Predicting Steady-State Behavior in Complex Networks with Graph Neural Networks

Amit Reza; Priodyuti Pradhan

REVIEW 4 major objections 4 minor 41 references

Predicting Steady-State Behavior in Complex Networks with Graph Neural Networks

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Graph neural networks trained on small graphs can predict the steady-state regime of linear dynamics on much larger networks.

desk verdict Legitimate new application of GNNs to eigenvector localization prediction, with honest limitations, but the scale-invariance headline is fragile because the class labels shift with network size under fixed thresholds. read the letter →

arxiv 2502.01693 v3 pith:734ODZVT submitted 2025-02-02 cs.LG cs.AInlin.AO

Priodyuti Pradhan , Amit Reza This is my paper · ORCID

classification cs.LGcs.AInlin.AO

keywords graphneuralnetworkinverseparticipationratioprincipaleigenvectorlocalizationlineardynamicalsystemscomplexnetworksregressionscaleinvariance

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

This paper aims to establish that a graph neural network can read only the adjacency structure of a network and predict the steady-state behavior of a linear dynamical process running on it, without simulating the process or computing eigenvectors. The behavior is summarized by the inverse participation ratio (IPR) of the principal eigenvector, a scalar that separates delocalized, weakly localized, and strongly localized states. Trained on synthetic networks of a few hundred nodes, the model is reported to reach about 95% binary and 96% three-state classification accuracy on unseen synthetic networks of 5,000 to 15,000 nodes, a property the paper calls scale invariance. The same pipeline is also applied to real-world benchmark graphs, where accuracy is lower because the datasets are imbalanced across the three states.

What carries the argument

The central object is the inverse participation ratio of the principal eigenvector, $y=\sum_i u_i^4/(\sum_i u_i^2)^2$, defined on each graph's adjacency matrix; it ranges from $1/n$ for a delocalized state to a value near 1 for a strongly localized one. The carrying mechanism is message passing: each node begins with centrality features, three graph convolution layers (or two attention layers) aggregate neighborhood information, a mean-pooling readout forms a graph-level vector, and a linear layer regresses the log-scaled IPR. The threshold function $r(y,\tau_1,\tau_2,\epsilon)$ in Eq. (5) is what turns the continuous prediction into the classification of steady-state behavior.

What would settle it

Train the model on the synthetic families used in the paper and then test it on 5,000 to 15,000-node networks from a family absent from training, such as small-world or lattice graphs; if classification accuracy against the $\tau_1=0.05$, $\tau_2=0.2$ partition drops toward the majority class, the scale-invariance claim is specific to the training families rather than a general property of IPR learning.

Watch

Extended reading notes

Core claim

The paper's central claim is that the localization class of the long-time state of the linear system $\frac{dx}{dt}=(\alpha I+\beta A)x$ can be predicted from network topology alone by a graph neural network. Because the steady state is the principal eigenvector of the adjacency matrix, the dynamical question reduces to a spectral regression: predict the inverse participation ratio $y=\sum_i u_i^4/(\sum_i u_i^2)^2$ from the adjacency matrix $A$. A three-layer GCN with mean-pooling readout, or a two-layer GAT with learned attention weights, is trained under a log-compressed mean-squared error on small graphs and then applied to larger graphs; the piecewise threshold map with $\tau_1=0.05$, $\tau_2=0.2$, $\epsilon=10^{-6}$ converts the predicted scalar into the dynamical-state label. The key reported result is that models trained on networks of size 500 to 1500 nodes classify unseen networks of size 5000 to 15000 nodes with 95% accuracy for delocalized versus localized and 96% accuracy for the three-state problem.

Load-bearing premise

The load-bearing premise is that the hand-picked thresholds $\tau_1=0.05$ and $\tau_2=0.2$ divide the IPR continuum into physically meaningful classes, so accuracy measured against this partition is a valid measure of predictive skill.

