PIMPC-GNN: Physics-Informed Multi-Phase Consensus Learning for Enhancing Imbalanced Node Classification in Graph Neural Networks

TL;DR

PIMPC-GNN targets imbalanced node classification by embedding three physics-inspired views—thermodynamic diffusion, Kuramoto synchronization, and spectral embedding—into a unified, differentiable GNN. The three-phase consensus is fused via confidence-aware ensembles and adaptive decision thresholds, optimized with an imbalance-aware loss that couples classification with physics-consistency terms. The framework yields theoretical guarantees for convergence and minority-class performance, and empirically outperforms 16 baselines across five datasets with notable gains in minority recall and balanced accuracy. This work provides interpretable insights into consensus dynamics on graphs and offers a principled, scalable approach to robust minority detection in real-world imbalanced graph tasks.

Abstract

Graph neural networks (GNNs) often struggle in class-imbalanced settings, where minority classes are under-represented and predictions are biased toward majorities. We propose \textbf{PIMPC-GNN}, a physics-informed multi-phase consensus framework for imbalanced node classification. Our method integrates three complementary dynamics: (i) thermodynamic diffusion, which spreads minority labels to capture long-range dependencies, (ii) Kuramoto synchronisation, which aligns minority nodes through oscillatory consensus, and (iii) spectral embedding, which separates classes via structural regularisation. These perspectives are combined through class-adaptive ensemble weighting and trained with an imbalance-aware loss that couples balanced cross-entropy with physics-based constraints. Across five benchmark datasets and imbalance ratios from 5-100, PIMPC-GNN outperforms 16 state-of-the-art baselines, achieving notable gains in minority-class recall (up to +12.7\%) and balanced accuracy (up to +8.3\%). Beyond empirical improvements, the framework also provides interpretable insights into consensus dynamics in graph learning. The code is available at \texttt{https://github.com/afofanah/PIMPC-GNN}.

Figures (4)

• Figure 1: Illustration of the multi-phase consensus process. (a) Initial graph with uniform weights, biased toward majority nodes (blue). (b) Consensus-driven update via central aggregation, a standard operation in GNNs that reduces noise. (c) Physics-informed refinement, the novel contribution of this paper, where diffusion, synchronisation, and spectral dynamics rebalance majority (green) and minority (orange) nodes. (d) Balanced consensus output that preserves graph structure while improving minority representation.
• Figure 2: Overall architecture of the proposed PIMPC-GNN framework: where $\mathbf{H}^{(0)}$ is the initial feature projection, $\mathbf{H}_{\text{th}}^{(0)}$, $\mathbf{H}_{\text{sync}}^{(0)}$, and $\mathbf{H}_{\text{spec}}^{(0)}$ are the thermodynamic diffusion, Kuramoto oscillator synchronisation, and structural spectral phases of the PIMPC-GNN, respectively. Th, Sync, and Spec are the three phase weights, $\tau_i$ is the adaptive thresholding for decision making, $\mathbf{w}^{(y)}$ is the confidence-aware weighting, $\mathbf{H}_{\text{fused}}$ is the multi-phase feature fusion, and $\mathbf{a}_{\text{ensm.}}$ is the ensemble weights, $\hat{y}_i^{\text{class}}$ is the final classification weights, $\mathcal{L}_{\text{class}}$ the classifier loss function, and $\mathcal{L}_{\text{total}}$ the total loss function of PIMPC-GNN.
• Figure 3: Node-level characteristic analysis of PIMPC-GNN: (a-b) resilience to connectivity variations and feature noise, (c-d) effective optimisation of physics constraints and training dynamics, and (e-f) favourable computational scaling and identifiable optimal operating parameters that align with theoretical expectations.
• Figure 4: Imbalanced learning performance analysis: (a) Minority class F1-score versus minority class ratio demonstrating effectiveness under extreme imbalance; (b) Training loss convergence showing faster and more stable optimisation; (c) Spectral properties analysis correlating eigenvalue gap with clustering quality; (d) Performance variance during training (log scale) validating theoretical stability guarantees.

Theorems & Definitions (16)

• Theorem 1: Computational Complexity
• proof
• Lemma 1: Thermodynamic Phase Convergence
• proof
• Lemma 2: Synchronisation Phase Stability
• proof
• Lemma 3: Spectral Embedding Consistency
• proof
• Theorem 2: Three-Phase Consensus Convergence
• proof