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Edge-Wise Graph-Instructed Neural Networks

Francesco Della Santa, Antonio Mastropietro, Sandra Pieraccini, Francesco Vaccarino

TL;DR

This work introduces Edge-Wise Graph-Instructed (EWGI) layers to augment Graph-Instructed Neural Networks (GINNs) for Regression on Graph Nodes (RoGN). By adding per-node incoming weights to the GI framework, EWGI layers increase expressive power while keeping the weight count low, defined as \widehat{W} = \text{diag}(\boldsymbol{w}^{\rm out}) \hat{A} \text{diag}(\boldsymbol{w}^{\rm in}) with \hat{A} = A + I_n. Empirical evaluation on stochastic max-flow RoGN tasks over Barabási–Albert and Erdos–Rényi graphs shows EWGINNs can outperform GINNs on BA graphs and offer improved regularization on ER graphs, highlighting the trade-off between increased capacity and the need for careful hyperparameter tuning. The work provides a formal definition of EWGI, analyzes weight counts, and demonstrates practical benefits for graph-structured regression problems with potential applications in physics-based simulations and network flow modeling.

Abstract

The problem of multi-task regression over graph nodes has been recently approached through Graph-Instructed Neural Network (GINN), which is a promising architecture belonging to the subset of message-passing graph neural networks. In this work, we discuss the limitations of the Graph-Instructed (GI) layer, and we formalize a novel edge-wise GI (EWGI) layer. We discuss the advantages of the EWGI layer and we provide numerical evidence that EWGINNs perform better than GINNs over some graph-structured input data, like the ones inferred from the Barabasi-Albert graph, and improve the training regularization on graphs with chaotic connectivity, like the ones inferred from the Erdos-Renyi graph.

Edge-Wise Graph-Instructed Neural Networks

TL;DR

This work introduces Edge-Wise Graph-Instructed (EWGI) layers to augment Graph-Instructed Neural Networks (GINNs) for Regression on Graph Nodes (RoGN). By adding per-node incoming weights to the GI framework, EWGI layers increase expressive power while keeping the weight count low, defined as \widehat{W} = \text{diag}(\boldsymbol{w}^{\rm out}) \hat{A} \text{diag}(\boldsymbol{w}^{\rm in}) with \hat{A} = A + I_n. Empirical evaluation on stochastic max-flow RoGN tasks over Barabási–Albert and Erdos–Rényi graphs shows EWGINNs can outperform GINNs on BA graphs and offer improved regularization on ER graphs, highlighting the trade-off between increased capacity and the need for careful hyperparameter tuning. The work provides a formal definition of EWGI, analyzes weight counts, and demonstrates practical benefits for graph-structured regression problems with potential applications in physics-based simulations and network flow modeling.

Abstract

The problem of multi-task regression over graph nodes has been recently approached through Graph-Instructed Neural Network (GINN), which is a promising architecture belonging to the subset of message-passing graph neural networks. In this work, we discuss the limitations of the Graph-Instructed (GI) layer, and we formalize a novel edge-wise GI (EWGI) layer. We discuss the advantages of the EWGI layer and we provide numerical evidence that EWGINNs perform better than GINNs over some graph-structured input data, like the ones inferred from the Barabasi-Albert graph, and improve the training regularization on graphs with chaotic connectivity, like the ones inferred from the Erdos-Renyi graph.
Paper Structure (9 sections, 9 equations, 4 figures)

This paper contains 9 sections, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Graph-Instructed Layers
  3. Edge-Wise Graph-Instructed Layers
  4. Preliminary Results
  5. Maximum Flow Regression for Stochastic Flow Networks
  6. Performance Measures
  7. Data, Model Architectures, and Hyperparameters
  8. Analysis of the Results
  9. Conclusion

Figures (4)

  • Figure 1: Visual representation of \ref{['eq:ewginn_node_action']}. Example with $n=4$ nodes (non-directed graph), $i=1$; for simplicity, the bias is not illustrated.
  • Figure 2: $\mathcal{G}_{\rm BA}$. Performances of GINN and EWGINN models in the $(\mathrm{MRE}_{av}, \mathrm{MRE}_\varphi)$ plane. Marker sizes are proportional to the number of NN weights.
  • Figure 3: $\mathcal{G}_{\rm ER}$. Performances of GINN and EWGINN models in the $(\mathrm{MRE}_{av}, \mathrm{MRE}_\varphi)$ plane. Marker sizes are proportional to the number of NN weights.
  • Figure 4: Training and validation loss of the EWGINN corresponding to the top-rightmost dots in \ref{['fig:ERres']}.

Theorems & Definitions (3)

  • Definition 2.1: GI Layer - General form GINN
  • Remark 3.1: EWGI Layers - Advantages of the Formulation
  • Definition 3.1: EWGI Layer - General form