Editorial extensions

If this is right

A network's steady-state class can be obtained from its adjacency matrix in near real time, without computing the principal eigenvector or simulating the linear dynamics.
Because models are trained on networks of size 200 to 1500 and test on networks up to 15,000 nodes, the marginal cost of state prediction does not grow with network size beyond reading the graph.
The feature ablation implies that betweenness and closeness centralities carry most of the predictive signal, so cheaper feature engineering may be possible.
The entropy variant shows that the same framework applies to Shannon entropy as a localization measure, with about 90% three-state accuracy.
On real-world datasets the approach works best for weakly localized states, while delocalized and strongly localized regimes are harder to separate.

Reading between the lines

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

The reported scale invariance is demonstrated only across the synthetic families used in training; whether it transfers to real-world or structurally different graphs is an open question, and the paper's own real-world results show a drop in accuracy.
If the IPR prediction is trustworthy, the framework becomes a cheap screening tool: an adjacency snapshot alone would classify a network's diffusion regime, which could inform epidemic-response or infrastructure planning before running any simulation.
Because the Rényi entropy of order 2 equals $-\ln y$, the predicted IPR directly supplies an entropy value, so the same trained model could serve both localization measures without retraining.
A stronger test of the claim would be to train on the synthetic families and test on a held-out generative family; that experiment would reveal whether the model has learned a general spectral functional or per-family signatures.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a legitimate new application—GNN regression of the IPR of the principal eigenvector—with code and an honest limitations section. The scale-invariance claim is real but fragile because the class labels themselves shift with n given the fixed thresholds in Eq. (5). Worth refereeing, but it needs a threshold sweep and baselines before the 95/96% headline is believable.

What's new: first GNN model to regress IPR and classify localization states, trained on small synthetic graphs and tested on larger ones. That's a plausible utility: eigendecomposition is expensive, a GNN surrogate is fast. The paper does a fair amount of diligence: ablation over node features, hyperparameter optimization, feature importance ranking, and an entropy-based alternative. Code is on GitHub. The authors are also honest that real-world three-class separation is weak and that GCN underperforms on ER/SF, which is why they move to GAT. That transparency counts.

Soft spots, in order of severity: 1. The 95/96% accuracy is measured against labels from hand-set thresholds τ1=0.05, τ2=0.2. For delocalized states IPR=1/n, so on the large test networks (n=5000–15000) the delocalized class sits orders of magnitude below τ1, making the extreme classes easy to separate. The weakly localized band has absolute width 0.15, so the same physical state can cross the boundary as n changes. This makes the 'scale-invariant' claim partly an artifact of the label function. A threshold sweep and a size-normalized IPR (e.g., n·IPR) would settle whether the model's predictions are actually scale-consistent. I'd want that before trusting Fig. 6. 2. No baselines: no comparison to direct IPR computation, power iteration, or a simple degree-based predictor. No error bars, no RMSE/MAE for the regression, just scatter plots and classification accuracy. The classification might be riding on a few graph families with distinct IPR ranges. 3. The backprop derivation is standard and the 'explainability' framing oversells it; it's just gradients. Minor.

The central claim—that a GNN can predict IPR of the leading eigenvector—probably holds for the extreme classes. The weakly localized regime is where the evaluation is weakest, and that's exactly where the thresholds bite.

Who should read this: network scientists and graph-ML folks who want a fast surrogate for spectral localization. It's a proof-of-concept, not a new mechanism. My recommendation: send it to review, but condition acceptance on a threshold sensitivity analysis, at least one baseline (power iteration or a mean-degree predictor), and regression error metrics.

