significant accuracy gain.
GCN
GCN+GDC GCN+SDRF GCN + QDC GCN + MultiScaleeDDC
Cornell
5.25
5.28
5.18
19.41
40.87
Texas
5.41
5.27
6.55
8.46
4.52
Wisconsin
8.17
8.98
12.02
31.88
22.56
Chameleon
7.42
39.45
52.92
20.22
257.33
Squirrel
36.65
32.42
160.73
77.12
115.47
Actor
12.98
10.80
332.40
309.55
299.91
Cora
20.08
83.71
42.13
79.52
166.70
Citeseer
36.90
59.25
51.02
59.25
130.39
Pubmed
81.92
84.62
4230.08
2366.10
2377.68
Table 3: parameerr ranges that we optimized over for our GCN.
Parameters
Distribution
Values
Number of Layers
Categorical
1, 2]
Hidden Dim Size
Categorical
[2, 4, 8, 16, 32, 64, 128
Dropout percentage
Uniform
[0, 0.99]
Learning Rate
Loguniform
[1e-4, 1e-1]
Weight Decay
uniform
[0.0, 0.9
Table Hyper-parameter ranges that we optimized over for our GCN+GDC.
Parameters
Distribution
Values
Number of Layers
Categorical
1, 2]
Hidden Dim Size
Categorical
12, 4, 8, 16, 32, 64, 128)
Dropout percentage
Uniform
[0, 0.99]
GDC-a
uniform
[0.001, 0.5]
GDC-
uniform
[1e-7, le-1
Learning Rate
Loguniform
[1e-4, 1e-1
Weight Decay
uniform
[0.0, 0.9
Table 5: Hyper-parameter ranges that optimized over for our GCN+QDC.
Parameters
Distribution
Values
Number of Layers
Categorical
[1, 2]
Hidden Dim Size
Categorical
[2, 4, 8, 16, 32, 64, 128)
Dropout percentage
Uniform
[0, 0.99)
QDC-u
uniform
[-1,
QDC-a
uniform
[0.1, 1.0
QDC-c
loguniform
11--7, 1e-1
Learning Rate
Loguniform
[1e-4, le-1]
Weight Decay
uniform
00.0, 0.9
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Comell
Texas
Wisconsin
0.55
335
639
338
0.0
030
17
05
1.00 22.02 21500. ao0
1.0a 22500
0,1
0.1
0z
630
111 Squirrel
Chameleon
Actor
050
3483
6275
0.15
0220
136
0373
655
220
6200
050
1123
0199
1.00 52.39 2500
Cora
Citeseer
Pubmed
30
16
25
53
-10 075
34
027 0.0
100
325 250
L33
6m
a25
850
677
L00
Figure 8: Triangulation surface plots of mean test accuracy plotted against hyperparameters μ and In(a) for GCN+QDC (orange) and GCN+MultiScaeeDDCC (blue). We observe that each of QDC and MultiScaleQDCC are robust with respect to deviations in each of the hyperparameters because the high performance regions tend to be quite large.
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Cornell
Texas
Wisconsin
635
0.50
0.9
615
010
650
005
03
0.
2332 200 am
22 000 am
050 am ce
mm Chameleon
Squirrel
Actor
0.00
340
0.52
136
225
522
22.0 20.00
1.4 50
50
Les Pubmed
Cora
Citeseer
03
04
18
93399222 2000
2950 000 025 c20 072
10035 0000 azt 020 am se0
622 020 c35 120
L33
Figure 9: Triangulation surface plots mean test accuracy plotted against hyperparameters μ and In(a) for GAT+QDC (orange) and GATT-MuliSSalleDDCC (blue). We observe that each of QDC and MultiScaleQDC are robust with respect to deviations in each of the hyperparameters because the high performance regions tend to be quite large.
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Table 6: Hyper-parameter ranges that we optimized over for our MulliScaleQDC.
