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arxiv: 2605.23708 · v1 · pith:VOO3ETYTnew · submitted 2026-05-22 · 💻 cs.LG · cs.SY· eess.SY· nlin.AO

Learning Dynamic Stability Landscapes in Synchronization Networks

Christian Nauck , Junyou Zhu , Michael Lindner , Frank Hellmann This is my paper

Pith reviewed 2026-05-25 05:02 UTC · model grok-4.3

classification 💻 cs.LG cs.SYeess.SYnlin.AO
keywords synchronization networksstability landscapesgraph neural networksgraph-to-image predictionpower gridsdynamical systemsnetwork robustness
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The pith

Stability landscapes in synchronization networks can be learned directly from graph topology via a graph-to-image task.

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

The paper sets out to establish that detailed per-node stability landscapes, which capture how synchronization robustness varies across a network, are predictable from topology alone even though conventional network science cannot reach them. A sympathetic reader would care because these landscapes contain more information than the usual scalar stability indices and allow multiple such indices to be derived from a single learned output. The authors support the claim by releasing two large graph datasets with landscape labels generated from an oscillator model, then training an end-to-end GNN encoder plus CNN decoder that achieves good in-distribution accuracy and generalizes across graph sizes and to realistic power-grid topologies.

Core claim

We pioneer a graph-to-image prediction paradigm in which a GNN encodes network topology and a CNN decoder renders per-node image-like stability landscapes; trained on 10,000-graph datasets at 20 and 100 nodes generated from a conceptual oscillator model, the model attains good accuracy within distribution, generalizes across sizes, and transfers to realistic power-grid topologies, showing that stability landscapes are learnable from topology.

What carries the argument

The graph-to-image prediction paradigm, in which a GNN encodes topology and a CNN decoder produces per-node landscape images as targets.

If this is right

  • Many scalar per-node stability indices can be derived from the learned landscape images.
  • The approach generalizes across graph sizes from 20 to 100 nodes.
  • The model transfers to realistic power-grid topologies outside the training distribution.
  • The method opens avenues for moving beyond scalar indices in biology, neuroscience, and power grids.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the learned landscapes prove reliable on real grids, operators could use them to identify nodes whose stability is most sensitive to specific topological changes.
  • The same graph-to-image framing might be tested on other dynamical properties such as transient response or basin structure in networked oscillators.
  • Topology optimization routines could be coupled to the model to search for graphs that produce desired landscape shapes rather than single scalar targets.

Load-bearing premise

The conceptual oscillator model used to generate the per-node landscape labels accurately represents the synchronization dynamics of interest in real power grids and other systems.

What would settle it

Direct numerical simulation of the oscillator dynamics on a held-out realistic power-grid topology, followed by comparison of the resulting per-node stability landscapes against those predicted by the trained GNN-CNN model.

Figures

Figures reproduced from arXiv: 2605.23708 by Christian Nauck, Frank Hellmann, Junyou Zhu, Michael Lindner.

Figure 1
Figure 1. Figure 1: Top: Contour plot of the asymptotic deviation from synchrony. Bottom: its Monte Carlo origin landscape for 10,000 perturbations at the same node i. The axes represent the pertur￾bations in p = {ϕ, ϕ˙}. The color encodes the maximum final frequency deviation from the set point, with darker colors indicat￾ing more stable regions. SNBSi(G) = Z BS(G, i, p) ρ(p) dp, (1) where BS(G, i, p) ∈ {0, 1} indicates whet… view at source ↗
Figure 2
Figure 2. Figure 2: Examples of basin landscapes, original MC samples (top) and derived Basin Landscape with 20x20 grid cells (bottom). Unstable = [0,1]N Y SNBS  Generation of Synthetic Grids (a) Previous task and datasets Unstable = [0,1]N Y SNBS  Generation of Synthetic (a) Non-ML Physics Methods (c) Proposed New Task & Dataset Stability=? ? ? ? ? Probabilistic Stability = ? ? ? ? Stable Unstable 0.95 1.0 0.32 0.85 0.63 R… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of stability-assessment workflows. (a) Traditional physics-based methods rely on costly Monte Carlo simulations to evaluate single-node basin stability, requiring large amounts of CPU hours. (b) Prior ML approaches (Nauck et al., 2022b;a; 2023; Zhu et al., 2026) replace simulation with a trained GNN that predicts a probabilistic stability value per node in seconds, but sacrifices spatial detail.… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of true labels and predicted heatmaps for the TAG-MLP model trained and evaluated on dataset20 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visualization of critical contingencies (true labels on the top and predictions on the bottom) of a node from dataset20. The predictions are from TAG-MLP. 8 [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Histograms of the downstream task SNBS. The top panel corresponds to the synthetic topologies: dataset20, dataset100 and Texas. The bottom represents the real-world topologies. Figures from (Nauck et al., 2024c) [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Distribution of final frequencies for the first 100 grids of dataset20. The linear histogram is limited to samples withf > 0.1. The legend probabilities show the fractions of samples with f > 0.1 and 0.1 < f < 2.5, confirming that these results do not affect the reported statistics. The red dashed vertical line marks the synchronization threshold at f = 0.1. 0 or 1 will have almost no dependence on the wit… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of true labels and predicted heatmaps for the TAG-MLP model trained and evaluated on dataset100 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of true labels and predicted heatmaps for the TAG-MLP model trained on dataset20 and evaluated on dataset100. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Visualization of critical contingencies for node 16 in the first grid of the test set from dataset20. True labels are shown on the top, and predictions are shown on the bottom. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the grid size of the heatmaps with the following number per axis: 5, 10, 20, 30. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Training curves of DBGNN-MLP and TAG-MLP show the smoothness of TAG, which may show the strong out-of-distribution generalization. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_12.png] view at source ↗
read the original abstract

