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REVIEW 3 major objections 4 minor 49 references

Adapting to Heterophilic Graph Data with Structure-Guided Neighbor Discovery

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A two-layer GNN's prediction error is bounded by feature noise plus the number of false-positive edges, so rewiring the graph along structural attributes can improve node classification.

desk verdict Empirically strong and interpretable multi-graph method for heterophily, but Theorem 1 needs an injectivity assumption before the paper's central theoretical claim holds. read the letter →

arxiv 2506.08871 v1 pith:W2XWYIH7 submitted 2025-06-10 cs.LG eess.SP

Victor M. Tenorio , Madeline Navarro , Samuel Rey , Santiago Segarra , Antonio G. Marques This is my paper · ORCID
classification cs.LGeess.SP MSC 68T0705C8268R10
keywords GraphNeuralNetworksHeterophilyNeighbordiscoveryMulti-graphStructuralattributesNodeclassificationrewiringAdaptiveaggregation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Most GNNs implicitly assume that connected nodes tend to share labels, which fails on heterophilic graphs where the opposite happens. This paper tries to establish that the fix can be made in the graph itself instead of in the architecture: build new graphs by linking nodes with similar structural roles or global centrality, and feed those alongside the original graph. Its theoretical anchor is a bound showing that a two-layer GNN's output error relative to the ideal, perfectly homophilic case grows with the number of false-positive edges, so graphs with fewer cross-class connections should yield better label recovery. It also proves that considering several candidate graphs raises the probability that at least one is free of false positives. The paper then introduces SG-GNN, which processes the original and the derived graphs in parallel with learned weights, and reports top-tier accuracy on ten of eleven benchmarks.

What carries the argument

The engine of the argument is the decomposition of the observed adjacency matrix into a perfectly homophilic part $A^*$ (edges between same-class nodes only) and a false-positive difference $\Delta=A-A^*$; Theorem 1 isolates $\|\Delta\|_F$ as the graph-dependent error term. To produce such favorable graphs, the paper computes per-node structural attribute vectors, including role-based features such as degree, triangle counts, and egonet statistics, and global features such as centrality measures, and builds $k$-NN and $\epsilon$-ball graphs from their squared Euclidean distances. SG-GNN then passes the input through parallel GNN branches, one per graph, weights the branch outputs by softmax coefficients $\alpha_r$, concatenates them, and feeds them to an MLP; node-specific and multi-layer extensions follow the same idea. The claimed role of this machinery is to translate structural similarity into label homophily without labels and without expensive learned embeddings.

What would settle it

Train the two-layer GNN in Eq. (10) with $\Theta^{(1)}=0$ on any dataset, using $X^*$ as class means and $A^*$ as the perfectly homophilic edge-filtered graph. The theorem concludes that $Z^*=\Phi(X^*;A^*,\Theta)$ recovers $y$, but with $\Theta^{(1)}=0$ every node receives an identical representation, so no classifier can separate classes; this single configuration shows that the missing distinctness-preservation condition is load-bearing.

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Extended reading notes

Core claim

The central claim is Theorem 1 (Eq. 11): for a two-layer GNN $\Phi$ with row-normalized adjacency, nonexpansive activations, and weight matrices $\Theta^{(1)},\Theta^{(2)}$, if the class-mean feature matrix $X^*$ recovers $y$ and $A^*$ is the observed adjacency with all false-positive edges removed, then $\|Z^*-\hat Z\|_F \le \rho_1\rho_2(\alpha\sqrt N + 2(1+\sqrt N)\|\Delta\|_F\|X\|_F)$. Here $\Delta=A-A^*$ is the false-positive difference matrix, so the bound says plainly that prediction error shrinks as cross-class edges disappear. On the construction side, the paper claims that $k$-NN and $\epsilon$-ball graphs formed from role-based and global structural attributes consistently have higher edge homophily and lower $\|\Delta\|_F/N$ than the original graph on heterophilic datasets, and Propositions 1 and 2 show that more candidate graphs raise the chance of finding a perfectly or locally homophilic graph. Combining these pieces, SG-GNN learns nonnegative weights over parallel graph branches and achieves the best or second-best accuracy on 10 of the 11 datasets evaluated.