Parameters
Distribution
Values
GCN Number of Layers
Categorical
[1, 2]
GCN Hidden Dim Size
Categorical
[2, 4, 8, 16, 32, 64, 128)
GCN Dropout percentage
Uniform
[0, 0.99]
QDC Number Layers
Categorical
[1, 2]
QDC Hidden Dim Size
Calegorical
[2, 4, 8, 16, 32, 64, 128)
QDC Dropout percentage
Uniform
[0. 0.99]
QDC-μ
Uniform
[-1,
QDC-a
Uniform
[0.1, 1.0)
QDC-E
loguniform
[1e-7, 1e-1]
Combinator
Categorical
[ concat, add]
Learning Rate
Loguniform
[1c-4, 1c-1]
Weight Decay
Uniform
[0.0, 0.9)
Table 7: paree-earreeter ranges that optimized over for our GAT.
Parameters
Distribution
Values
Number of Layers
Categorical
[1, 2]
Hidden Dim Size
Categorical
[2, 4, 8, 16, 32, 64, 128)
Number of Heads
Categorical
[1, 2, 3, 4, 5
Dropout percentage
Uniform
[0, 0.99
Learning Rate
Loguniform
[le-4, 1e-1]
Weight Decay
Uniform
0.0, 0.9
Table Hyper-parameter ranges that we optimized over for our GAT+GDC.
Parameters
Distribution
Values
Number of Layers
Categorical
1, 2
Hidden Dim Size
Categorical
12, 4, 8, 16, 32, 64, 128]
Number of Heads
Categorical
[1, 2, 3, 5]
Dropout percentage
Uniform
[0. 0.99]
GDC-a
Uniform
[0.001, 0.5
GDC-
Uniform
[1e-7, le-1
Learning Rate
Loguniform
[1e-4, 1e-1
Weight Decay
Uniform
0.00, 0.9
Table Alyper-paramette ranges that we optimized over for our GAT+QDC.
Parameters
Distribution
Values
Number of Layers
Categorical
1, 2
Hidden Dim Size
Catogorical
[2, 4, 8, 16, 32, 64, 128)
Number of Heads
Categorical
[1, 2, 3, 4, 5]
Dropout percentage
Uniform
[0, 0.99
QDC-u
Uniform
[-1,
QDC-a
Uniform
[0.1, 1.0]
QDC-e
loguniform
[1e-7, 1e-1
Learning Rate
Loguniform
[le-4, 1e-1
Weight Decay
Uniform
00.0, 0.9
19

---

Published in Transactions on Machine Learning Research (02/2024)
Table 10: Hyper-parameter ranges that we optimized over for our Multiscale GAT+QDCG
Paramcters
Distribution
Values
GAT Number of Layers
Categorical
[1 2]
GAT Hidden Dim Size
Categorical
[2, 4, 8, 16, 32, 64, 128
GAT Number of Heads
Categorical
[1, 2, 3, 4, 5]
GAT Dropout percentage
Uniform
[0, 0.99]
QDC Number of Layers
Categorical
[1, 2]
QDC Hidden Dim Size
Categorical
[2, 4, 8, 16, 32, 64, 128)
QDC Number Heads
Categorical
[1. 2, 3, 4, 5]
QDC Dropout percentage
Uniform
[0, 0.99]
QDC-H
Uniform
[-1, 1]
QDC-a
Uniform
[0.1, 1.0)
QDC-
loguniform
[1e-7, 1e-1]
Combinator
Categorical
concat, add]
Learning Rate
Loguniform
[le-4, le-1]
Weight Decay
Uniform
[0.0, 0.9
20


Reviews:
[{"review_id": "q1ObpTuI8x", "reviewer": "Reviewer_MqtD", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The exposition motivating both the reason for a diffusion perspective on the design of GCNs and the utility for the Schrodinger equation in this context is well-founded. The document is overall very readable in this sense, other than for the related works section (which I discuss in more detail below) and a few things in Section 3.\n- The experimental results clearly highlight that the preconditioned GCN with the Schrodinger method is performant. \n\n\nExpository Weaknesses:\n- Numerous acronyms introduced without any clarification in the related works. e.g., the labelling +FA is undefined in relation to the paper of Alon and Yahav 2020 that is cited. Same goes for \"SDRL\", \"GRAND\" method, \"BLEND\", \"PDE-GCN\", etc.\n- The name ChebNet is introduced in the paragraph below equation (5) without any definition.\n- Caption of Figure 2 does not seemingly support the evidence of the figure. While it is true that the the top cluster on the bar-bell still has some mixing at time t= 1000, the bottom cluster is entirely uniform. I see what the authors intend to stress with information bottleneck of the head diffusion, but they should just clarify the text in the caption to be a bit more nuanced.\n- Typo in first sentence of Section 4: Laplaician --> Laplacian.\n\nTechnical Weaknesses:\n- It's important to point out that this paper does not actually model any sort of quantum dynamics. Indeed, this is a bit of a distraction from, say, just specifying that you write down the Schrodinger equation (just another PDE) and consider the infinite time limiting dynamics of the solution to the Schrodinger equation for a free particle. Really, the rest of this does nothing quantum. It's more distracting than elucidating to label it as a quantum diffusion -- there is no entanglement, seemingly no \n- The authors point out (which is useful to state!) that a priori having a dense kernel is a blessing (because it helps overcome message passing bottlenecks) and a curse (because it makes all diagonalizations etc very costly). This isn't really a weakness in the paper, it's just being realistic, which is a plus in my book.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The aims of this paper of this paper are to devise a new diffusion kernel to be used in graph neural networks. Graph diffusions were introduced as a means of overcoming expressivity issues in the spatial convolutions used in graph convolutional networks.