Frequency-Guided Graph Structure Learning

• FgGSL is an end-to-end framework that models both homophilic and heterophilic node relationships via frequency-aware spectral filtering.
• It employs dual masking functions and polynomial filters to construct and combine refined graph topologies for improved information propagation.
• The method offers theoretical stability guarantees and consistently outperforms traditional GNNs on low-homophily datasets.

Frequency-Guided Graph Structure Learning (FgGSL) is an end-to-end graph structure inference framework designed to address the challenges posed by heterophilic graphs in node classification tasks. Unlike traditional GNNs that often assume homophily, FgGSL explicitly models both homophilic and heterophilic relationships through complementary learned graph structures and combines them using distinct spectral filter banks. A label-based structural loss supervises mask learning, ensuring graph rewiring is task-driven and frequency-aware. The approach demonstrates significant empirical improvements and provides theoretical robustness guarantees (Raghuvanshi et al., 29 Dec 2025).

1. Motivation and Problem Setting

The primary challenge addressed by FgGSL is the poor performance of conventional GNNs on heterophilic graphs, where most connected node pairs have dissimilar labels and structural cues from features alone are insufficient for discriminative node representation. Most previous approaches either rely on fixed adjacency matrices or limited rewiring mechanisms; these do not robustly accommodate both homophilic (similarly labeled neighbors) and heterophilic (differently labeled neighbors) edge patterns. FgGSL introduces joint learning of two refined graph topologies—one promoting homophily and the other heterophily—and processes each with frequency-aware graph filtering, enabling effective information propagation for classification in the presence of heterophily (Raghuvanshi et al., 29 Dec 2025).

2. Architecture and Data Flow

FgGSL operates in a multi-branch framework. Inputs consist of a node feature matrix $X \in \mathbb{R}^{N \times F}$ and an optional initial adjacency $A_f$ (commonly fully connected or observed edges). The main data flow steps are:

• Structure Learning:

Two symmetric, feature-driven masking functions $S_{\theta_1}(x_i, x_j)$ and $S_{\theta_2}(x_i, x_j)$, parameterized by small MLPs, produce weighted adjacency matrices $A_{\mathrm{Ho}}$ (homophilic) and $A_{\mathrm{Ht}}$ (heterophilic), respectively, using

$$S_\theta(x_i, x_j) = \sigma(\Phi_\theta(x_i)^\top \Phi_\theta(x_j)),$$

with $A_{\mathrm{Ho}} = A_f \odot W^{(1)}$, $A_{\mathrm{Ht}} = A_f \odot W^{(2)}$ ($\odot$ denotes entrywise product).

• Spectral Encoding:

The normalized Laplacians $L_{\mathrm{Ho}}$ and $L_{\mathrm{Ht}}$ are computed for each graph. A bank of predefined polynomial low-pass filters $\{h_L^{(j)}\}$ is applied to the homophilic branch and high-pass filters $\{h_H^{(j)}\}$ to the heterophilic branch:

$$H_L^{(j)}X = h_L^{(j)}(L_{\mathrm{Ho}}) X, \quad H_H^{(j)}X = h_H^{(j)}(L_{\mathrm{Ht}}) X.$$

Concatenated outputs across scales yield $Z_L$ and $Z_H$, which are combined as $Z = [Z_L\mid Z_H]$.

• Classification:

A linear classifier maps $Z$ to class logits via $\phi \in \mathbb{R}^{2(J-1)F \times C}$ and softmax.

• Backpropagation:

Task and structural losses jointly supervise the classifier and mask parameters.

The following table summarizes key architectural steps:

Component	Operation (summary)	Output
Structure Learner	Feature-masked graph construction ($A_{\mathrm{Ho}}$, $A_{\mathrm{Ht}}$)	Two weighted adjacencies
Spectral Encoder	Polynomial filter banks over $L_{\mathrm{Ho}}$, $L_{\mathrm{Ht}}$	Embedding $Z$
Classifier	Linear + softmax	Predicted classes

3. Frequency-Aware Filtering

FgGSL leverages two distinct filter banks for frequency decomposition:

• Low-pass filters ($h_L^{(j)}$) for $A_{\mathrm{Ho}}$, capturing smooth (homophilic) structural signals.
• High-pass filters ($h_H^{(j)}$) for $A_{\mathrm{Ht}}$, targeting discriminative, heterophilic frequency content.

For $j=2,\dots,J$, the filters are defined as:

$$h_L^{(j)}(\lambda) = (0.5\,\lambda)^{2^{j-1}} - (0.5\,\lambda)^{2^j}, \ h_H^{(j)}(\lambda) = (1-0.5\,\lambda)^{2^{j-1}} - (1-0.5\,\lambda)^{2^j},$$

where $\lambda$ denotes Laplacian eigenvalues. These filters are implemented as polynomial operators, enabling efficient message passing via repeated sparse-dense multiplication.

The concatenated representations from both spectral branches ($Z = [Z_L \mid Z_H]$) provide complementary signal extraction, leading to improved class separability, especially in heterophilic regimes.