Referee Report

4 major / 4 minor

Summary. The paper proposes a graph neural network framework (GCN and GAT) to predict the inverse participation ratio (IPR) of the principal eigenvector of the adjacency matrix for undirected, unweighted, connected graphs, and then to assign each graph to a steady-state regime via a hand-set threshold function (Eq. 5). The authors train on small synthetic networks from six or seven families and test on larger synthetic networks, reporting 95% binary and 96% three-class accuracy as evidence of scale-invariant performance (Section III.C, Fig. 6). They also apply the model to real-world benchmark graphs, perform an ablation over node features, include a derivation of forward and backward propagation for a single GCN layer, and extend the analysis to Shannon entropy. The central empirical claim is that a GNN trained on small model networks predicts the localization class of larger unseen networks.

Significance. If the claims are validated, the paper would provide a useful learned proxy for a spectral quantity that normally requires eigen-decomposition, with potential applicability to large networks and a clear practical motivation. The paper has tangible strengths: the code is made publicly available, the experiments cover multiple synthetic graph families, and the out-of-sample size transfer (training on 500-1500 nodes, testing on 5000-15000 nodes) is a nontrivial positive result. The analytical backpropagation derivation, while standard, is clearly presented. However, the headline accuracy is measured against labels created by an arbitrary threshold partition, there are no baselines to establish that a GNN is needed, and regression performance is never quantified with RMSE/MAE. These gaps currently make the central 'scale-invariant performance' claim substantially weaker than the paper's framing.

major comments (4)

[§III.C, Eq. (5), Fig. 6] The 95% and 96% accuracy figures are computed against labels produced by the fixed thresholds τ1=0.05, τ2=0.2, ε=1e-6. This partition is not size-consistent: for an n-node delocalized state IPR=1/n, so for the reported test sizes n=5000-15000 the delocalized class lies orders of magnitude below τ1, while the weakly localized band has a fixed absolute width of 0.15. The thresholds are chosen with reference to the known IPR values of star and regular graphs, so the reported accuracy may partly reflect separation of archetypal graph families rather than physically meaningful steady-state classes. A threshold sensitivity analysis, and ideally a size-dependent definition of the classes, is required before the scale-invariance claim can be accepted.
[§III.C, Fig. 6] The quantitative claims rest on single accuracy numbers with no error bars, confidence intervals, or repeated-seed variation. Given the randomness in network generation and in neural network training, the paper should report means and standard deviations over multiple runs, together with per-class sample counts so the confusion matrices can be interpreted.
[§III, 'Methodology and Results'] There are no baselines. The paper does not compare GCN/GAT against simple graph-statistic regressors (e.g., degree heterogeneity, clustering, or spectral moments fed to a random forest or linear model), nor does it report any regression metric such as RMSE, MAE, or R² for the IPR prediction task. Without these, it is impossible to judge whether the graph neural network architecture contributes predictive skill beyond what classical network features already provide, or whether the reported classification accuracy is a meaningful measure of regression quality.
[§V and Appendix F] The real-world results and the entropy-based extension do not provide independent support for the main claim. Section V explicitly acknowledges that distinguishing three states on real-world graphs poses a significant challenge, and Appendix F states that the entropy thresholds are numerically calibrated so that the entropy-based partitioning 'mirrors' the IPR-based partitioning. Consequently, the high entropy classification accuracy is partly circular and should be presented only as a consistency check, not as additional evidence of predictive skill.

minor comments (4)

[§III.C] There is a typo: 'learning rare to 1e-5' should read 'learning rate to 1e-5'.
[§III.C, Fig. 6] The caption says 'Scale invariant' but only two size ranges are compared, both within the same synthetic families; this is more precisely a size-transfer result than scale invariance in the usual sense.
[§IV, Fig. 11] The description of the gray color in the weight distribution plots is ambiguous; the caption says 'epochs 2-9' but it is not clear whether all intermediate epochs are shown or only some.
[Appendix B, Example 1] The example uses a generic matrix A that is not a normalized adjacency matrix, which may confuse readers since the main text reserves A for adjacency; a brief note explaining that this is a purely illustrative matrix would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the model's IPR predictions are out-of-sample and the thresholding is an explicit, not fitted, labeling convention.