Load-bearing premise

The proof of Theorem 1 needs the class-mean rows to stay pairwise distinct after the GNN's weight matrices and nonlinearities are applied, but the paper states no injectivity or margin condition that would guarantee this; degenerate weights, such as $\Theta^{(1)}=0$, collapse all rows to one vector and break the recovery conclusion.

Editorial extensions

If this is right

  • Rewiring input graphs from cheap structural attributes can replace expensive embedding-based neighbor discovery without sacrificing accuracy, since role- and global-attribute graphs are competitive with DeepWalk, Node2Vec, Struc2Vec, and GraphWave constructions in the paper's Table IV.
  • The theoretical bound gives a concrete design criterion: prefer candidate graphs that minimize $\|\Delta\|_F/N$, which on heterophilic datasets favors the structurally informed $k$-NN and $\epsilon$-ball graphs over the original topology.
  • Multiple graph views hedge against the unknown best graph; Proposition 1 formalizes this, and the learned $\alpha_r$ coefficients let practitioners see which structure type carries the signal.
  • SG-GNN's single-layer, node-specific, and multi-layer variants are reported to reach the best or second-best accuracy on 10 of 11 datasets, including strongly heterophilic ones, which implies that adaptive fusion of the original graph with structure-based views is a workable strategy across homophily regimes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to use the empirical false-positive estimate $\hat q$ (Table III) as a model-selection criterion for choosing structural attributes before seeing labels; the paper computes $\hat q$ for analysis but does not propose using it that way.
  • Because Propositions 1 and 2 assume independent graphs while SG-GNN derives all views from the same original graph, varying the number of input views $R$ on a fixed dataset should show saturating returns as correlations among views dominate; the paper mentions the tradeoff but does not quantify it.
  • If node-specific weights really work as local attention, they should correlate with per-node homophily $h_{\text{node}}(i)$; checking that correlation would turn interpretability into a quantitative claim.
  • The bound is proven for a two-layer network; a natural extension is to ask whether deeper stacks preserve the linear dependence on $\|\Delta\|_F$, since the multi-layer SG-GNN is validated only empirically.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes Structure-Guided GNN (SG-GNN), a multi-graph architecture that augments the observed graph with k-NN and epsilon-ball graphs built from interpretable structural attributes (role-based and global), and adaptively combines these views with per-graph or per-node weights. The authors provide a theoretical analysis: Theorem 1 claims that a two-layer GNN on a graph with fewer false-positive edges produces embeddings closer to an ideal label-recoverable representation, while Propositions 1 and 2 claim that multiple graph views increase the probability of finding a homophilic graph globally or locally. Empirically, the paper reports that structural-attribute graphs improve plain GCN and FBGNN baselines on heterophilic datasets, and that SG-GNN variants are competitive with or better than several heterophily-specialized baselines on 11 benchmark datasets.

Significance. If the theoretical results are repaired, the paper makes a useful and timely contribution: it connects interpretable, low-cost structural graph construction to GNN performance under heterophily, and the experiments (Tables IV-VI) are extensive, with a clear comparison against learned embedding-based graph constructions and a practical computational-efficiency argument (Table V). The adaptive weighting scheme adds interpretability by revealing which graph views matter for each dataset. However, the central theoretical support is currently incomplete: the proof of Theorem 1 relies on an unstated non-collapse condition, and the proof of Proposition 2 contains an unjustified independence step. The empirical claims therefore currently stand without a fully valid theoretical explanation.