The robustness of synchronization is typically characterized by scalar, per-node stability indices whose dependence on topology is studied via network science or graph neural networks (GNNs). We propose a novel upstream task, learning stability landscapes, which provide deeper insights into synchronization behavior and from which many such scalar indices can be derived. Crucially, we pioneer a graph-to-image prediction paradigm: learning image-like landscapes as per-node targets directly from graph topology, a formulation we are not aware of having been established elsewhere in the literature. To support this task, we release two datasets of 10,000 graphs each at 20 and 100 nodes with per-node landscape labels, based on a conceptual oscillator model, capturing power grid synchronization behavior. A GNN encodes topology and a CNN decoder renders per-node images, learned end-to-end with good in-distribution accuracy, generalizing across graph sizes and to realistic power grid topologies. This demonstrates that stability landscapes, while beyond the reach of conventional network science, are learnable from topology and open new avenues for moving beyond scalar stability indices in biology, neuroscience, and power grids.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes a graph-to-image prediction task to learn per-node stability landscapes in synchronization networks directly from topology. A GNN encodes the graph and a CNN decoder produces image-like per-node landscape targets, trained end-to-end on two released datasets of 10,000 graphs each (20-node and 100-node) whose labels are generated by a conceptual oscillator model claimed to capture power-grid synchronization. The approach is reported to achieve good in-distribution accuracy, to generalize across graph sizes, and to extend to realistic power-grid topologies, thereby enabling derivation of scalar stability indices and providing deeper insight than conventional network-science methods.

Significance. If the empirical results and model assumptions hold, the work establishes a new upstream task that moves synchronization analysis beyond scalar per-node indices and demonstrates that topology-to-landscape mappings are learnable. The public release of the two 10,000-graph datasets constitutes a concrete, reusable contribution. The significance is limited, however, by the absence of any reported validation that the conceptual oscillator labels match the structural properties of established higher-fidelity models (swing equation, Kuramoto with measured parameters) on benchmark grids.

major comments (2)
  1. [Abstract, dataset construction paragraph] Abstract, dataset-construction paragraph: the claim that the datasets 'capture power grid synchronization behavior' and that the method generalizes 'to realistic power grid topologies' rests on an unverified assumption that the conceptual oscillator model produces landscapes structurally consistent with swing-equation or Kuramoto dynamics. No comparison to measured-parameter Kuramoto, no sensitivity analysis to coupling form or damping, and no quantitative match to known stability indices on benchmark grids are supplied. This assumption is load-bearing for the central claim that the learned mapping addresses the synchronization phenomenon of interest rather than an artifact of the label generator.
  2. [Abstract] Abstract: the assertion of 'good in-distribution accuracy' and generalization is presented without any quantitative metrics, baselines, error bars, ablation studies, or cross-validation details. Because the soundness of the empirical demonstration cannot be assessed from the supplied information, the strength of the central learnability result remains unevaluable.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below, indicating planned revisions where the manuscript requires clarification or additional discussion.

read point-by-point responses

  1. Referee: [Abstract, dataset construction paragraph] Abstract, dataset-construction paragraph: the claim that the datasets 'capture power grid synchronization behavior' and that the method generalizes 'to realistic power grid topologies' rests on an unverified assumption that the conceptual oscillator model produces landscapes structurally consistent with swing-equation or Kuramoto dynamics. No comparison to measured-parameter Kuramoto, no sensitivity analysis to coupling form or damping, and no quantitative match to known stability indices on benchmark grids are supplied. This assumption is load-bearing for the central claim that the learned mapping addresses the synchronization phenomenon of interest rather than an artifact of the label generator.