full rationale

The paper's target variable is the IPR of the principal eigenvector of the adjacency matrix (Eq. 4), computed directly from the graph; the GNN is trained on these computed labels for small graphs and then evaluated on held-out graphs of different sizes (Section III.C, Fig. 6). Because test labels are never used to set model parameters or thresholds, the reported accuracies are genuine out-of-sample assessments of a learned map from adjacency structure to IPR. The threshold function in Eq. (5) with τ1=0.05, τ2=0.2 is a hand-chosen discretization of the continuous IPR scale; it defines the classification task rather than being fitted to the test set, and its arbitrariness is a robustness/validation concern, not a circular reduction. The Appendix F entropy thresholds are calibrated to mirror the IPR partition, but this is a secondary illustration, and the entropy values themselves are independently computed from the same eigenvectors; it does not feed back into the central IPR regression. Prior citations to the authors' own IPR-localization work provide background and analytic examples, but the load-bearing learning claim is not justified by those citations. No equation reduces to its input by construction, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard linear algebra (PEV determines steady state), the choice of IPR and hand-set thresholds as the target, and an implicit assumption that synthetic training distributions transfer to larger real-world networks. No new entities are introduced.

free parameters (2)

IPR class thresholds (τ1=0.05, τ2=0.2, ϵ=1e-6) = τ1=0.05, τ2=0.2, ϵ=1e-6
Chosen by hand in Eq. (5) to separate known delocalized (regular/cycle) from strongly localized (star) IPR values; all classification accuracy figures depend on these values.
Entropy thresholds for Appendix F = τ1=0.7, τ2=0.9, ϵ=1e-6
Calibrated numerically to mirror the IPR-based partitioning, so entropy classification inherits the same hand-set boundaries.

assumptions (4)

standard math The steady state of x'=(αI+βA)x is proportional to the principal eigenvector of A (Appendix A).
Standard eigendecomposition result; requires symmetric diagonalizable A, true for undirected simple graphs.
domain assumption IPR of the principal eigenvector is an adequate scalar summary of localization/delocalization (Eq. 4).
The paper chooses IPR as the target; the validity of classifying dynamical states by IPR thresholds is assumed, not derived.
domain assumption Model networks (cycle, star, ER, SF, RGG, etc.) in training sizes 200-300 are representative of larger networks up to 15000 nodes.
Scale-invariance claim rests on this transferability; only tested within same synthetic families.
ad hoc to paper The threshold scheme Eq. (5) with fixed τ1, τ2 partitions the continuum of IPR values into physically meaningful regimes.
Thresholds are chosen by hand, and accuracy numbers are reported against this partition.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Predicting Steady-State Behavior in Complex Networks with Graph Neural Networks." pith.science (2026). https://pith.science/paper/734ODZVT

@misc{pith2026250201693,
  author       = {Pith},
  title        = {Pith review of: Predicting Steady-State Behavior in Complex Networks with Graph Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/734ODZVT}},
  note         = {Machine review of arXiv:2502.01693}
}

read the original abstract

In complex systems, information propagation can be defined as diffused or delocalized, weakly localized, and strongly localized. This study investigates the application of graph neural network models to learn the behavior of a linear dynamical system on networks. A graph convolution and attention-based neural network framework has been developed to identify the steady-state behavior of the linear dynamical system. We reveal that our trained model distinguishes the different states with high accuracy. Furthermore, we have evaluated model performance with real-world data. In addition, to understand the explainability of our model, we provide an analytical derivation for the forward and backward propagation of our framework.

Figures

Figures reproduced from arXiv: 2502.01693 by the authors.