major comments (3)
  1. [Section III, Theorem 1; Appendix B] The proof's step that 'because Θ(1) and Θ(2) perform column-wise linear transformations and σ1 and σ2 are element-wise operations, the matrices σ1(X*Θ(1)) and σ2(σ1(X*Θ(1))Θ(2)) also recover y' is invalid without an additional non-collapse condition. For example, taking Θ(1)=0, σ1=ReLU, Θ(2)=1, σ2=identity makes every row of Z* equal to the same vector, violating Definition 1(b). Thus the proof does not establish that Z* recovers y, and the bound in (11) may measure distance to a target that is not label-discriminative. Please add an explicit assumption (e.g., Θ(1) and Θ(2) have full column rank and σ1, σ2 are strictly increasing, or a margin condition preserving pairwise distinctness of class-mean rows) and re-derive the theorem; as stated, the conclusion does not follow from the assumptions.
  2. [Appendix B, proof of Theorem 1] The proof uses the inequality ||σ1(A*_rw X Θ(1))||_F ≤ ||A*_rw X Θ(1)||_F, which requires σ1(0)=0 in addition to nonexpansiveness. Nonexpansiveness alone does not control the image of zero; without this assumption the constant in (11) should include a term involving ||σ1(0)||_F. Please either state that the nonlinearities satisfy σ(0)=0 or track this additional term in the bound.
  3. [Appendix D, Proposition 2] In the proof of Proposition 2, the equality P[γ] = ∏_{i=1}^N P[γ_i] is not justified by independence of the graphs. For a fixed graph, the events γ_i and γ_j share potential edges (both require the absence of a cross-class edge between i and j), so they are not independent. The lower bound in (14) may still be true via positive association (FKG) for monotone events, but the present proof is invalid as written. Please provide a correct argument or replace the bound with a union bound, e.g., P[γ] ≥ 1 − N(1 − (1−q)^{N(C−1)/C})^R, and adjust the subsequent discussion accordingly.
minor comments (4)
  1. [Section VI-C, Table VI] The sentence claiming that SG-GNN variants 'achieve the best performance on 10 out of the 11 datasets evaluated' is not supported by Table VI: on Cora the top accuracy is MixHop (88.36) and the best SG-GNN variant (SG-GCNL, 87.81) ranks fourth. Please correct the claim to state precisely the rank achieved on each dataset.
  2. [Section V, Propositions 1 and 2] The phrase 'without loss of generality, we assume classes are equally sized' is inaccurate; equal class sizes is a simplifying assumption, not a matter of lossless generality. Please rephrase.
  3. [Section VI-B, Table III] The empirical estimate q̂ is defined using ordered pairs (V×V), while Proposition 1 counts unordered inter-class pairs; please clarify the convention or align the two definitions.
  4. [Section IV, Table I and Section VI-B] The value of k (stated as k=3 in Section VI-C) and any ε values used for ε-ball graphs should be stated clearly in the captions of Tables I, II, and IV, since the main-text discussion refers to these tables before the experimental setup is given.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the perturbation bound and probability propositions are self-contained; the only self-citation ([21]) is transparent and non-load-bearing, and the unproven injectivity step in Theorem 1 is a proof gap rather than a circular reduction.

full rationale

The derivation chain is not circular. Theorem 1's bound (Eq. 11) is a nonexpansiveness and triangle-inequality argument: with X* as the class-mean matrix (Eq. 9) and A* as the false-positive-free adjacency (Eq. 8), the proof bounds ||Phi(X;A)-Phi(X*;A*)||_F by alpha sqrt(N) plus 2(1+sqrt(N))||Delta||_F ||X||_F; no term is fitted to the accuracies that the paper later reports. Propositions 1 and 2 are direct union and intersection bounds under explicit independence and equal-class-size assumptions, and the empirical qhat in Table III is an illustrative estimate of the assumption parameter q, not a fitted constant used to generate predictions. The only self-citation, [21], is transparently identified as a prior short conference paper introducing a similar neighbor-discovery idea, and it is not used to prove Theorem 1 or to justify the SG-GNN architecture. One non-circular caveat is that Appendix B asserts without proof that column-wise linear maps and pointwise nonlinearities preserve the pairwise distinctness of class-mean rows, so the statement "Z* recovers y" requires an unstated injectivity or margin condition on Theta and sigma; this is a proof gap and a correctness risk, not a circular reduction. Accordingly, no circular step is exhibited, and the central theoretical and empirical claims do not reduce to their own inputs by construction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