4. Supervised Structural Loss and Objective

In addition to cross-entropy loss on labeled nodes,

$$\ell_{\mathrm{CE}} = - \sum_{i \in \mathcal V_\mathrm{train}} \sum_{c=1}^C Y_{i,c} \log \widehat Y_{i,c},$$

FgGSL directly supervises mask learning through a label-driven structural loss comprising two penalties:

• Homophilic penalty:

$$\ell_{\mathrm{Ho}}(W^{(1)}, \widehat Y) = \sum_{(i,j)\in E_f} w^{(1)}_{ij}(1 - \cos(\widehat Y_i, \widehat Y_j)),$$

• Heterophilic penalty:

$$\ell_{\mathrm{Ht}}(W^{(2)}, \widehat Y) = \sum_{(i,j)\in E_f} w^{(2)}_{ij}\cos(\widehat Y_i, \widehat Y_j),$$

where $\cos(u, v) = \frac{u^\top v}{\|u\|_2\|v\|_2}$ is the cosine similarity between class probabilities.

The total objective is

$$\min_{\theta_1, \theta_2, \phi} \ \ell_{\mathrm{CE}}\left(\widehat Y_\mathrm{train}, Y_\mathrm{train}\right) + \alpha\,\ell_{\mathrm{Ho}}\left(W^{(1)}, \widehat Y\right) + \beta\,\ell_{\mathrm{Ht}}\left(W^{(2)}, \widehat Y\right),$$

with $\alpha, \beta > 0$ controlling structural supervision. Gradients propagate from both losses through classifier, filters, and masks, ensuring topology adapts to task objectives.

5. Theoretical Analysis: Stability and Robustness

FgGSL provides explicit stability guarantees:

• Structural-Loss Stability:

If $Y_i$ and $\widehat Y_i$ are close in $\ell_2$ norm, then the gap between oracle and surrogate cosine similarities is bounded:

$$\left|\cos(Y_i, Y_j) - \cos(\widehat Y_i, \widehat Y_j)\right| \le 2\sqrt{C}(\epsilon_i + \epsilon_j),$$

where $\|Y_i - \widehat Y_i\|_2 \le \epsilon_i$.

• Filter Stability:

For polynomial filters $h_j$ of Lipschitz constant $K_h$, if Laplacians deviate by at most $\epsilon$, then

$$\left\|h_j(L^\star) - h_j(L)\right\| \le 2K_h(1 + \delta\sqrt{N})\epsilon + O(\epsilon^2),$$

with $\delta$ as the eigenvector misalignment measure. For FgGSL's filters, $K_h = 2^{j-1}$, so error scales as $O(2^{j-1} \epsilon)$. This demonstrates robustness of the filter banks and the learned representations to moderate perturbations in the graph structure.

6. Implementation Considerations and Complexity

FgGSL's main hyperparameters include filter bank scale ($J = 3$–5), mask-MLP dimension ($D = 16$–64), structural loss weights ($\alpha, \beta \approx 0.1$–$1.0$), learning rate ($10^{-3}$–$10^{-2}$), and training epochs ($T = 100$–500).

• Per-epoch computational complexity:
  • Mask computation: $O(|E_f| D)$, or $O(N^2 D)$ for a dense base graph.
  • Each filter operation: $O(r|E|F)$, with $r$ the polynomial order; total $O(J|E|F)$ for $2(J-1)$ filters.
  • Dense graph case: $O(JN^2 F)$.
• Memory:

Dominated by storing two $N \times N$ masks for fully-connected initializations. Pruning is used in practice to exploits sparsity.

FgGSL's algorithm can be expressed as a sequence of mask parameter updates, spectral filtering, classification, and loss computation; see the provided pseudocode for an exact stepwise specification (Raghuvanshi et al., 29 Dec 2025).

7. Empirical Evaluation

FgGSL has been benchmarked on six standard heterophilic datasets (Texas, Wisconsin, Cornell, Squirrel, Actor, Chameleon), characterized by low homophily (high heterophily ratios, $R_\mathrm{het}=0.76$–0.88). The reported mean node classification accuracy across ten splits demonstrates FgGSL's consistent outperformance over baselines such as GraphSAGE, GAT, MLP, H2GCN, Geom-GCN, MixHop, SG-GCN, and FAGCN:

Model	Texas	Wisconsin	Cornell	Squirrel	Actor	Chameleon
FgGSL	0.94±0.08	0.96±0.05	0.94±0.08	0.58±0.09	0.41±0.02	0.79±0.09
GraphSAGE	0.74±0.08	0.74±0.08	0.69±0.05	0.37±0.02	0.34±0.01	0.50±0.01
GAT	0.52±0.06	0.49±0.04	0.61±0.05	0.40±0.01	0.27±0.01	0.60±0.02
MLP	0.79±0.04	0.85±0.03	0.75±0.02	0.35±0.02	0.35±0.01	0.50±0.02
H2GCN	0.80±0.05	0.84±0.05	0.70±0.05	0.59±0.01	0.35±0.01	0.69±0.01
Geom-GCN	0.78±0.07	0.80±0.06	0.61±0.08	0.56±0.02	0.35±0.01	0.65±0.02
MixHop	0.81±0.09	0.83±0.08	0.78±0.09	0.35±0.03	0.34±0.01	0.53±0.02
SG-GCN	0.83±0.01	0.83±0.01	0.72±0.01	0.60±0.02	0.36±0.01	0.67±0.03
FAGCN	0.83±0.01	0.82±0.01	0.71±0.01	0.31±0.02	0.35±0.01	0.46±0.03

Ablation studies indicate that removing masking or either filter-bank reduces accuracy by up to 10%, confirming both components are essential. Cosine-similarity distributions of learned node embeddings show enhanced separation of intra-class and inter-class examples, even in highly heterophilic environments.

FgGSL thus constitutes a principled solution for task-driven, frequency-aware graph structure learning, with empirical and theoretical support for its applicability and effectiveness in challenging, low-homophily settings (Raghuvanshi et al., 29 Dec 2025).