**Figure 1.** FIG. 1. Steady-state behavior in linear dynamical systems on complex networks. We depict the nodes in the graphs with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2. The architecture of the Graph Neural Network (GNN) for the regression task over the graphs. The [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Neural message passing in Graph Convolutional Networks (GCN) model for the regression task. Each node’s features [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: FIG. 4. We train the GCN model with two types of structures associated with delocalized and strongly localized states. The [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: FIG. 5. We use scale-free networks for training and testing. (a) We can observe very low accuracy during the test time for the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. Scale invariant. Confusion matrices for the GAT [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. Prediction of Graph Attention Networks (GAT) on real world data sets ( [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: FIG. 8. Confusion matrices for the GAT model trained [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: FIG. 9. Comparison of prediction performance using different subsets of node features. (a–c) show the results for the GAT [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: FIG. 10. Prediction performance of the GAT model after hyperparameter optimization using Optuna. (a) represent the predicted [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]

**Figure 11.** Figure 11: FIG. 11. Portray the distribution of weight matrices ( [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 34 canonical work pages

[1]

S. H. Strogatz, Exploring complex networks, nature410, 268 (2001)

work page 2001
[2]

D. S. Bassett, N. F. Wymbs, M. A. Porter, P. J. Mucha, 13 J. M. Carlson, and S. T. Grafton, Dynamic reconfiguration of human brain networks during learning, Proceedings of the National Academy of Sciences108, 7641 (2011)

work page 2011
[3]

H. J. Kimble, The quantum internet, Nature453, 1023 (2008)

2008
[4]

Z. Yang, F. Liu, Z. Gao, H. Sun, J. Zhao, D. Janssens, and G. Wets, Estimating the influence of disruption on highway networks using gps data, Expert Systems with Applications187, 115994 (2022)

work page 2022
[5]

Multilevel Digital Contact Tracing

G. Mahapatra, P. Pradhan, A. Khan, S. K. Setua, R. K. Pal, and A. Rathor, Multilevel digital contact tracing, arXiv preprint arXiv:2007.05637 (2020)

work page Pith review arXiv 2020
[6]

Filoche and S

M. Filoche and S. Mayboroda, Universal mechanism for anderson and weak localization, Proceedings of the National Academy of Sciences109, 14761 (2012)

work page 2012
[7]

Elsner, V

U. Elsner, V. Mehrmann, F. Milde, R. A. Römer, and M. Schreiber, The anderson model of localization: a challenge for modern eigenvalue methods, SIAM Journal on Scientific Computing20, 2089 (1999)

work page 1999
[8]

Pradhan, C

P. Pradhan, C. Angeliya, and S. Jalan, Principal eigenvector localization and centrality in networks: Revisited, Physica A: Statistical Mechanics and its Applications554, 124169 (2020)

work page 2020

Show all 41 references

[9]

Zhang, Robust spectral detection of global structures in the data by learning a regularization, Advances in Neural Information Processing Systems29(2016)

P. Zhang, Robust spectral detection of global structures in the data by learning a regularization, Advances in Neural Information Processing Systems29(2016)

2016
[10]

D. F. Gleich and M. W. Mahoney, Using local spectral methods to robustify graph-based learning algorithms, inProceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (2015) pp. 359–368

2015
[11]

Pradhan and S

P. Pradhan and S. Jalan, From spectra to localized networks: A reverse engineering approach, IEEE Transactions on Network Science and Engineering7, 3008 (2020)

2020
[12]

Vespignani, Modelling dynamical processes in complex socio-technical systems, Nature physics8, 32 (2012)

A. Vespignani, Modelling dynamical processes in complex socio-technical systems, Nature physics8, 32 (2012)

2012
[13]

Eubank, H

S. Eubank, H. Guclu, V. Anil Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai, and N. Wang, Modelling disease outbreaks in realistic urban social networks, Nature429, 180 (2004)

2004
[14]

I. M. Longini Jr, A. Nizam, S. Xu, K. Ungchusak, W. Hanshaoworakul, D. A. Cummings, and M. E. Halloran, Containing pandemic influenza at the source, Science309, 1083 (2005)

2005
[15]