All quantities are either existing graph-theoretic measures (degree, centrality, etc.) or learned/constructed from data; no new physical or mathematical entities are postulated. The main free parameters are design choices (k, epsilon, R, L) and the central theoretical premises are the unstated non-collapse condition and the independence/equal-class assumptions.

free parameters (4)
  • k in k-NN graph construction = 3
    Used for all k-NN graphs (Tables I-II, IV, VI); not derived or tuned per dataset, affects homophily and accuracy.
  • epsilon in epsilon-ball graph construction = not reported
    Epsilon-ball graphs are used in Tables IV-VI, but no values or tuning procedure are given; the constructed topology and results depend on this threshold.
  • number of graph views R = 15 (original + k-NN and epsilon-ball variants from feat, role, global, DW, N2V, S2V, GW)
    The input graph set for SG-GNN is fixed by hand; R controls the probability bounds in Propositions 1-2 and the model's capacity.
  • number of layers L in SG-GNNL = 2
    Only L=2 is evaluated; depth is an unstated design choice influencing expressiveness and results.
assumptions (6)
  • domain assumption Class means of node features are distinct (X* recovers y, cf. Definition 1 and Eq. 9)
    Invoked in Theorem 1 as the tractability condition; analogous to Assumption 1 of [8], empirically verified there, not in this paper.
  • ad hoc to paper Nonlinearities and weight matrices preserve the distinctness of class-mean rows (implicit)
    The proof of Theorem 1 (Appendix B) asserts Z* recovers y because X* does; this requires an unstated non-collapse condition, which can fail (e.g., zero weight matrix).
  • domain assumption Candidate graphs G_r are independent
    Needed for Propositions 1 and 2; the structural graphs are all derived from the same original graph G and are dependent, as acknowledged after Proposition 1.
  • domain assumption Classes are equally sized with K=N/C
    Used to compute the exponents in (13) and (14); stated as 'without loss of generality' but real datasets violate it, changing the bound quantitatively.
  • domain assumption Nodes with similar structural attributes share labels
    The empirical premise behind the whole method; supported on 11 datasets in Tables I-II but not guaranteed elsewhere.
  • standard math Eigenvalues of the normalized augmented Laplacian lie approximately in [0,1.5]
    Used in Section III (Eq. 7) to argue GNNs are low-pass; standard result from [8], [35].

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Pith. "Pith review of Adapting to Heterophilic Graph Data with Structure-Guided Neighbor Discovery." pith.science (2026). https://pith.science/paper/W2XWYIH7

@misc{pith2026250608871,
  author       = {Pith},
  title        = {Pith review of: Adapting to Heterophilic Graph Data with Structure-Guided Neighbor Discovery},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W2XWYIH7}},
  note         = {Machine review of arXiv:2506.08871}
}
read the original abstract

Graph Neural Networks (GNNs) often struggle with heterophilic data, where connected nodes may have dissimilar labels, as they typically assume homophily and rely on local message passing. To address this, we propose creating alternative graph structures by linking nodes with similar structural attributes (e.g., role-based or global), thereby fostering higher label homophily on these new graphs. We theoretically prove that GNN performance can be improved by utilizing graphs with fewer false positive edges (connections between nodes of different classes) and that considering multiple graph views increases the likelihood of finding such beneficial structures. Building on these insights, we introduce Structure-Guided GNN (SG-GNN), an architecture that processes the original graph alongside the newly created structural graphs, adaptively learning to weigh their contributions. Extensive experiments on various benchmark datasets, particularly those with heterophilic characteristics, demonstrate that our SG-GNN achieves state-of-the-art or highly competitive performance, highlighting the efficacy of exploiting structural information to guide GNNs.

Figures

Figures reproduced from arXiv: 2506.08871 by the authors.