    Authors: We acknowledge that the manuscript relies on a conceptual oscillator model for label generation without providing direct comparisons to the swing equation or Kuramoto models fitted to measured parameters on benchmark grids, nor sensitivity analyses. The model was chosen to enable scalable dataset creation while capturing core synchronization features. We will add an explicit limitations section stating that all results are with respect to this conceptual model, revise phrasing in the abstract and dataset section to avoid implying direct equivalence to higher-fidelity power-grid models, and note that empirical validation against established models remains future work. revision: partial

  2. Referee: [Abstract] Abstract: the assertion of 'good in-distribution accuracy' and generalization is presented without any quantitative metrics, baselines, error bars, ablation studies, or cross-validation details. Because the soundness of the empirical demonstration cannot be assessed from the supplied information, the strength of the central learnability result remains unevaluable.

    Authors: We agree the abstract is too high-level. The body of the manuscript reports specific metrics (MSE on predicted landscapes, derived scalar index accuracy), comparisons against baselines (e.g., topology-only predictors and non-CNN decoders), error bars across random seeds, and cross-size / cross-topology generalization results with the described train/validation splits. We will revise the abstract to include the key quantitative figures and a brief statement of the evaluation protocol. revision: yes

Circularity Check

0 steps flagged

No circularity; empirical ML task on externally generated labels

full rationale

The paper defines an upstream ML task of predicting per-node stability landscape images from graph topology using a GNN encoder and CNN decoder, trained end-to-end on two synthetic datasets of 10,000 graphs each. Labels are produced by a separate conceptual oscillator model whose outputs serve as fixed targets; the learned mapping is evaluated by in-distribution accuracy and generalization, with no equations or claims that reduce a derived quantity to a fitted parameter or self-citation by construction. No self-definitional steps, fitted-input predictions, or load-bearing self-citations appear in the provided text. The result is a standard supervised prediction experiment whose validity rests on the external label generator rather than internal tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper depends on a conceptual oscillator model to produce labels; no explicit free parameters or invented entities are described in the abstract.

axioms (1)
  • domain assumption The conceptual oscillator model captures relevant power-grid synchronization behavior for generating landscape labels
    Used to create the two datasets and per-node targets.

pith-pipeline@v0.9.0 · 5731 in / 1104 out tokens · 23798 ms · 2026-05-25T05:02:22.918354+00:00 · methodology

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Reference graph

Works this paper leans on

294 extracted references · 294 canonical work pages · 12 internal anchors

  1. [1]

    Need to Develop Grid-Forming Statcom Systems -

    50Hertz and Amprion and Tennet and Transnet BW , year = 2020, url =. Need to Develop Grid-Forming Statcom Systems -

    work page 2020

  2. [2]

    Probabilistic

    Abujubbeh, Mohammad and Munikoti, Sai and Pahwa, Anil and Natarajan, Balasubramaniam , year = 2023, month = may, journal =. Probabilistic. doi:10.1109/TPWRS.2022.3186349 , urldate =

    work page doi:10.1109/tpwrs.2022.3186349 2023

  3. [3]

    Acebr. The. Reviews of Modern Physics , volume =. doi:10.1103/RevModPhys.77.137 , urldate =

    work page doi:10.1103/revmodphys.77.137

  4. [4]

    Physical Review

    Synchronization in Populations of Globally Coupled Oscillators with Inertial Effects , author =. Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics , volume =. doi:10.1103/physreve.62.3437 , abstract =

    work page doi:10.1103/physreve.62.3437

  5. [5]

    Hypocoercivity in

    Achleitner, Franz and Arnold, Anton and Mehrmann, Volker , year = 2023, month = dec, journal =. Hypocoercivity in. doi:10.1007/s10884-023-10327-6 , urldate =

    work page doi:10.1007/s10884-023-10327-6 2023

  6. [6]

    Climate Economics:

    Ackerman, Frank and Stanton, Elizabeth , year = 2013, publisher =. Climate Economics:

    work page 2013

  7. [7]