H. Tian, S. Hu, B. Cazelles, G. Chowell, L. Gao, M. Laine, Y. Li, H. Yang, Y. Li, Q. Yang,et al., Urbanization prolongs hantavirus epidemics in cities, Proceedings of the National Academy of Sciences115, 4707 (2018)

2018
[16]

B. D. Dalziel, S. Kissler, J. R. Gog, C. Viboud, O. N. Bjørnstad, C. J. E. Metcalf, and B. T. Grenfell, Urbanization and humidity shape the intensity of influenza epidemics in us cities, Science362, 75 (2018)

2018
[17]

Jalan and P

S. Jalan and P. Pradhan, Wheel graph strategy for pev localization of networks, Europhysics Letters129, 46002 (2020)

2020
[18]

S. H. Strogatz,Nonlinear dynamics and chaos: with ap- plications to physics, biology, chemistry, and engineering (CRC press, 2018)

2018
[19]

Aguirre, D.Papo, and J.M

J. Aguirre, D.Papo, and J.M. Buldú, Successful strategies for competing networks, Nature Physics9, 230 (2013)

2013
[20]

Pradhan, A

P. Pradhan, A. Yadav, S. K. Dwivedi, and S. Jalan, Optimized evolution of networks for principal eigenvector localization, Physical Review E96, 022312 (2017)

2017
[21]

Van Mieghem,Graph spectra for complex networks (Cambridge university press, 2023)

P. Van Mieghem,Graph spectra for complex networks (Cambridge university press, 2023)

2023
[22]

Pradhan and S

P. Pradhan and S. Jalan, Network construction: A learning framework through localizing principal eigenvector, arXiv preprint arXiv:1802.00202 (2018)

2018 arXiv
[23]

L. V. Tran, V. H. Vu, and K. Wang, Sparse random graphs: Eigenvalues and eigenvectors, Random Structures & Algorithms42, 110 (2013)

2013
[24]

A. V. Goltsev, S. N. Dorogovtsev, J. G. Oliveira, and J. F. Mendes, Localization and spreading of diseases in complex networks, Physical review letters109, 128702 (2012)

2012
[25]

W. L. Hamilton,Graph representation learning(Morgan & Claypool Publishers, 2020)

2020
[26]

Ivanov, S

S. Ivanov, S. Sviridov, and E. Burnaev, Understanding iso- morphism bias in graph data sets (2019), arXiv:1910.12091 [cs.LG]

2019 arXiv
[27]

Morris, N

C. Morris, N. M. Kriege, F. Bause, K. Kersting, P. Mutzel, and M. Neumann, Tudataset: A collection of benchmark datasets for learning with graphs, arXiv preprint arXiv:2007.08663 (2020)

2020 arXiv
[28]

Glorot and Y

X. Glorot and Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, in Proceedings of the thirteenth international conference on artificial intelligence and statistics(JMLR Workshop and Conference Proceedings, 2010) pp. 249–256

2010
[29]

Veličković, G

P. Veličković, G. Cucurull, A. Casanova, A. Romero, P. Lio, and Y. Bengio, Graph attention networks, arXiv preprint arXiv:1710.10903 (2017)

2017 arXiv
[30]

T. N. Kipf and M. Welling, Semi-supervised classification with graph convolutional net- works, arXiv preprint arXiv:1609.02907 https://doi.org/10.48550/arXiv.1609.02907 (2016)

2016 doi arXiv
[31]

J. Zhu, Y. Yan, L. Zhao, M. Heimann, L. Akoglu, and D. Koutra, Beyond homophily in graph neural networks: Current limitations and effective designs, Advances in neural information processing systems33, 7793 (2020)

2020
[32]

Z. Wu, S. Pan, F. Chen, G. Long, C. Zhang, and P. S. Yu, A comprehensive survey on graph neural networks, IEEE transactions on neural networks and learning systems32, 4 (2020)