\n\nThe main idea stems from considering a means of constructing a band pass filter on a graph. The authors motivate that common parametric realizations of filters, e.g. such as the Chebyshev expansion given in equation 5, can be seen as the generators governing some pde, connecting these methods to diffusion equations. This draws direct analogy to the heat equation, which, unfortunately, has an asymptotic time evolution which equilibrates unfavorably with the respect to the graph. They seek to highlight (and what they should express more explicitly) is that the choice of diffusion can have broad effect on the information propagation on the graph. There are many such diffusions that could be realized on the graph (and they mention some, but mainly focus their proposal in juxtaposition with the heat kernel which seems especially ill-suited for the task). \n\nIn lieu of the heat diffusion, they consider dynamics under the Schrodinger equation. They state this equation gives more favorable message-passing dynamics. Time evolution under the schrodinger equation is governed by a unitary operator which is the matrix exponential of the Hamiltonian of the system. They study the case for the hamiltonian of a free particle, which has eigenstates which are linear superpositions of basis vectors $\\ket{\\phi_i}$ weights by coefficients $c_i$ and a phase factor $e^{iE_it}$ (which, in computing likelihoods via the Born rule, should vanish only for terms corresponding to $i=j$ in the inner products $\\bra{\\phi_j}\\ket{\\phi_i}$). \n\nSimulating the time evolution of quantum systems is an open challenge in the field of quantum physics (when scaling with respect to the number of particles in the many-body system and with respect to the extent of the time evolution, as in practice, most time-evolutions of hamiltonians need to be Trotterized, introducing finite error which compounds wrt to the time evolution T), so it is a bit interesting to see folks suggest it as a means of getting at graph neural networks. \n\nThe reason they can get away with this and the quantum community cannot is because they circumvent the need to do the time-evolution by considering the infinite time dynamics of the system, which is made possible by the fact that they are considering the hamiltonian of a free particle.\n\nHowever, this means that there are non-zero transition probabilities for all vertices and the transition matrix is dense. Diagonalizing this transition matrix is a challenge (as someone who is less familiar with graph conv nets, how large is N in practice?), but they can get away with iterative methods that give them access to a subset of the eigenvalues and vectors needed for the \"frequencies\" (energy states) associated with their filter.\n\nThey benchmark their method using some standard datasets and compare it to GCNs. I am not an expert in GCNs and do not know if there are other methods they should reasonably be comparing to other than the ones they've listed. What they do compare to shows that their method performs quite well.\n\n\nI detail what I see to be the strengths and weaknesses in the other sections, but, if they authors address my remarks made there, I think the paper should be **accepted.**", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Address the typos described in expository weaknesses\n- Consider removing the stress on this being a \"quantum\" method -- indeed there is very little quantum about it (many-body entanglement, meaningful potentials, etc) other than using the PDE associated with quantum mechanics and deriving things under unitary evolution (this problem is entirely classical, for all intents and purposes). \n- State more clearly that what Hamiltonian you are actually using -- you discuss the free particle but because of my own failings or because of lack of specification, I was a bit lost when you jumped to experiments (but assumed you stuck with the free particle b/c you needed to get to equation 10)\n- State clearly how many basis vectors you are assuming to use in equation 8 :)", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "XWYzqHSVqc", "reviewer": "Reviewer_bvxn", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Section 3 about the background and the classical definition of convolutions for graphs is well-written and should probably be shortened. But Section 4 is completely unclear to me and the authors need to provide more details about their methodology. The authors introduce concepts from physics without explaining them, like \"ket\". I think that these concepts are not necessary. As graphs are finite, we are always dealing with finite dimensional states or operators, hence, it should be able to rephrase these concepts in terms of matrices and vectors.