    Distributed Generation: A Definition1 , shorttitle =

    Ackermann, Thomas and Andersson, G. Distributed Generation: A Definition1 , shorttitle =. Electric Power Systems Research , volume =. doi:10.1016/S0378-7796(01)00101-8 , urldate =

    work page doi:10.1016/s0378-7796(01)00101-8

  8. [8]

    Design of

    Adachi, Ryosuke and Yamashita, Yuh and Kobayashi, Koichi , year = 2020, month = jan, journal =. Design of. doi:10.1016/j.ifacol.2020.12.1496 , urldate =

    work page doi:10.1016/j.ifacol.2020.12.1496 2020

  9. [9]

    AIP Publishing LLC , url =

    Advances in. AIP Publishing LLC , url =

    work page

  10. [10]

    , year = 2012, month = mar, journal =

    Agresti, Alan and Coull, Brent A. , year = 2012, month = mar, journal =. Approximate Is

    work page 2012

  11. [11]

    Ainsworth, Nathan and Grijalva, Santiago , year = 2013, month = nov, journal =. A. doi:10.1109/TPWRS.2013.2257887 , urldate =

    work page doi:10.1109/tpwrs.2013.2257887 2013

  12. [12]

    Balanced

    Aka, Julius and Brunnemann, Johannes and Eiden, J. Balanced. doi:10.48550/arXiv.2410.10174 , urldate =. arXiv , keywords =:2410.10174 , primaryclass =

    work page doi:10.48550/arxiv.2410.10174

  13. [13]

    Efficient

    Aka, Julius and Brunnemann, Johannes and Freund, Svenne and Speerforck, Arne , year = 2023, month = dec, journal =. Efficient. doi:10.3384/ecp204107 , urldate =

    work page doi:10.3384/ecp204107 2023

  14. [14]

    Akiba, Takuya and Sano, Shotaro and Yanase, Toshihiko and Ohta, Takeru and Koyama, Masanori , year = 2019, month = jul, number =. Optuna:. doi:10.48550/arXiv.1907.10902 , urldate =. arXiv , keywords =:1907.10902 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1907.10902 2019

  15. [15]

    Energy , volume =

    A New Flexible Model for Generation Scheduling in a Smart Grid , author =. Energy , volume =. doi:10.1016/j.energy.2019.116438 , urldate =

    work page doi:10.1016/j.energy.2019.116438 2019

  16. [16]

    Nature Climate Change , volume =

    Decentralized Energy Systems for Clean Electricity Access , author =. Nature Climate Change , volume =. doi:10.1038/nclimate2512 , urldate =

    work page doi:10.1038/nclimate2512

  17. [17]

    Anbar, A. D. and Romaniello, S. J. and Allenby, B. R. and Broecker, W. S. , year = 2016, journal =. Addressing the

    work page 2016

  18. [18]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =

    Multiscale Dynamics in Communities of Phase Oscillators , author =. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =. doi:10.1063/1.3672513 , urldate =

    work page doi:10.1063/1.3672513

  19. [19]

    The European Physical Journal Special Topics , volume =

    Moving the Epidemic Tipping Point through Topologically Targeted Social Distancing , author =. The European Physical Journal Special Topics , volume =. doi:10.1140/epjs/s11734-021-00138-5 , urldate =

    work page doi:10.1140/epjs/s11734-021-00138-5

  20. [20]

    Nature Communications , volume =

    Data-Driven Load Profiles and the Dynamics of Residential Electricity Consumption , author =. Nature Communications , volume =. doi:10.1038/s41467-022-31942-9 , urldate =

    work page doi:10.1038/s41467-022-31942-9

  21. [21]

    Erratum: ``

    Anvari, Mehrnaz and Hellmann, Frank and Zhang, Xiaozhu , year = 2021, month = apr, journal =. Erratum: ``. doi:10.1063/5.0052803 , urldate =

    work page doi:10.1063/5.0052803 2021

  22. [22]

    Introduction to

    Anvari, Mehrnaz and Hellmann, Frank and Zhang, Xiaozhu , year = 2020, month = jun, journal =. Introduction to. doi:10.1063/5.0016372 , urldate =

    work page doi:10.1063/5.0016372 2020

  23. [23]

    Physics Reports , volume =

    Synchronization in Complex Networks , author =. Physics Reports , volume =. doi:10.1016/j.physrep.2008.09.002 , urldate =

    work page doi:10.1016/j.physrep.2008.09.002 2008

  24. [24]

