Graph Homophily and Network Learning

• Graph homophily is the tendency of nodes with similar attributes to connect, underpinning network modeling and GNN performance.
• It is quantified using edge and node homophily ratios along with composite metrics like Tri-Hom to capture label, structural, and feature similarities.
• Enhancing homophily through graph rewiring, feature transformation, and cluster-aware structure learning can boost model robustness and clustering quality.

Graph homophily is the tendency of nodes with similar attributes to connect via edges in a graph. This property underpins many phenomena in network science, social modeling, and contemporary machine learning—especially the design and performance of Graph Neural Networks (GNNs). Homophily can manifest with respect to discrete labels (e.g., class, region), structural position, observed node features, or behavioral patterns. Accurate quantification, modeling, and enhancement of homophily remain active research areas due to its central role in learning, clustering, anomaly detection, and transfer tasks on graph data.

1. Fundamental Definitions and Metrics

Homophily in graphs is most commonly measured by the proportion of edges connecting same-label nodes. For a graph $G = (V, E)$ with node labels $y_v$, the edge homophily ratio is

$$H(G) = \frac{1}{|E|}\sum_{(u,v) \in E} \mathbb{I}(y_u = y_v)$$

where $\mathbb{I}$ is the indicator function. The node homophily ratio averages (over all nodes) the fraction of their neighbors sharing the same label:

$$h_{node} = \frac{1}{|V|}\sum_{v \in V} \frac{|\{u \in N(v): y_u = y_v\}|}{deg(v)}$$

Alternative, degree-weighted and “adjusted” homophily metrics (e.g., Newman’s assortativity coefficient) seek to account for class imbalance and ensure comparability across datasets (Platonov et al., 2022).

For machine learning tasks, feature-based homophily is also critical; it quantifies the average similarity of node features across edges. Additionally, structural homophily considers the purity of label distributions within neighborhoods:

$$H_s(G) = \frac{1}{|V|} \sum_{i} \max_c \frac{|\{j \in N(i)\cup \{i\}: y_j = c\}|}{|N(i)|+1}$$

In hypergraphs, the Perplexity-Homophily Index associates each hyperedge’s entropy of attribute composition with a degree-preserving random baseline, yielding a normalized diversity gap per interaction (Kumar et al., 24 Nov 2025). In heterogeneous (multi-type) graphs, classical metrics fail; recently introduced Cross-Type Homophily Ratio (CHR) measures the information compatibility of cross-type edges via label or feature-propagated similarities (Tao et al., 24 Jan 2025).

2. Desiderata, Theoretical Properties, and Recent Advances

Robust homophily metrics should satisfy:

• Maximal/minimal agreement: Achieve constant values on fully homophilic/heterophilic graphs.
• Constant baseline: Return a fixed value (e.g., zero) on random-label networks, independent of class count/balance.
• Empty-class tolerance: Unaffected by the addition of labels without nodes.
• Monotonicity: Increase strictly with added homophilic edges, decrease with added heterophilic edges (Mironov et al., 12 Dec 2024, Platonov et al., 2022).

No classic metric satisfies all properties, particularly when class sizes vary or graphs are directed. Mironov & Prokhorenkova’s unbiased homophily meets all desiderata for undirected graphs, providing a normalized, continuous, and scale-invariant measure (Mironov et al., 12 Dec 2024). For directed graphs, there exists a provable conflict between baseline invariance and minimal agreement; thus, no measure achieves all desired axioms.

Beyond quantification, homophily's theoretical connection to GNN performance is established via Dirichlet energy—measuring the smoothness of node embeddings over the graph Laplacian. Lower energy (i.e., more homophily) permits simultaneous smoothing and class separation by message passing (Ajorlou et al., 18 Dec 2025, Mendelman et al., 18 May 2025). Empirically, GNN accuracy, ARI, and NMI improve monotonically with increased homophily, but only composite metrics (see below) robustly predict performance across diverse benchmarks (Zheng et al., 27 Jun 2024).

3. Composite, Contextual, and Aspect-Specific Metrics

Current research disentangles homophily into multiple interacting aspects:

• Label Homophily: Fraction of same-label edges.
• Structural Homophily: Neighborhood label purity.
• Feature Homophily: Feature similarity within neighborhoods and across clusters.

The Tri-Hom metric is the geometric mean of these three axes:

$$\mathrm{TriHom}(G) = \Big(H_l(G)\cdot H_s(G)\cdot H_f(G)\Big)^{1/3}$$

Generative models (CSBM-3H) enable independent tuning of each aspect, revealing that GNN classification accuracy is tightly coupled to Tri-Hom, not any single metric in isolation (Zheng et al., 27 Jun 2024). Correlation analyses across 31 real-world datasets show Tri-Hom outperforms 17 previous metrics in predictiveness of model accuracy.