2020
[33]

Akiba, S

T. Akiba, S. Sano, T. Yanase, T. Ohta, and M. Koyama, Optuna: A next-generation hyperparameter optimization framework, inProceedings of the 25th ACM SIGKDD international conference on knowledge discovery & data mining(2019) pp. 2623–2631

2019
[34]

Hsiao, R

Y.-C. Hsiao, R. Yue, and A. Dutta, Derivation of back-propagation for graph convolutional networks using matrix calculus and its application to explainable artificial intelligence, arXiv preprint arXiv:2408.01408 (2024)

2024 arXiv
[35]

F. L. Metz and I. Neri, Localization and universality of eigenvectors in directed random graphs, Physical Review Letters126, 040604 (2021)

2021
[36]

Zhang, Z

M. Zhang, Z. Cui, M. Neumann, and Y. Chen, An end-to- end deep learning architecture for graph classification, inProceedings of the AAAI conference on artificial intelligence, Vol. 32 (2018)

2018
[37]

Pham, Graph convolutional networks (gcn), TOP- BOTS https://www.topbots.com/graph-convolutional- networks/ (2020)

C. Pham, Graph convolutional networks (gcn), TOP- BOTS https://www.topbots.com/graph-convolutional- networks/ (2020). 14

2020
[38]

Fey and J

M. Fey and J. E. Lenssen, Fast graph representation learning with PyTorch Geometric, inICLR Workshop on Representation Learning on Graphs and Manifolds(2019)

2019
[39]

Pósfai and A.-L

M. Pósfai and A.-L. Barabási,Network science(Citeseer, 2016)

2016
[40]

Dall and M

J. Dall and M. Christensen, Random geometric graphs, Physical review E66, 016121 (2002)

2002
[41]

Anand and G

K. Anand and G. Bianconi, Entropy measures for networks: Toward an information theory of complex topologies, Physical Review E80, 045102 (2009). Appendix A: Linear Dynamics We can write Eq. (1) in matrix form as dx(t) dt =Mx(t)(A1) whereMis a transition matrix given byM= αI+ β...

2009

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

[1] [1]

S. H. Strogatz, Exploring complex networks, nature410, 268 (2001)

work page 2001

[2] [2]

D. S. Bassett, N. F. Wymbs, M. A. Porter, P. J. Mucha, 13 J. M. Carlson, and S. T. Grafton, Dynamic reconfiguration of human brain networks during learning, Proceedings of the National Academy of Sciences108, 7641 (2011)

work page 2011

[3] [3]

H. J. Kimble, The quantum internet, Nature453, 1023 (2008)

2008

[4] [4]

Z. Yang, F. Liu, Z. Gao, H. Sun, J. Zhao, D. Janssens, and G. Wets, Estimating the influence of disruption on highway networks using gps data, Expert Systems with Applications187, 115994 (2022)

work page 2022

[5] [5]

Multilevel Digital Contact Tracing

G. Mahapatra, P. Pradhan, A. Khan, S. K. Setua, R. K. Pal, and A. Rathor, Multilevel digital contact tracing, arXiv preprint arXiv:2007.05637 (2020)

work page Pith review arXiv 2020

[6] [6]

Filoche and S

M. Filoche and S. Mayboroda, Universal mechanism for anderson and weak localization, Proceedings of the National Academy of Sciences109, 14761 (2012)

work page 2012

[7] [7]

Elsner, V

U. Elsner, V. Mehrmann, F. Milde, R. A. Römer, and M. Schreiber, The anderson model of localization: a challenge for modern eigenvalue methods, SIAM Journal on Scientific Computing20, 2089 (1999)

work page 1999

[8] [8]

Pradhan, C

P. Pradhan, C. Angeliya, and S. Jalan, Principal eigenvector localization and centrality in networks: Revisited, Physica A: Statistical Mechanics and its Applications554, 124169 (2020)

work page 2020