    Arghir, Catalin and D. The. IEEE Transactions on Power Electronics , volume =. doi:10.1109/TPEL.2019.2939710 , urldate =

    work page doi:10.1109/tpel.2019.2939710 2019

  25. [25]

    Linear Algebra and its Applications , volume =

    On Sectorial Matrices , author =. Linear Algebra and its Applications , volume =. doi:10.1016/S0024-3795(03)00388-4 , urldate =

    work page doi:10.1016/s0024-3795(03)00388-4

  26. [26]

    Communications Physics , volume =

    Emergence of Explosive Synchronization Bombs in Networks of Oscillators , author =. Communications Physics , volume =. doi:10.1038/s42005-022-01039-2 , urldate =

    work page doi:10.1038/s42005-022-01039-2

  27. [27]

    Review of Existing Literature on Methodologies to Model Non-Linearity, Thresholds and Irreversibility in High-Impact Climate Change Events in the Presence of Environmental Tipping Points , author =

    work page

  28. [28]

    Nature Machine Intelligence , volume =

    Geometric Deep Learning on Molecular Representations , author =. Nature Machine Intelligence , volume =. doi:10.1038/s42256-021-00418-8 , urldate =

    work page doi:10.1038/s42256-021-00418-8

  29. [29]

    The European Physical Journal Special Topics , volume =

    The Impact of Model Detail on Power Grid Resilience Measures , author =. The European Physical Journal Special Topics , volume =. doi:10.1140/epjst/e2015-50265-9 , urldate =

    work page doi:10.1140/epjst/e2015-50265-9

  30. [30]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =

    Stability of Synchrony against Local Intermittent Fluctuations in Tree-like Power Grids , author =. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =. doi:10.1063/1.5001818 , urldate =

    work page doi:10.1063/1.5001818

  31. [31]

    Multitask

    Avelar, Pedro and Lemos, Henrique and Prates, Marcelo and Lamb, Luis , editor =. Multitask. Artificial. doi:10.1007/978-3-030-30493-5_63 , abstract =

    work page doi:10.1007/978-3-030-30493-5_63

  32. [32]

    Interaction of two systems with saddle-node bifurcations on invariant circles. I. Foundations and the mutualistic case

    Baesens, Claude and MacKay, Robert S. , year = 2013, month = dec, journal =. Interaction of Two Systems with Saddle-Node Bifurcations on Invariant Circles. doi:10.1088/0951-7715/26/12/3043 , urldate =. arXiv , langid =:1309.6954 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0951-7715/26/12/3043 2013

  33. [33]

    Science Advances , volume =

    Structure Motif--Centric Learning Framework for Inorganic Crystalline Systems , author =. Science Advances , volume =. doi:10.1126/sciadv.abf1754 , urldate =

    work page doi:10.1126/sciadv.abf1754

  34. [34]

    Linear and Multilinear Algebra , volume =

    Generalized Matrix Tree Theorem for Mixed Graphs , author =. Linear and Multilinear Algebra , volume =. doi:10.1080/03081089908818623 , urldate =

    work page doi:10.1080/03081089908818623

  35. [35]

    Network Science , author =

    work page

  36. [36]

    Disciplinarity, Epistemic Friction, and the `

    Barber, Jacob , year = 2018, publisher =. Disciplinarity, Epistemic Friction, and the `

    work page 2018

  37. [37]

    Toward a

    Bardi, Ugo and Falsini, Sara and Perissi, Ilaria , year = 2019, month = mar, journal =. Toward a. doi:10.1007/s41247-018-0049-0 , urldate =

    work page doi:10.1007/s41247-018-0049-0 2019

  38. [38]

    Bardi, Ugo , year = 2013, month = mar, journal =. Mind. doi:10.3390/su5030896 , urldate =

    work page doi:10.3390/su5030896 2013

  39. [39]

    doi:10.48550/arXiv.2603.13922 , urldate =

    On the. doi:10.48550/arXiv.2603.13922 , urldate =. arXiv , keywords =:2603.13922 , primaryclass =

    work page doi:10.48550/arxiv.2603.13922

  40. [40]

    doi:10.48550/arXiv.2503.13367 , urldate =

    Mixed. doi:10.48550/arXiv.2503.13367 , urldate =. arXiv , keywords =:2503.13367 , primaryclass =

    work page doi:10.48550/arxiv.2503.13367

  41. [41]

    Dynamical Processes on Complex Networks , author =

    work page

  42. [42]