In heterophilous regimes, Label Informativeness (LI)—the mutual information between neighbor and node labels normalized by label entropy—captures predictive signal even when raw homophily is low (Platonov et al., 2022). For multi-type graphs, cross-type measures like CHR (based on soft label or feature propagation) enable performance gains by targeted subgraph pruning (Tao et al., 24 Jan 2025).

4. Modeling and Enhancing Homophily

Homophily can be actively modeled and enhanced, especially to improve GNNs and clustering:

• Graph Rewiring: Edges are added or deleted to increase homophily, guided by reference graphs constructed via feature and label-driven diffusion. The theoretical guarantee is that adding edges from a more homophilic reference increases expected homophily and, correspondingly, GNN accuracy (Mendelman et al., 18 May 2025).
• Feature-Nodes Transformation: Techniques like GRAPHITE introduce auxiliary nodes for each feature, ensuring nodes sharing features are only two-hops apart, directly increasing feature-based homophily without quadratic edge overhead (Qiu et al., 16 Sep 2025).
• Cluster-Aware Structure Learning: Methods such as HoLe iteratively refine the graph structure by recovering intra-cluster edges (via latent similarity in high-confidence cluster assignments) and removing inter-cluster edges, increasing edge homophily and clustering quality (Gu et al., 2023).
• Similarity-Enhanced Homophily: In multi-view clustering, combining neighbor-pattern, node-feature, and global similarity measures allows the recovery or construction of homophilous graphs even when direct label homophily is weak, yielding robust clustering under both homophilic and heterophilic conditions (Chen et al., 4 Oct 2024).

5. Homophily in Directed, Bipartite, and Heterogeneous Graphs

Directed graphs present unique challenges, as edge direction impacts the composition and informational content of neighborhoods. The Popularity-Homophily Index (PH Index) weights homophilic edges by the target node’s centrality, capturing both local and global tendencies (Oswal, 2021). Directional homophily-aware GNNs (DHGNN) decouple message passing for forward and backward edges, employing resettable gating and structure-aware fusion to adapt to fluctuating, asymmetric homophily patterns; empirical analysis shows up to 15% link prediction improvement over baselines (Zhang et al., 28 May 2025).

For bipartite networks, homophily is not directly observed along edges but via two-paths or shared affiliations. Curved exponential-family random graph models with tunable exponents ($\alpha, \beta$) interpolate between linear counts and thresholded indicators of shared partners, enabling flexible modeling and estimation via MCMC (Bomiriya et al., 2023).

Heterogeneous graphs (HGs) require metrics like CHR, which propagate “target information” (labels or soft predictions) across types and measure the similarity along cross-type edges. Selective pruning of low-CHR edges enhances clustering and classification performance by increasing target-node separation in HGNN embeddings (Tao et al., 24 Jan 2025).

6. Homophily, Learning Robustness, and Practical Implications

Variability in local homophily—the per-node neighborhood composition—can induce substantial accuracy discrepancies in GNNs. Nodes whose local homophily deviates from the graph’s global level suffer predictably worse classification, as formalized by bias terms in aggregated logits (Loveland et al., 2023). Models that decouple ego and neighbor embeddings (e.g., LINKX, H2GCN) mitigate this sensitivity. Grouping neighbors by estimated neighborhood homophily (NH) and aggregating along separate channels allows for fairer, higher-fidelity learning on both homophilic and heterophilic graphs (Gong et al., 2023).

In clustering and anomaly detection, contrastive learning schemes must account for variable homophily. Adaptive methods (NeuCGC) interpolate between positive and neutral pairs based on empirical homophily, robustly aligning node distributions and representations across homophily regimes (Peng et al., 17 Dec 2025). Robust learning filters (AdaFreq) learn spectral multipliers for channel-specific and cross-channel homophily presentations, outperforming fixed-filter methods in anomaly detection (Ai et al., 18 Jun 2025).

For domain adaptation across graphs, aligning distributions of homophily (and heterophily) between source and target networks is critical; mixed filters (low-pass, high-pass, full) allow for robust transfer even under significant distributional shifts (Fang et al., 26 May 2025).

7. Broader Impact, Challenges, and Open Questions

Graph homophily deeply impacts learning performance, model selection, generalization bounds, and interpretability. Recent measures such as unbiased homophily (Mironov et al., 12 Dec 2024) and composite Tri-Hom (Zheng et al., 27 Jun 2024) facilitate robust, fair comparisons across graphs and settings. Enhancing homophily via rewiring, feature expansion, and similarity-based augmentation enables stronger clustering, prediction, and anomaly detection, especially in challenging heterophilic regimes (Mendelman et al., 18 May 2025, Qiu et al., 16 Sep 2025, Gu et al., 2023, Chen et al., 4 Oct 2024). Ongoing research seeks scalable solutions for very large graphs, finer modeling in directed and multi-type networks, and more nuanced theoretical connections between various types of homophily and downstream learning tasks. The complex interplay between label, structural, and feature homophily, along with their context-specific implications, marks homophily as a central, multifaceted concept in graph learning research.