Geometry-Induced Diffusion on Graphs: A Learnable Weighted Laplacian for Spectral GNNs

Mia Zosso; Ali Hariri; Victor Kawasaki-Borruat; Pierre-Gabriel Berlureau; Pierre Vandergheynst

cs.LGFeb 20, 2026

Geometry-Induced Diffusion on Graphs: A Learnable Weighted Laplacian for Spectral GNNs

Authors: Mia Zosso, Ali Hariri, Victor Kawasaki-Borruat, Pierre-Gabriel Berlureau, Pierre Vandergheynst

Organizations: École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland · École Normale Supérieure – PSL, Paris, France

Abstract

Long-range graph tasks are challenging for Graph Neural Networks (GNNs): global mechanisms such as attention or rewiring schemes can be computationally expensive, while deep local propagation is prone to vanishing gradients, oversmoothing, and oversquashing. The introduced mu-ChebNet architecture is a simple spectral GNN that learns a node-wise weight function mu before applying ChebNet-style filters. The learned weighting mu induces a modified graph Laplacian which effectively changes the propagation geometry without altering the graph topology. This task-dependent geometry promotes preferred routes for information propagation, thereby helping long-range signals avoid highly contractive bottlenecks, and obviating the need for repeated layer stacking. In practice, we replace the fixed graph Laplacian L by a learned operator L_mu, keeping the proposed mu-ChebNet architecture lightweight while making propagation task-adaptive. Furthermore, we provide a spectral analysis demonstrating how mu modulates propagation dynamics, and empirically observe improved performance on both synthetic long-range reasoning tasks and real-world graph benchmarks. The learned weight function is not only interpretable, but also offers a lightweight alternative to attention and rewiring for adaptive graph propagation.

Figures & tables

Figure 2: Minimal two-community graph.

Table 3

Model	Diameter $\downarrow$	SSSP $\downarrow$	Eccentricity $\downarrow$
GCN	$0.7424\pm 0.0466$	$0.9499\pm 0.0001$	$0.8468\pm 0.0028$
GAT	$0.8221\pm 0.0752$	$0.6951\pm 0.1499$	$0.7909\pm 0.0222$
GraphSAGE	$0.8645\pm 0.0401$	$0.2863\pm 0.1843$	$0.7863\pm 0.0207$
GIN	$0.6131\pm 0.0990$	$-0.5408\pm 0.4193$	$0.9504\pm 0.0007$
GCNII	$0.5287\pm 0.0570$	$-1.1329\pm 0.0135$	$0.7640\pm 0.0355$
DGC	$0.6028\pm 0.0050$	$0.1483\pm 0.0231$	$0.8261\pm 0.0032$

Table 3: Mean test set

\log_{10}(\mathrm{MSE})

and standard deviation averaged over 4 random weight initializations for each configuration on the Graph Property Prediction dataset. The lower the better.

Method	London	Paris	Shanghai	Los Angeles
ChebNet	39.10 $\pm$ 1.79	34.34 $\pm$ 0.31	40.00 $\pm$ 0.39	41.31 $\pm$ 0.48
Stable-ChebNet	40.11 $\pm$ 0.20	38.11 $\pm$ 0.25	41.45 $\pm$ 0.27	42.17 $\pm$ 0.16
$\mu$ -ChebNet (ours)	41.88 $\pm$ 0.11	35.14 $\pm$ 0.12	41.98 $\pm$ 0.96	43.33 $\pm$ 0.33
+ Stability	41.44 $\pm$ 0.07	37.49 $\pm$ 0.26	42.63 $\pm$ 0.70	43.10 $\pm$ 0.60

Table 4: Accuracy (%) on City-Networks dataset. We use this dataset to fairly compare ChebNet architectures by setting them to the same receptive field

K=10

and isolating the effect of the weighted Laplacian

L_{\mu}

Model	ROC-AUC $\uparrow$
MLP	$72.04\pm 0.48$
GCN	$72.51\pm 0.35$
ChebNet	$77.55\pm 0.43$
Stable-ChebNet	$79.55\pm 0.34$
SGC	$70.31\pm 0.23$
GCN-NSAMPLER	$73.51\pm 1.31$

Table 5: Results on ogbn-proteins .

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Ring graph configuration. (Left) The query node is in purple and the answer node in yellow. Nodes on the rightward path contain noisy features. (Right) The resulting learned weight function has broken the symmetry of the configuration, routing information on the most informative path.

Panel of Figure 4 — Figure 4: Spectral analysis on the simple community graph.

Figure 5: 2-community Graph

Figure 6: Singular values of the propagated feature matrix restricted to the target cluster. While standard diffusion rapidly collapses all but one direction, the learned weighted diffusion geometry opens a transient regime in which multiple independent directions survive long enough to traverse the bottleneck.

Hyper-parameter	Values in grid	Diam	SSSP	Ecc
Hidden dimension $d$	20, 30, 50	50	30	30
Number of layers	1, 2, 3, 5, 10, 20	20	5	5
Polynomial order $K$	3, 5, 10	4	10	10
Step size $\epsilon$	0.01, 0.10, 0.20, 0.30	0.40	0.30	0.30
Dissipative force $\gamma$	0, 0.01, 0.50, 1	0.01	0.00	0.00
Activation function	tanh, relu	relu	relu	relu

Appendix

Table 6: Hyper-parameter grid and best settings for our model on three synthetic graph-property benchmarks.

Hyper-parameter	Sweep
Hidden dim $d$	256, 512, 1024
Polynomial order $K$	5, 10, 15
Num of layers	3, 5, 7
MLP layers	1, 2, 3
Step size $\epsilon$	[0.1, 1.0]
Dissipative force $\gamma$	0.01, 0.05, 0.1

Appendix

Table 7: Hyper-parameter sweep ranges for our model on ogbn-proteins .

Figure 7: Illustration of a Barbell graph composed of two densely connected node clusters

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University of Southern California.

Spectral Graph Convolution Graph Signal Processing Long-Range Graph Learning

Open paper AskPDF arXiv

Despite their theoretical advantages, spectral methods based on the graph Fourier transform (GFT) are seldom used in graph neural networks (GNNs) due to the cost of computing the eigenbasis and the lack of vertex-domain locality in the resulting representations. As a result, most GNNs rely on local approximations such as polynomial Laplacian filters or message passing, which limit their ability to model long-range dependencies. In this paper, we introduce an exact factorization of the GFT into operators acting on subgraphs, which are then combined via a sequence of Cauchy matrices. Building on this factorization, we propose a new class of spectral GNNs, termed L2G-Net (Local to Global Net). Unlike existing spectral methods, which are either fully global (when using the GFT) or local (when using polynomial filters), L2G-Net operates by processing the spectral representations of subgraphs and then combining them via structured matrices. Our algorithm avoids full eigendecompositions, exploiting graph topology to construct the factorization with quadratic complexity in the number of nodes, scaled by the maximum cut size between subgraphs. Experiments stressing long-range dependencies on large graphs show that L2G-Net scales to regimes out of reach for the standard GFT, and is competitive with state-of-the-art methods with orders of magnitude fewer learnable parameters.

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