Figure 1
Figure 1. Histograms of node homophily measurements [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Structure-based neighbor discovery on a toy graph. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Proposed architecture exemplified using three input [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Learned adaptive coefficients αr for the single-layer SG-GNN architecture using (a) GCN base layers and (b) FBGNN base layers. Each row represents a dataset, and columns correspond to different input graph types: origi￾nal graph (G), and ϵ-ball or k-NN graphs derived f…

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Works this paper leans on

49 extracted references · 44 canonical work pages

  1. [1]

    A comprehensive survey on graph neural networks,

    Z. Wu, S. Pan, F. Chen, G. Long, C. Zhang, and P. S. Yu, “A comprehensive survey on graph neural networks,”IEEE Trans. Neural Netw. and Learning Syst., vol. 32, no. 1, pp. 4–24, 2021

    work page 2021

  2. [2]

    Geometric deep learning: Going beyond Euclidean data,

    M. M. Bronstein, J. Bruna, Y . LeCun, A. Szlam, and P. Vandergheynst, “Geometric deep learning: Going beyond Euclidean data,”IEEE Signal Process. Mag., vol. 34, no. 4, pp. 18–42, 2017

    work page 2017

  3. [3]

    A survey of convolutional neural networks: Analysis, applications, and prospects,

    Z. Li, F. Liu, W. Yang, S. Peng, and J. Zhou, “A survey of convolutional neural networks: Analysis, applications, and prospects,”IEEE Trans. Neural Netw. and Learning Syst., vol. 33, no. 12, pp. 6999–7019, 2022

    work page 2022

  4. [4]

    Exploiting the structure of two graphs with graph neural networks,

    V . M. Tenorio and A. G. Marques, “Exploiting the structure of two graphs with graph neural networks,”IEEE Trans. Signal and Info. Process. over Netw., 2024

    work page 2024

  5. [5]

    Beyond homophily in graph neural networks: Current limitations and effective designs,

    J. Zhu, Y . Yan, L. Zhao, M. Heimann, L. Akoglu, and D. Koutra, “Beyond homophily in graph neural networks: Current limitations and effective designs,” inConf. Neural Inform. Process. Syst., vol. 33, 2020, pp. 7793–7804

    work page 2020

  6. [6]

    Is homophily a necessity for graph neural networks?

    Y . Ma, Z. Liu, N. Shah, and J. Tang, “Is homophily a necessity for graph neural networks?” inIntl. Conf. Learning Representations (ICLR), 2022

    work page 2022

  7. [7]

    Geom- GCN: Geometric graph convolutional networks,

    H. Pei, B. Wei, K. C.-C. Chang, Y . Lei, and B. Yang, “Geom- GCN: Geometric graph convolutional networks,”Intl. Conf. Learning Representations (ICLR), 2020

    work page 2020

  8. [8]

    Revisiting graph neural networks: Graph filtering perspective,

    H. NT, T. Maehara, and T. Murata, “Revisiting graph neural networks: Graph filtering perspective,” inIntl. Conf. on Pattern Recognition, 2021, pp. 8376–8383

    work page 2021

Show all 49 references
  1. [9]

    MixHop: Higher-order graph convolutional architectures via sparsified neighborhood mixing,

    S. Abu-El-Haija, B. Perozzi, A. Kapoor, N. Alipourfard, K. Lerman, H. Harutyunyan, G. Ver Steeg, and A. Galstyan, “MixHop: Higher-order graph convolutional architectures via sparsified neighborhood mixing,” inIntl. Conf. Machine Learning (ICML). PMLR, 2019, pp. 21–29

    2019

  2. [10]

    Beyond low-frequency infor- mation in graph convolutional networks,

    D. Bo, X. Wang, C. Shi, and H. Shen, “Beyond low-frequency infor- mation in graph convolutional networks,” inAAAI Conf. Artif. Intell. AAAI Press, 2021, pp. 3950–3957

    2021

  3. [11]

    Large scale learning on non-homophilous graphs: new benchmarks and strong simple methods,

    D. Lim, F. Hohne, X. Li, S. L. Huang, V . Gupta, O. Bhalerao, and S.-N. Lim, “Large scale learning on non-homophilous graphs: new benchmarks and strong simple methods,” inConf. Neural Inform. Process. Syst., 2021

    2021

  4. [12]

