Homophily-Aware Graph Spectral Network

• The paper introduces a spectral GNN that modulates filtering operations using quantitative homophily measures to improve performance across diverse graph regimes.
• It presents architectures like HW-GNN and NewtonNet which employ Gaussian-window constraints and polynomial interpolation to adapt low, mid, and high-frequency filters based on homophily levels.
• Empirical evaluations show substantial gains in tasks such as bot detection and node classification, validating the model’s effectiveness in both homophilic and heterophilic settings.

A homophily-aware graph spectral network is a spectral graph neural network (GNN) architecture that systematically modulates its spectral filtering operations according to the global or localized homophily structure of the input graph. By explicitly linking the design and learning of spectral filters to quantitative measures of homophily—typically the fraction of edges connecting same-label node pairs—these models achieve improved adaptability and discriminative power across a wide spectrum of graph regimes, ranging from highly homophilic to strongly heterophilic. This paradigm has catalyzed recent advances in node classification, anomaly detection, cross-graph transfer, and federated representation learning on real-world graphs with diverse and often non-uniform homophily structures.

1. Homophily, Spectral Graph Convolutions, and the Frequency Domain

Homophily is defined as the proportion of edges connecting nodes within the same class, denoted for graphs $G=(V,E)$ as

$$h = \frac{\left|\{(u,v)\in E: y_u = y_v\}\right|}{|E|}.$$

Empirically and theoretically, strong homophily ($h\to1$) results in graph signals and labels that are locally smooth, with energy concentrated in the lower end of the Laplacian spectrum. Heterophilic graphs ($h\ll1$) exhibit sharp label and feature transitions across edges, emphasizing higher-frequency spectral components (Xu et al., 2023). Consequently, optimal node representation learning requires the ability to emphasize distinct frequency bands conditioned on the homophily of the graph.

In spectral GNNs, filtering proceeds by decomposing the normalized graph Laplacian $L = I - D^{-1/2}AD^{-1/2} = U\Lambda U^T$, and applying a frequency-domain function $g(\lambda)$ to the eigenvalues $\Lambda$: $g(L)X = U g(\Lambda) U^T X,$ where $X$ is the node feature matrix. By shaping $g(\lambda)$ to amplify or suppress frequency intervals according to $h$, the network can reconcile signal smoothness with discriminative capacity in both homophilic and heterophilic settings (Liu et al., 27 Nov 2025, Xu et al., 2023, Zou et al., 6 Jan 2025).

2. Homophily-Aware Spectral Filter Architectures

Several architectures instantiate homophily awareness in their spectral filtering design:

a) Gaussian-Window Constrained Spectral Network (HW-GNN)

HW-GNN employs a filter bank of S Gaussian windows,

$$g_s(\lambda; \omega_s, \sigma_s) = \exp\left(-\frac{(\lambda-\omega_s)^2}{2\sigma_s^2}\right),$$

approximated via polynomial expansion. Centers and widths $(\omega_s, \sigma_s)$ are dynamically steered toward the target region dictated by the observed graph homophily: $\bar{\omega}(h) = 2(1-h),$ via learnable MLPs and a frequency-distribution loss enforcing proximity of the learned filter centers to the homophily-driven target. This construction allows HW-GNN to flexibly focus on low, mid, or high-frequency bands as required (Liu et al., 27 Nov 2025).

b) Newton-Interpolation Filter with Shape-Aware Regularization (NewtonNet)

NewtonNet interpolates a polynomial $g$ through selected spectral nodes $\{q_i\}$, with amplitudes $\{t_i\}$ regularized according to the estimated $h$. The regularizer

$$\mathcal{L}_{\mathrm{SR}} = \gamma_1(\frac{1}{C} - h)\|t_{\mathrm{low}}\|_2^2 + \gamma_2|h - \frac{1}{C}|\|t_{\mathrm{mid}}\|_2^2 + \gamma_3(h - \frac{1}{C})\|t_{\mathrm{high}}\|_2^2$$

adapts the filter shape—low-pass, band-pass, or high-pass—according to the graph's homophily ratio (Xu et al., 2023).

c) Dual- or Multi-Band Hybrid Backbones

DFGNN and HS-GPPT combine multiple parallel spectral filters (low-, mid-, high-pass), each implemented via distinct polynomials or Beta-wavelet bases, with learnable fusion weights. In HS-GPPT, per-filter prompt graphs further align the spectral energy of downstream tasks under different homophily levels (Yang et al., 18 Nov 2024, Luo et al., 15 Aug 2025).

