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Optimality of Message-Passing Architectures for Sparse Graphs

Aseem Baranwal, Kimon Fountoulakis, Aukosh Jagannath

TL;DR

The paper studies node classification on extremely sparse, feature-decorated graphs in the fixed-feature regime and defines asymptotic local Bayes optimality to benchmark classifiers. It formalizes a CSBM data model with $q_{ij}=b_{ij}/n$ and arbitrary feature distributions, derives the Bayes-optimal local classifier, and proves that a decoupled, message-passing GNN architecture can implement it. Through a Gaussian-feature specialization, it characterizes generalization error and shows the optimal method interpolates between an MLP (low graph signal) and a convolutional style GCN (high graph signal), with a non-asymptotic analysis confirming robustness on finite graphs. The results imply that, in sparse graphs, MP-GNNs are theoretically optimal for node classification, guiding architecture design and offering insights into when graph information is advantageous. The work advances understanding of when and how graph structure should be leveraged in learning, and provides precise performance guarantees under local weak convergence and non-asymptotic conditions.

Abstract

We study the node classification problem on feature-decorated graphs in the sparse setting, i.e., when the expected degree of a node is $O(1)$ in the number of nodes, in the fixed-dimensional asymptotic regime, i.e., the dimension of the feature data is fixed while the number of nodes is large. Such graphs are typically known to be locally tree-like. We introduce a notion of Bayes optimality for node classification tasks, called asymptotic local Bayes optimality, and compute the optimal classifier according to this criterion for a fairly general statistical data model with arbitrary distributions of the node features and edge connectivity. The optimal classifier is implementable using a message-passing graph neural network architecture. We then compute the generalization error of this classifier and compare its performance against existing learning methods theoretically on a well-studied statistical model with naturally identifiable signal-to-noise ratios (SNRs) in the data. We find that the optimal message-passing architecture interpolates between a standard MLP in the regime of low graph signal and a typical convolution in the regime of high graph signal. Furthermore, we prove a corresponding non-asymptotic result.

Optimality of Message-Passing Architectures for Sparse Graphs

TL;DR

The paper studies node classification on extremely sparse, feature-decorated graphs in the fixed-feature regime and defines asymptotic local Bayes optimality to benchmark classifiers. It formalizes a CSBM data model with and arbitrary feature distributions, derives the Bayes-optimal local classifier, and proves that a decoupled, message-passing GNN architecture can implement it. Through a Gaussian-feature specialization, it characterizes generalization error and shows the optimal method interpolates between an MLP (low graph signal) and a convolutional style GCN (high graph signal), with a non-asymptotic analysis confirming robustness on finite graphs. The results imply that, in sparse graphs, MP-GNNs are theoretically optimal for node classification, guiding architecture design and offering insights into when graph information is advantageous. The work advances understanding of when and how graph structure should be leveraged in learning, and provides precise performance guarantees under local weak convergence and non-asymptotic conditions.

Abstract

We study the node classification problem on feature-decorated graphs in the sparse setting, i.e., when the expected degree of a node is in the number of nodes, in the fixed-dimensional asymptotic regime, i.e., the dimension of the feature data is fixed while the number of nodes is large. Such graphs are typically known to be locally tree-like. We introduce a notion of Bayes optimality for node classification tasks, called asymptotic local Bayes optimality, and compute the optimal classifier according to this criterion for a fairly general statistical data model with arbitrary distributions of the node features and edge connectivity. The optimal classifier is implementable using a message-passing graph neural network architecture. We then compute the generalization error of this classifier and compare its performance against existing learning methods theoretically on a well-studied statistical model with naturally identifiable signal-to-noise ratios (SNRs) in the data. We find that the optimal message-passing architecture interpolates between a standard MLP in the regime of low graph signal and a typical convolution in the regime of high graph signal. Furthermore, we prove a corresponding non-asymptotic result.
Paper Structure (21 sections, 17 theorems, 46 equations, 2 figures)

This paper contains 21 sections, 17 theorems, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Work.
  3. Our Contributions.
  4. Architecture
  5. Theoretical Analysis and Discussion
  6. Asymptotic Local Bayes Optimality
  7. Data Model
  8. Optimal Classifier
  9. Comparative Study
  10. Non-asymptotic Setting
  11. Proofs
  12. Preliminary Results
  13. Bayes Optimal Classifier
  14. Computing the Classifier
  15. Generalization Error
  16. ...and 6 more sections

Key Result

Theorem 1

For any $\ell\geq 1$, the asymptotically $\ell$-locally Bayes optimal classifier of the root for the sequence $(G_n,u_n)\sim \mathrm{CSBM}(n,d,\mathds{P},\mathbf{Q})$ is where $\{\rho_i\}_{i\in[C]}$ are the densities associated with the distributions $\mathds{P}_i\in \mathds{P}$, and $\boldsymbol{\rho} = (\rho_i)_{i\in[C]}$.

Figures (2)

  • Figure 1: Comparison of \ref{['arch']} against an MLP and a vanilla GCN kipf:gcn.
  • Figure 2: Demonstration of \ref{['thm:gen-error-extremes']} for extreme graph signals. In the case where $\Gamma=0$, the architecture reduces to an MLP (\ref{['fig:Gamma-0']}), while if $\Gamma=1$, it behaves the same as a GCN (\ref{['fig:Gamma-1']}).

Theorems & Definitions (29)

  • Definition 3.1: $\ell$-local classifier
  • Definition 3.2
  • Theorem 1: Bayes optimal message-passing
  • Corollary 1.1: Optimal classifier for binary symmetric CSBM
  • Theorem 2: Generalization error
  • Theorem 3: Extreme graph signals
  • Proposition 3.1: Tree neighbourhoods
  • Theorem 4: Misclassification error for fixed $n$
  • Lemma 4.1
  • Lemma 4.2
  • ...and 19 more