\nAs I understand it, the notion of Quantum Diffusion Convolution Kernels is a small modification from previous work. It looks like the \"ket\" $\\phi$ in equations (8), (9) and (10) are the eigenvectors of $\\mathcal{H}=-\\Delta$ the combinatorial Laplacians. However, I am unsure how to interpret the $x_i$ in the matrix $\\mathcal{Q}(x_i,x_j)$.\nI do not understand at all how this QDC can be encoded as a message-passing algorithm. This is perhaps explained in the paragraph Sparsification on page 7, but this is very unclear.\nIn the experiments, the authors claim that QDC has multiple hyperparameters, but they are not defined. To me, there are only 2 parameters $\\mu$ and $\\sigma$ appearing in equation (10) which are tuned thanks to SGD like the parameters $\\theta_k$ in equation (5).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a new notion of convolution on graphs. The classical definition of convolution on graphs is done thanks to the (combinatorial) Laplacian, where the eigenvectors of the Laplacian define the Fourier modes. As a result, we can define filters on graphs as polynomials of the Laplacian. Here, the authors propose to replace the Laplacian with a Quantum Diffusion Kernel.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "As explained above, the paper requires major changes before any possible publication. \nOther minor changes: in the definitions of matrices on page 3, use $\\times$ instead of $x$.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "8Lfn2fIJUk", "reviewer": "Reviewer_4R5C", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n- The research showcases that the proposed technique surpasses several other graph diffusion convolution methods, especially in tasks with heterophily.\n- The insights on the spectral independence of homophily are interesting, albeit not exhaustive.\n\n### Weaknesses\n- **Conceptual Complexity**: The authors approach their method through the quantum dynamics prism for message passing kernels. While this viewpoint offers novelty, it might add layers of intricacy that could pose challenges in both understanding and implementation, especially for those lacking a quantum mechanics foundation. Simplifying the presentation of this quantum diffusion, which essentially utilizes Gaussian filters instead of the polynomial filters as seen in Gasteiger et al. (2019), could enhance clarity.\n    \n- **Presentation and Writing**: The introductory query, \"Can we improve message passing in graph neural networks with a quantum mechanical message passing kernel?\", lacks persuasive force and doesn't compellingly justify the need for quantum diffusion. Additionally, the introduction's explication of quantum diffusion and its nuances with the heat equation appears somewhat opaque. This may pose comprehension challenges for readers not well-versed with the topic. Moreover, the experiments section would benefit from a summarizing segment highlighting primary outcomes, limitations, and prospective avenues.\n    \n- **Baseline Comparisons**: The empirical evaluations indicate that the band filters from the proposed quantum diffusion can notably improve GNNs, particularly in contexts of heterophily or low homophily. Given that this theme has been the focus of recent works, including [H2GCN](https://arxiv.org/abs/2006.11468) and [Bodnar et al. 2022](https://proceedings.neurips.cc/paper_files/paper/2022/file/75c45fca2aa416ada062b26cc4fb7641-Paper-Conference.pdf), a broader comparative analysis would enhance the discernment of this method's practical value.\n    \n- **Hyperparameters Concern**: In comparison to the heat kernel, the introduced quantum diffusion kernel necessitates two extra hyperparameters, $\\mu$ and $\\sigma$. It is clear from Figure 4 that the model's efficacy is considerably influenced by these hyperparameters, which implies the need for meticulous hyperparameter optimization.\n    \n- **Scalability Issues**: Calculating the quantum diffusion kernel, as proposed, seems computationally demanding for large graphs.\n    \n- **Analytical Gaps**: A more in-depth discourse or exploration into why BPDC's efficiency is predominantly less than QDC's would be valuable. Further, understanding the implications of the spectral dependence of homophily on model generalizability or potential biases would be insightful.