Model	Filter Construction	Homophily Adaptation Mechanism
HW-GNN (Liu et al., 27 Nov 2025)	Bank of Gaussian windows	Homophily-driven window targeting (MLP, loss)
NewtonNet (Xu et al., 2023)	Newton polynomial	Shape regularization per homophily
DFGNN (Yang et al., 18 Nov 2024)	Low/high-pass parallel	Self-aware dynamic fusion
HS-GPPT (Luo et al., 15 Aug 2025)	Beta-wavelet hybrid	Prompt-tuned spectral matching

3. Homophily Estimation and Adaptation Strategies

Exact homophily is available only if node labels are fully observed. Real-world settings address this through (i) sampling and partial label estimation; (ii) heuristic graph statistics; or (iii) LLMs, which can infer edge-level class agreement via natural language reasoning on node attributes (Lu et al., 17 Jun 2025). LLM-discovered homophily priors can then be injected into spectral polynomial parameters (basis mixing, coefficient modulation), producing filter families that rapidly adapt in low-label or weakly supervised environments.

For HW-GNN, the adaptation loop computes $h$, selects a spectrally-appropriate target center, and regularizes the filter family around that frequency. NewtonNet directly computes $h$ and penalizes deviations in the filter amplitude profile, while in DFGNN and LOHA, the partitioning and aggregation of frequency-specific representations rely on implicit or data-driven homophily-conditioning (Zou et al., 6 Jan 2025, Yang et al., 18 Nov 2024).

4. Optimization, Training, and Modular Integration

Homophily-aware spectral networks are typically trained via cross-entropy or task-specific supervision, with auxiliary regularizers or losses enforcing homophily-informed filter shapes or distributions. HW-GNN adopts a two-term loss combining classification (e.g., FocalLoss) and a frequency-distribution term; NewtonNet and similar models use shape penalties controlled by global or block-wise homophily. Pseudocode provided in HW-GNN and LLM-SGNN variants illustrates that, modulo the homophily-adaptive filter layer, the overall structures of stacking, activation, and fusion mirror those of standard message-passing or spectral GNNs—enabling drop-in replacement or retrofitting (Liu et al., 27 Nov 2025, Lu et al., 17 Jun 2025).

The computational cost is dominated by polynomial filter application, which avoids explicit eigendecomposition by leveraging Chebyshev, Bernstein, Jacobi, or Newton bases. Practical settings recommend low-degree polynomials (e.g., $K=4$) and compact filter banks (e.g., $S=5$), striking a balance between localized frequency resolution and scalability (Liu et al., 27 Nov 2025, Xu et al., 2023).

5. Empirical Evaluations and Comparative Performance

Homophily-aware graph spectral networks consistently outperform homophily-agnostic baselines on both homophilic and heterophilic graphs across tasks:

• HW-GNN demonstrates average F1-score improvements of +4.3% over previous bests in Twitter-bot detection, with ablation revealing up to 10% F1 loss upon removing Gaussian windowing or homophily guidance (Liu et al., 27 Nov 2025).
• NewtonNet achieves 1–3% higher accuracy over established competitors by dynamically conforming filter shapes to the graph's homophily profile, with especially strong gains in low-supervision regimes (Xu et al., 2023).
• DFGNN and LOHA outperform state-of-the-art methods by 2.8% (average) and up to 11.9% (heterophilic datasets), establishing that explicit or contrastively-supervised frequency partitioning improves generalization to diverse label/graph regimes (Yang et al., 18 Nov 2024, Zou et al., 6 Jan 2025).
• RHO extends these findings to anomaly detection by robustly aligning per-channel homophily patterns, enabling substantial AUROC and AUPRC improvements, particularly when homophily varies between labeled exemplars and broader graph regions (Ai et al., 18 Jun 2025).
• In federated settings, FedGSP selectively shares low-frequency and complements high-frequency bases across clients, resulting in +3.28% mean absolute gain over prior art on heterophilic datasets (Yu, 19 Feb 2025).