    Adaptive universal generalized PageRank graph neural network,

    E. Chien, J. Peng, P. Li, and O. Milenkovic, “Adaptive universal generalized PageRank graph neural network,” inIntl. Conf. Learning Representations (ICLR), 2021

    2021

  5. [13]

    Simple and deep graph convolutional networks,

    M. Chen, Z. Wei, Z. Huang, B. Ding, and Y . Li, “Simple and deep graph convolutional networks,” inIntl. Conf. Machine Learning (ICML). PMLR, 2020, pp. 1725–1735

    2020

  6. [14]

    Diffusion improves graph learning,

    J. Gasteiger, S. Weißenberger, and S. G ¨unnemann, “Diffusion improves graph learning,” inAdvances in Neural Info. Process. Syst. (NeurIPS), vol. 32, 2019

    2019

  7. [15]

    Predict then propagate: Graph neural networks meet personalized PageRank,

    J. Gasteiger, A. Bojchevski, and S. G ¨unnemann, “Predict then propagate: Graph neural networks meet personalized PageRank,” inIntl. Conf. Learning Representations (ICLR), 2019

    2019

  8. [16]

    Graph neural networks: Architec- tures, stability, and transferability,

    L. Ruiz, F. Gama, and A. Ribeiro, “Graph neural networks: Architec- tures, stability, and transferability,”Proc. IEEE, vol. 109, no. 5, pp. 660–682, 2021

    2021

  9. [17]

    Stability properties of graph neural networks,

    F. Gama, J. Bruna, and A. Ribeiro, “Stability properties of graph neural networks,”IEEE Trans. Signal Process., vol. 68, pp. 5680–5695, 2020

    2020

  10. [18]

    S. Rey, M. Navarro, V . M. Tenorio, S. Segarra, and A. G. Marques, “Redesigning graph filter-based GNNs to relax the homophily assump- 12 G Gballϵ,feat Gnnk,featGballϵ,role Gnnk,roleGballϵ,glob Gnnk,globGballϵ,DW Gnnk,DWGballϵ,N2V Gnnk,N2VGballϵ,S2V Gnnk,S2V Gnnk,GW TexasWisco...

    2025

  11. [19]

    Breaking the limit of graph neural networks by improving the assortativity of graphs with local mixing patterns,

    S. Suresh, V . Budde, J. Neville, P. Li, and J. Ma, “Breaking the limit of graph neural networks by improving the assortativity of graphs with local mixing patterns,” inIntl. Conf. Knowledge Discovery & Data Mining (SIGKDD), 2021, pp. 1541–1551

    2021

  12. [20]

    Beyond homophily and homogeneity assumption: Relation-based frequency adaptive graph neural networks,

    L. Wu, H. Lin, B. Hu, C. Tan, Z. Gao, and Z. Liu, “Beyond homophily and homogeneity assumption: Relation-based frequency adaptive graph neural networks,”IEEE Trans. Neural Netw. and Learning Syst., vol. 35, no. 6, pp. 8497–8509, 2024

    2024

  13. [21]

    Structure-guided input graph for GNNs facing heterophily,

    V . M. Tenorio, M. Navarro, S. Rey, S. Segarra, and A. G. Marques, “Structure-guided input graph for GNNs facing heterophily,” inAsilomar Conf. Signals, Syst., and Computers, 2024, pp. 1080–1084

    2024

  14. [22]

    Role-oriented graph auto-encoder guided by structural information,

    X. Guo, W. Zhang, W. Wang, Y . Yu, Y . Wang, and P. Jiao, “Role-oriented graph auto-encoder guided by structural information,” inDatabase Syst. Advanced Appl., 2020, pp. 466–481

    2020

  15. [23]

    Recovering missing node features with local structure-based embeddings,

    V . M. Tenorio, M. Navarro, S. Segarra, and A. G. Marques, “Recovering missing node features with local structure-based embeddings,” inIEEE Intl. Conf. Acoust., Speech and Signal Process. (ICASSP). IEEE, 2024, pp. 9931–9935

    2024

  16. [24]

