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Spectral Neural Graph Sparsification

Angelica Liguori, Ettore Ritacco, Pietro Sabatino, Annalisa Socievole

TL;DR

Spectral Preservation Network (SpecNet) tackles the rigidity and scalability of graph neural networks by jointly evolving graph topology and node features through the Joint Graph Evolution layers and guiding sparsification with the Spectral Concordance loss. The JGE layer reparameterizes the graph Laplacian via bilinear transformations, while a non-expansive normalization ensures stable, scalable learning, and a lighter LJGE variant offers efficiency for undirected graphs. The Spectral Concordance loss enforces alignment of both the Laplacian spectrum and the feature-space Gram matrices, combined with a sparsity penalty to yield a compact yet spectrally faithful subgraph. Empirical results on five real-world datasets show that SpecNet preserves key structural and spectral properties (e.g., MASS, modularity, epidemic threshold) across varying sparsity levels and outperforms several baselines, supporting its potential for scalable, spectrum-driven graph learning and downstream tasks.

Abstract

Graphs are central to modeling complex systems in domains such as social networks, molecular chemistry, and neuroscience. While Graph Neural Networks, particularly Graph Convolutional Networks, have become standard tools for graph learning, they remain constrained by reliance on fixed structures and susceptibility to over-smoothing. We propose the Spectral Preservation Network, a new framework for graph representation learning that generates reduced graphs serving as faithful proxies of the original, enabling downstream tasks such as community detection, influence propagation, and information diffusion at a reduced computational cost. The Spectral Preservation Network introduces two key components: the Joint Graph Evolution layer and the Spectral Concordance loss. The former jointly transforms both the graph topology and the node feature matrix, allowing the structure and attributes to evolve adaptively across layers and overcoming the rigidity of static neighborhood aggregation. The latter regularizes these transformations by enforcing consistency in both the spectral properties of the graph and the feature vectors of the nodes. We evaluate the effectiveness of Spectral Preservation Network on node-level sparsification by analyzing well-established metrics and benchmarking against state-of-the-art methods. The experimental results demonstrate the superior performance and clear advantages of our approach.

Spectral Neural Graph Sparsification

TL;DR

Spectral Preservation Network (SpecNet) tackles the rigidity and scalability of graph neural networks by jointly evolving graph topology and node features through the Joint Graph Evolution layers and guiding sparsification with the Spectral Concordance loss. The JGE layer reparameterizes the graph Laplacian via bilinear transformations, while a non-expansive normalization ensures stable, scalable learning, and a lighter LJGE variant offers efficiency for undirected graphs. The Spectral Concordance loss enforces alignment of both the Laplacian spectrum and the feature-space Gram matrices, combined with a sparsity penalty to yield a compact yet spectrally faithful subgraph. Empirical results on five real-world datasets show that SpecNet preserves key structural and spectral properties (e.g., MASS, modularity, epidemic threshold) across varying sparsity levels and outperforms several baselines, supporting its potential for scalable, spectrum-driven graph learning and downstream tasks.

Abstract

Graphs are central to modeling complex systems in domains such as social networks, molecular chemistry, and neuroscience. While Graph Neural Networks, particularly Graph Convolutional Networks, have become standard tools for graph learning, they remain constrained by reliance on fixed structures and susceptibility to over-smoothing. We propose the Spectral Preservation Network, a new framework for graph representation learning that generates reduced graphs serving as faithful proxies of the original, enabling downstream tasks such as community detection, influence propagation, and information diffusion at a reduced computational cost. The Spectral Preservation Network introduces two key components: the Joint Graph Evolution layer and the Spectral Concordance loss. The former jointly transforms both the graph topology and the node feature matrix, allowing the structure and attributes to evolve adaptively across layers and overcoming the rigidity of static neighborhood aggregation. The latter regularizes these transformations by enforcing consistency in both the spectral properties of the graph and the feature vectors of the nodes. We evaluate the effectiveness of Spectral Preservation Network on node-level sparsification by analyzing well-established metrics and benchmarking against state-of-the-art methods. The experimental results demonstrate the superior performance and clear advantages of our approach.

Paper Structure

This paper contains 29 sections, 4 theorems, 50 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Spectral Preservation Network
  3. Joint Graph Evolution Layer
  4. Node Sparsification
  5. Spectral Concordance
  6. Experimental Test-bed and Results
  7. Related Work
  8. Graph Embeddings and Joint Structure–Feature Learning
  9. Graph Sparsification
  10. Conclusion and Future Work
  11. Stability of the Joint Graph Evolution Layer
  12. Properties of the Laplacian Matrix
  13. Laplacian Matrix for Directed Graphs
  14. On the Singularity of $L + \alpha I$
  15. Intuition and Analysis of the Spectral Preservation Network
  16. ...and 14 more sections

Key Result

Theorem 1

Let $Q\in\mathbb{R}^{r\times s}$ be any real matrix. Define its row– and column–absolute sums by Form the diagonal scaling matrices $U\in\mathbb{R}^{r\times r}$ and $V\in\mathbb{R}^{s\times s}$ via Then the normalized matrix satisfies i.e. $\widehat{Q}$ is non‑expansive in the Euclidean norm, indicating by $\| \widehat{Q} \|_{\mathrm{op}}$ the induced spectral operator norm, i.e., the square r

Figures (1)

  • Figure 1: Node-level sparsification Pipeline. The operator $\oplus$ denotes the concatenation of the vectorized (flattened) forms of $Q_T$ and $H_T$.

Theorems & Definitions (8)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof