Transferability of Graph Neural Networks using Graphon and Sampling Theories

Graph neural networks (GNNs) have become powerful tools for processing graph-based information in various domains. A desirable property of GNNs is transferability, where a trained network can swap in information from a different graph without retraining and retain its accuracy. A recent method of capturing transferability of GNNs is through the use of graphons, which are symmetric, measurable functions representing the limit of large dense graphs. In this work, we contribute to the application of graphons to GNNs by presenting an explicit two-layer graphon neural network (WNN) architecture. We prove its ability to approximate bandlimited signals within a specified error tolerance using a minimal number of network weights. We then leverage this result, to establish the transferability of an explicit two-layer GNN over all sufficiently large graphs in a sequence converging to a graphon. Our work addresses transferability between both deterministic weighted graphs and simple random graphs and overcomes issues related to the curse of dimensionality that arise in other GNN results. The proposed WNN and GNN architectures offer practical solutions for handling graph data of varying sizes while maintaining performance guarantees without extensive retraining

Non Commutative Convolutional Signal Models in Neural Networks: Stability to Small Deformations

In this paper we discuss the results recently published in~[1] about algebraic signal models (ASMs) based on non commutative algebras and their use in convolutional neural networks. Relying on the general tools from algebraic signal processing (ASP), we study the filtering and stability properties of non commutative convolutional filters. We show how non commutative filters can be stable to small perturbations on the space of operators. We also show that although the spectral components of the Fourier representation in a non commutative signal model are associated to spaces of dimension larger than one, there is a trade-off between stability and selectivity similar to that observed for commutative models. Our results have direct implications for group neural networks, multigraph neural networks and quaternion neural networks, among other non commutative architectures. We conclude by corroborating these results through numerical experiments

Fine-grained Expressivity of Graph Neural Networks

Numerous recent works have analyzed the expressive power of message-passing graph neural networks (MPNNs), primarily utilizing combinatorial techniques such as the $1$-dimensional Weisfeiler-Leman test ($1$-WL) for the graph isomorphism problem. However, the graph isomorphism objective is inherently binary, not giving insights into the degree of similarity between two given graphs. This work resolves this issue by considering continuous extensions of both $1$-WL and MPNNs to graphons. Concretely, we show that the continuous variant of $1$-WL delivers an accurate topological characterization of the expressive power of MPNNs on graphons, revealing which graphs these networks can distinguish and the level of difficulty in separating them. We identify the finest topology where MPNNs separate points and prove a universal approximation theorem. Consequently, we provide a theoretical framework for graph and graphon similarity combining various topological variants of classical characterizations of the $1$-WL. In particular, we characterize the expressive power of MPNNs in terms of the tree distance, which is a graph distance based on the concepts of fractional isomorphisms, and substructure counts via tree homomorphisms, showing that these concepts have the same expressive power as the $1$-WL and MPNNs on graphons. Empirically, we validate our theoretical findings by showing that randomly initialized MPNNs, without training, exhibit competitive performance compared to their trained counterparts. Moreover, we evaluate different MPNN architectures based on their ability to preserve graph distances, highlighting the significance of our continuous $1$-WL test in understanding MPNNs' expressivity

SC-MAD: Mixtures of Higher-order Networks for Data Augmentation

The myriad complex systems with multiway interactions motivate the extension of graph-based pairwise connections to higher-order relations. In particular, the simplicial complex has inspired generalizations of graph neural networks (GNNs) to simplicial complex-based models. Learning on such systems requires large amounts of data, which can be expensive or impossible to obtain. We propose data augmentation of simplicial complexes through both linear and nonlinear mixup mechanisms that return mixtures of existing labeled samples. In addition to traditional pairwise mixup, we present a convex clustering mixup approach for a data-driven relationship among several simplicial complexes. We theoretically demonstrate that the resultant synthetic simplicial complexes interpolate among existing data with respect to homomorphism densities. Our method is demonstrated on both synthetic and real-world datasets for simplicial complex classification.Comment: 5 pages, 1 figure, 1 tabl

Does Graph Distillation See Like Vision Dataset Counterpart?

Training on large-scale graphs has achieved remarkable results in graph representation learning, but its cost and storage have attracted increasing concerns. Existing graph condensation methods primarily focus on optimizing the feature matrices of condensed graphs while overlooking the impact of the structure information from the original graphs. To investigate the impact of the structure information, we conduct analysis from the spectral domain and empirically identify substantial Laplacian Energy Distribution (LED) shifts in previous works. Such shifts lead to poor performance in cross-architecture generalization and specific tasks, including anomaly detection and link prediction. In this paper, we propose a novel Structure-broadcasting Graph Dataset Distillation (SGDD) scheme for broadcasting the original structure information to the generation of the synthetic one, which explicitly prevents overlooking the original structure information. Theoretically, the synthetic graphs by SGDD are expected to have smaller LED shifts than previous works, leading to superior performance in both cross-architecture settings and specific tasks. We validate the proposed SGDD across 9 datasets and achieve state-of-the-art results on all of them: for example, on the YelpChi dataset, our approach maintains 98.6% test accuracy of training on the original graph dataset with 1,000 times saving on the scale of the graph. Moreover, we empirically evaluate there exist 17.6% ~ 31.4% reductions in LED shift crossing 9 datasets. Extensive experiments and analysis verify the effectiveness and necessity of the proposed designs. The code is available in the GitHub repository: https://github.com/RingBDStack/SGDD.Comment: Accepted by NeurIPS 202