    Newman,Networks

    M. Newman,Networks. Oxford University Press, 2018

    2018

  17. [25]

    Homophily- enhanced self-supervision for graph structure learning: Insights and directions,

    L. Wu, H. Lin, Z. Liu, Z. Liu, Y . Huang, and S. Z. Li, “Homophily- enhanced self-supervision for graph structure learning: Insights and directions,”IEEE Trans. Neural Netw. and Learning Syst., vol. 35, no. 9, pp. 12 358–12 372, 2024

    2024

  18. [26]

    Graph structure learning for robust graph neural networks,

    W. Jin, Y . Ma, X. Liu, X. Tang, S. Wang, and J. Tang, “Graph structure learning for robust graph neural networks,” inIntl. Conf. Knowledge Discovery & Data Mining (SIGKDD), 2020, pp. 66–74

    2020

  19. [27]

    Measuring and relieving the over-smoothing problem for graph neural networks from the topological view,

    D. Chen, Y . Lin, W. Li, P. Li, J. Zhou, and X. Sun, “Measuring and relieving the over-smoothing problem for graph neural networks from the topological view,”AAAI Conf. Artif. Intell., vol. 34, no. 04, p. 3438–3445, 2020

    2020

  20. [28]

    Node similarity preserving graph convolutional networks,

    W. Jin, T. Derr, Y . Wang, Y . Ma, Z. Liu, and J. Tang, “Node similarity preserving graph convolutional networks,” inACM Intl. Conf. Web Search and Data Mining. ACM, 2021, p. 148–156

    2021

  21. [29]

    Seeking similarities while removing differences: Graph neural networks based on node correlation,

    S. Li, B. Zhang, J. Song, Y . Xia, J. Xie, and C. Wang, “Seeking similarities while removing differences: Graph neural networks based on node correlation,” inIEEE Intl. Conf. Acoust., Speech and Signal Process. (ICASSP). IEEE, 2024, p. 4960–4964

    2024

  22. [30]

    Unbiased graph embedding with biased graph observations,

    N. Wang, L. Lin, J. Li, and H. Wang, “Unbiased graph embedding with biased graph observations,” inACM Web Conf.ACM, 2022, p. 1423–1433

    2022

  23. [31]

    struc2vec: Learning node representations from structural identity,

    L. F. Ribeiro, P. H. Saverese, and D. R. Figueiredo, “struc2vec: Learning node representations from structural identity,” inIntl. Conf. Knowledge Discovery & Data Mining (SIGKDD). ACM, 2017, p. 385–394

    2017

  24. [32]

    Semi-supervised classification with graph convolutional networks,

    T. N. Kipf and M. Welling, “Semi-supervised classification with graph convolutional networks,” inIntl. Conf. Learning Representations (ICLR), 2017

    2017

  25. [33]

    Graph attention networks,

    P. Veli ˇckovi´c, G. Cucurull, A. Casanova, A. Romero, P. Li`o, and Y . Ben- gio, “Graph attention networks,”Intl. Conf. Learning Representations (ICLR), 2018

    2018

  26. [34]

    Birds of a feather: Homophily in social networks,

    M. McPherson, L. Smith-Lovin, and J. M. Cook, “Birds of a feather: Homophily in social networks,”Annual Review of Sociology, vol. 27, no. 1, pp. 415–444, 2001

    2001

  27. [35]

    Simplifying graph convolutional networks,

    F. Wu, A. Souza, T. Zhang, C. Fifty, T. Yu, and K. Weinberger, “Simplifying graph convolutional networks,” inIntl. Conf. Machine Learning (ICML). PMLR, 2019, pp. 6861–6871

    2019

  28. [36]

    Is heterophily a real nightmare for graph neural networks to do node classification?

