Centrality Graph Shift Operators for Graph Neural Networks
Yassine Abbahaddou, Fragkiskos D. Malliaros, Johannes F. Lutzeyer,, Michalis Vazirgiannis
TL;DR
This paper introduces Centrality Graph Shift Operators (CGSOs) that normalize adjacency matrices using global centrality metrics, enhancing spectral analysis and improving graph neural network performance on real-world datasets.
Contribution
The paper proposes a novel class of Graph Shift Operators based on global centrality metrics, extending traditional local normalization methods in graph neural networks.
Findings
Spectral properties of CGSOs are characterized.
Spectral clustering with CGSOs performs well on synthetic and real datasets.
Graph neural networks using CGSOs outperform baselines on benchmarks.
Abstract
Graph Shift Operators (GSOs), such as the adjacency and graph Laplacian matrices, play a fundamental role in graph theory and graph representation learning. Traditional GSOs are typically constructed by normalizing the adjacency matrix by the degree matrix, a local centrality metric. In this work, we instead propose and study Centrality GSOs (CGSOs), which normalize adjacency matrices by global centrality metrics such as the PageRank, -core or count of fixed length walks. We study spectral properties of the CGSOs, allowing us to get an understanding of their action on graph signals. We confirm this understanding by defining and running the spectral clustering algorithm based on different CGSOs on several synthetic and real-world datasets. We furthermore outline how our CGSO can act as the message passing operator in any Graph Neural Network and in particular demonstrate strong…
Peer Reviews
Decision·ICLR 2026 Conference Withdrawn Submission
1. The idea and construction of CGSOs are simple but broadly applicable. 2. The paper provides theoretical properties of Markov Averaging Operators to explain when/why CGSOs should separate clusters. 3. The empirical results (many shown in Appendix) are comprehensive and presented with clarity.
1. The proof of Proposition 3.2 seems to assume no self-loops in the graph. In Section 3.2, the new parametrized CGSO adds self-loops to the adjacency matrix, so a brief discussion on how the theories/propositions can be extended would improve completeness. Also, it would be helpful to clearly state the assumptions and constants in propositions. 2. The hyperparameters in Appendix A.5 are from a grid search on the classical GCN, which might under-tune others. Although this is designed for compari
The idea of incorporating global node importance into the shift operator is simple, well-motivated The paper provides strong theoretical support for the CGSO design, including propositions on spectral characteristics and signal propagation. Experiments on node classification show that CGSO offers consistent performance gains.
Although improvements are consistent, the gains are sometimes marginal (especially on Pubmed) While the authors propose to use node centrality as a way to encode global importance, there is no principled theoretical justification for why these particular centrality metrics (e.g., betweenness, closeness, eigenvector) are most suitable for constructing GSOs. While CGSO is integrated into models like GCN, this integration is surface-level
1. It is interesting to study the class of generalized graph shift operators, and combine them with graph neural networks. 2. The paper is mostly well-written.
1. The motivation of this work remains unclear. Are the class of CGSOs intended to inject more global information to graph representations? If so, why not encoding these centrality metrics as node features? Or perhaps the proposed CGSOs enjoy provable guarantees on some problems (e.g. spectral clustering)? However, the established theoretical properties do not explicitly answer why or in what settings these CGSOs are preferred. 2. Limited novelty and effectiveness when incorporating CGSO in GNN
1. Uses several graph centralities to substitute degree in GSO/PGSO and induces global information into the message passing process. 2. Obtains performance improvements on node clustering and classification tasks against classic baselines. 3. Provides the performance landscape over the key trainable parameters (e2 and e3) for spectral clustering.
1. It is an incremental extension of PGSO. 2. The PageRank, k-core, and L-walk based CGSO do not show general advantage over degree based, which is basically PGSO, on node classification tasks (table2 and 3). I can accept that the paper does not compare the performance of CGSO with SOTA models like FAGCN, ChebNet, TEDGCN, GPRGNN, LINKX, ACM-GCN, and GloGNN, but CGSO should significant improvement over PGSO (CGSO-D in this paper), otherwise, what's point of spending extra computation on the othe
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Taxonomy
TopicsNeural Networks and Applications · Advanced Graph Neural Networks · Graph Theory and Algorithms
MethodsSoftmax · Attention Is All You Need · Spectral Clustering · Graph Neural Network