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Using Informational Connectivity to Measure the Synchronous Emergence of fMRI Multi-voxel Information Across Time

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VC dimension of Graph Neural Networks with Pfaffian activation functions.

Giuseppe Alessio D'Inverno¹, Monica Bianchini¹, Franco Scarselli¹

¹Department of Information Engineering and Mathematics, University of Siena, Via Roma 56, Siena, 53100, Italy.

Neural Networks : the Official Journal of the International Neural Network Society

|November 25, 2024

Summary

This study analyzes the generalization capability of Graph Neural Networks (GNNs) by extending Vapnik-Chervonenkis (VC) dimension analysis to sigmoid and hyperbolic tangent activation functions. Findings provide theoretical bounds related to GNN architecture and graph properties.

Keywords:

Graph Neural Networks Pfaffian functions VC dimension

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Area of Science:

Machine Learning
Graph Neural Networks
Theoretical Computer Science

Background:

Graph Neural Networks (GNNs) are powerful tools for graph-based learning, utilizing message passing mechanisms.
GNNs are theoretically equivalent to the Weisfeiler-Lehman (WL) test for graph isomorphism.
Previous research established GNNs as universal approximators and investigated their generalization via Vapnik-Chervonenkis (VC) dimension for specific activation functions.

Purpose of the Study:

Extend the analysis of GNNs' VC dimension to commonly used activation functions like sigmoid and hyperbolic tangent.
Provide theoretical bounds on GNN generalization capability.
Investigate the relationship between VC dimension, GNN architecture, and graph properties.

Main Methods:

Utilized Pfaffian function theory to analyze the VC dimension of GNNs.
Derived theoretical bounds concerning GNN architecture parameters (depth, neurons, input size).
Related bounds to the number of colors from the 1-WL test on graph domains.

Main Results:

Established theoretical bounds for the VC dimension of GNNs with sigmoid and hyperbolic tangent activation functions.
Demonstrated how these bounds depend on network architecture and graph characteristics.
Preliminary experimental results support the theoretical analysis.

Conclusions:

The study provides a theoretical framework for understanding the generalization capabilities of a broader range of GNNs.
The findings contribute to the theoretical understanding of GNNs' expressive power and limitations.
This research paves the way for designing more robust and generalizable GNN models.