Task	Model	Notable Gain/Strength
Bot detection	HW-GNN	+4.3% F1 (mean), strong gains for large/heterophilic graphs
Node classification	NewtonNet	1–3% accuracy gain, robust under weak supervision
Anomaly detection	RHO	+12.19 AUROC, +30.68 AUPRC (max)
Semi/self-supervised	LOHA	2.8% average improvement, outperforms supervised baselines on heterophilic
Federated learning	FedGSP	+3.28% on heterophilic, selective frequency sharing

6. Broader Algorithmic Landscape and Theoretical Underpinnings

A central theoretical result across multiple works is the direct relationship between edge homophily and the effective frequency band useful for learning: high $h$ induces low-pass preference; low $h$ pushes importance to higher frequencies; intermediate $h$ (e.g., Erdős–Rényi, $h\approx 1/C$) results in a maximal contribution from mid-spectrum modes (Xu et al., 2023).

These relationships are formalized via

• Laplacian quadratic forms and the Dirichlet energy of graph signals;
• spectral regression losses bounding node/classification risk relative to the overlap between graph signal energy $x^\top U g(\Lambda) U^\top x$ and filter profile $g(\lambda)$ (Luo et al., 15 Aug 2025);
• theoretical analyses guaranteeing that homophily-informed filter shaping yields smoother node clusters or aligns spectral properties for effective cross-client, cross-domain, or few-shot transfer (Luo et al., 15 Aug 2025, Yu, 19 Feb 2025);
• guarantees that rewiring or spectral prompt tuning can systematically transform the intrinsic spectrum of input graphs for better downstream adaptation (Li et al., 2022, Luo et al., 15 Aug 2025).

A plausible implication is that spectral GNN architectures parameterized by, or regularized with respect to, accurately estimated or inferred homophily achieve more stable and generalizable representations under distribution shift, label scarcity, and domain heterogeneity.

7. Challenges, Limitations, and Open Directions

Estimating or updating homophily in partially labeled or dynamic graphs remains nontrivial, although recent advances—e.g., LLM-based estimation requiring minimal labels—provide practical solutions with minimal overhead (Lu et al., 17 Jun 2025). Selecting appropriate polynomial order, number and spacing of filter bands/windows, and regularizer strength is data-dependent, though cross-validation and empirical sensitivity analyses provide robust guidelines (Liu et al., 27 Nov 2025, Xu et al., 2023).

Homophily-aware spectral approaches are compatible with retrofitting to standard polynomial-based SGNNs, and the modularity of filter implementation (separation of filter design and structural adaptation) facilitates plug-in use across diverse application domains, including social bot detection, anomaly detection in transactional networks, and federated multi-center learning. Ongoing research addresses (i) local homophily adaption, (ii) adaptive multi-band architectures beyond global homophily, and (iii) further principled integration with label-efficient, self-supervised, and cross-lingual graph transfer frameworks.

References:

• HW-GNN: Homophily-Aware Gaussian-Window Constrained Graph Spectral Network for Social Network Bot Detection (Liu et al., 27 Nov 2025)
• Shape-aware Graph Spectral Learning (Xu et al., 2023)
• Semi-supervised Graph Anomaly Detection via Robust Homophily Learning (Ai et al., 18 Jun 2025)
• Dual-Frequency Filtering Self-aware Graph Neural Networks for Homophilic and Heterophilic Graphs (Yang et al., 18 Nov 2024)
• LOHA: Direct Graph Spectral Contrastive Learning Between Low-pass and High-pass Views (Zou et al., 6 Jan 2025)
• Generalize across Homophily and Heterophily: Hybrid Spectral Graph Pre-Training and Prompt Tuning (Luo et al., 15 Aug 2025)
• Homophily Heterogeneity Matters in Graph Federated Learning: A Spectrum Sharing and Complementing Perspective (Yu, 19 Feb 2025)
• Can LLMs Improve Spectral Graph Neural Networks? (Lu et al., 17 Jun 2025)
• Restructuring Graph for Higher Homophily via Adaptive Spectral Clustering (Li et al., 2022)