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Gasteiger et al. 2019 presented an innovative method named \"graph diffusion convolution\" that primarily restructures a graph through generalized graph diffusion combined with sparsification and subsequent message passing on the altered graph (rewired graph). Building on this foundational approach, the current research introduces a new \"quantum diffusion\" technique for the graph rewiring process. This adaptation employs a Quantum diffusion kernel for graph diffusion, unlike Gasteiger et al.'s heat kernel-based method. This modification results in the formation of band-pass filters (Gaussian filters), as opposed to the low-pass filters (polynomial filters) created by the heat kernel.  Importantly, the study reveals that this quantum diffusion approach can synergize with existing low-pass filters, like the Laplacian, to conceive a multiscale method. Through empirical evaluation, the paper underscores the superiority of their proposed method compared to other graph diffusion convolution techniques.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- **Writing Refinements**:\n    \n    - The introduction could be supported with a more compelling rationale.\n    - A more comprehensive background on quantum diffusion and the heat equation is warranted in the introduction.\n    - Positioning the method within the framework of Gasteiger et al. (2019) might not only simplify the presentation but also clearly delineate this research's contributions.\n    - A concluding segment in the experiments section, detailing main takeaways and limitations would be advantageous.\n- **Extended Comparisons**: Incorporating contemporary techniques that address heterophily in graphs would certainly underline this research's real-world relevance.\n    \n- **Addressing Issues**: The authors should explore potential solutions to the challenges posed by the additional hyperparameters and scalability concerns.\n    \n- **Additional Analysis**: A deeper investigation into the disparities between BPDC's and QDC's efficiencies is recommended. Understanding the implications of spectral dependence in homophily, especially in terms of model generalizability or potential biases, would also add depth to the analysis.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: The paper studies GNNs, in particular graph convolutions based on graph diffusions, as studied by Gasteiger et al. 2019 and other works. In order overcome the over-smoothing limitations of heat diffusions that were considered in most prior work, they consider a different physical model inspired by quantum diffusion, leading to new diffusion-based graph convolutions that are more localized/band-pass.

The reviewers appreciated the method, its empirical effectiveness, and its motivation, though they generally found the quantum jargon to be a bit too extensive and in some cases unnecessary. After discussion, the author addressed this in large part, making the manuscript significantly stronger. As a result, the paper is now in much better shape, and I recommend acceptance.

In his final decision, reviewer 4R5C provided the following additional requests, which I ask the author to address in a final minor revision:
> * Clarity and Rationale in Writing: The authors have made attempts to enhance the manuscript's clarity in this revision. Nonetheless, further improvements are necessary, especially in the introduction. The question posed, "Can we improve graph neural networks by considering a different physical model?" still seems insufficiently justified. The core of this study appears to be the application of band-pass filters (Gaussian filters) instead of the low pass filters (polynomial filters) in the graph rewiring process. This distinction needs to be more prominently articulated, rather than motivating the propose method from a physical perspective.
> * Quantum Mechanics Integration: Both Reviewer MqtD and bvxn have noted, and I agree, that the introduction of quantum mechanics into the methodology seems to add unnecessary complexity. The approach largely builds upon Gasteiger et al.'s 2019 framework, and this connection seems to be sufficient to present the proposed method.
> * Experimental Section Structure: The experimental section currently jumps directly into the setup without adequately outlining the objectives or summarizing key findings.
> * Hyperparameter Optimization: Figure 4 indicates a significant impact of hyperparameters on the model's performance, suggesting a need for sophisticated optimization. The method also appears to use more exhaustive search grids (as per Tables 6-10) compared to baseline methods, potentially leading to unfair comparison.
> * Scalability and Computational Complexity: The manuscript should address concerns regarding scalability and computational complexity. The method's application to large real-world graphs seems computationally intractable, and the additional time required for hyperparameter tuning exacerbates this issue.

Also, please respond to reviewer bvxn.
Decision: Accept (poster)