    S. Luan, C. Hua, Q. Lu, J. Zhu, M. Zhao, S. Zhang, X.-W. Chang, and D. Precup, “Is heterophily a real nightmare for graph neural networks to do node classification?”arXiv preprint arXiv:2109.05641, 2021

    2021 arXiv

  29. [37]

    Discrete signal processing on graphs: Frequency analysis,

    A. Sandryhaila and J. M. F. Moura, “Discrete signal processing on graphs: Frequency analysis,”IEEE Trans. Signal Process., vol. 62, no. 12, pp. 3042–3054, 2014

    2014

  30. [38]

    RolX: structural role extraction & mining in large graphs,

    K. Henderson, B. Gallagher, T. Eliassi-Rad, H. Tong, S. Basu, L. Akoglu, D. Koutra, C. Faloutsos, and L. Li, “RolX: structural role extraction & mining in large graphs,” inIntl. Conf. Knowledge Discovery & Data Mining (SIGKDD), 2012, pp. 1231–1239

    2012

  31. [39]

    A set of measures of centrality based on betweenness,

    L. C. Freeman, “A set of measures of centrality based on betweenness,” Sociometry, vol. 40, no. 1, pp. 35–41, 1977

    1977

  32. [40]

    Automat- ing the construction of internet portals with machine learning,

    A. K. McCallum, K. Nigam, J. Rennie, and K. Seymore, “Automat- ing the construction of internet portals with machine learning,”Info. Retrieval, vol. 3, no. 2, p. 127–163, 2000

    2000

  33. [41]

    CiteSeer: An automatic citation indexing system,

    C. L. Giles, K. D. Bollacker, and S. Lawrence, “CiteSeer: An automatic citation indexing system,” inACM Conf. on Digital Libraries. ACM, 1998, p. 89–98

    1998

  34. [42]

    Tensor graph convolutional networks for multi-relational and robust learning,

    V . N. Ioannidis, A. G. Marques, and G. B. Giannakis, “Tensor graph convolutional networks for multi-relational and robust learning,”IEEE Trans. Signal Process., vol. 68, pp. 6535–6546, 2020

    2020

  35. [43]

    How powerful are graph neural networks?

    K. Xu, W. Hu, J. Leskovec, and S. Jegelka, “How powerful are graph neural networks?” inIntl. Conf. Learning Representations (ICLR), 2019

    2019

  36. [44]

    Collective classification in network data,

    P. Sen, G. Namata, M. Bilgic, L. Getoor, B. Galligher, and T. Eliassi- Rad, “Collective classification in network data,”AI Magazine, vol. 29, no. 3, pp. 93–93, 2008

    2008

  37. [45]

    DeepWalk: Online learning of social representations,

    B. Perozzi, R. Al-Rfou, and S. Skiena, “DeepWalk: Online learning of social representations,” inIntl. Conf. Knowledge Discovery & Data Mining (SIGKDD). ACM, 2014, pp. 701–710

    2014

  38. [46]

    node2vec: Scalable feature learning for net- works,

    A. Grover and J. Leskovec, “node2vec: Scalable feature learning for net- works,” inIntl. Conf. Knowledge Discovery & Data Mining (SIGKDD). ACM, 2016, pp. 855–864

    2016

  39. [47]

    Learning structural node embeddings via diffusion wavelets,

    C. Donnat, M. Zitnik, D. Hallac, and J. Leskovec, “Learning structural node embeddings via diffusion wavelets,” inIntl. Conf. Knowledge Discovery & Data Mining (SIGKDD). ACM, 2018, pp. 1320–1329

    2018

  40. [48]

    Edge directionality improves learning on heterophilic graphs,

    E. Rossi, B. Charpentier, F. D. Giovanni, F. Frasca, S. G ¨unnemann, and M. M. Bronstein, “Edge directionality improves learning on heterophilic graphs,” inLearning on Graphs Conf., ser. Proc. Mach. Learning Res., vol. 231. PMLR, 2024, pp. 25:1–25:27

    2024

  41. [49]

    Simple spectral graph convolution,

    H. Zhu and P. Koniusz, “Simple spectral graph convolution,” inIntl. Conf. Learning Representations (ICLR), 2021. 13 APPENDIXA STRUCTURALFEATUREDEFINITIONS Below we provide a brief definition for each of the structural attributes computed for nodes based on the original graphG=...

    2021

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.