    Relational inductive biases, deep learning, and graph networks

    Relational Inductive Biases, Deep Learning, and Graph Networks , author =. doi:10.48550/arXiv.1806.01261 , urldate =. arXiv , keywords =:1806.01261 , primaryclass =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1806.01261

  43. [43]

    Networks beyond Pairwise Interactions:

    Battiston, Federico and Cencetti, Giulia and Iacopini, Iacopo and Latora, Vito and Lucas, Maxime and Patania, Alice and Young, Jean-Gabriel and Petri, Giovanni , year = 2020, month = aug, journal =. Networks beyond Pairwise Interactions:. doi:10.1016/j.physrep.2020.05.004 , urldate =

    work page doi:10.1016/j.physrep.2020.05.004 2020

  44. [44]

    Normalized Graph

    Bauer, Frank , year = 2012, month = jun, journal =. Normalized Graph. doi:10.1016/j.laa.2012.01.020 , urldate =

    work page doi:10.1016/j.laa.2012.01.020 2012

  45. [45]

    Bechhoefer, John , year = 2021, month = feb, edition =. Control. doi:10.1017/9780511734809 , urldate =

    work page doi:10.1017/9780511734809 2021

  46. [46]

    Power Flow Forecasts at Transmission Grid Nodes Using

    Beinert, Dominik and Holzh. Power Flow Forecasts at Transmission Grid Nodes Using. Energy and AI , volume =. doi:10.1016/j.egyai.2023.100262 , urldate =

    work page doi:10.1016/j.egyai.2023.100262 2023

  47. [47]

    Electric Power Systems Research , volume =

    A Causality Based Feature Selection Approach for Data-Driven Dynamic Security Assessment , author =. Electric Power Systems Research , volume =. doi:10.1016/j.epsr.2021.107537 , urldate =

    work page doi:10.1016/j.epsr.2021.107537 2021

  48. [48]

    Chaos , volume =

    Synchronization in Asymmetrically Coupled Networks with Node Balance , author =. Chaos , volume =. doi:10.1063/1.2146180 , abstract =

    work page doi:10.1063/1.2146180

  49. [49]

    Comparing

    Bento, Nuno and Rebelo, Joana and Barandas, Mar. Comparing. Sensors (Basel, Switzerland) , volume =. doi:10.3390/s22197324 , urldate =

    work page doi:10.3390/s22197324

  50. [50]

    Saturated

    Benzaouia, Abdellah , year = 2012, series =. Saturated. doi:10.1007/978-1-4471-2900-4 , urldate =

    work page doi:10.1007/978-1-4471-2900-4 2012

  51. [51]

    and Hill, D.J

    Bergen, A.R. and Hill, D.J. , year = 1981, month = jan, journal =. A. doi:10.1109/TPAS.1981.316883 , abstract =

    work page doi:10.1109/tpas.1981.316883 1981

  52. [52]

    Improving Voltage and Frequency Control of

    Bernal, Rodrigo and Milano, Federico , year = 2024, month = oct, journal =. Improving Voltage and Frequency Control of. doi:10.1016/j.epsr.2024.110768 , urldate =

    work page doi:10.1016/j.epsr.2024.110768 2024

  53. [53]

    Adaptive

    Berner, Rico and Gross, Thilo and Kuehn, Christian and Kurths, J. Adaptive. doi:10.48550/arXiv.2304.05652 , urldate =. arXiv , keywords =:2304.05652 , primaryclass =

    work page doi:10.48550/arxiv.2304.05652

  54. [54]

    Physical Review Letters , volume =

    Desynchronization Transitions in Adaptive Networks , author =. Physical Review Letters , volume =

    work page

  55. [55]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =

    Generalized Splay States in Phase Oscillator Networks , author =. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =. doi:10.1063/5.0056664 , urldate =

    work page doi:10.1063/5.0056664

  56. [56]

    The European Physical Journal Special Topics , volume =

    Solitary States in Adaptive Nonlocal Oscillator Networks , author =. The European Physical Journal Special Topics , volume =. doi:10.1140/epjst/e2020-900253-0 , urldate =

    work page doi:10.1140/epjst/e2020-900253-0

  57. [57]

    Frontiers in Applied Mathematics and Statistics , volume =

    Synchronization in Networks with Heterogeneous Adaptation Rules and Applications to Distance-Dependent Synaptic Plasticity , author =. Frontiers in Applied Mathematics and Statistics , volume =

    work page

  58. [58]

    Physical Review E , volume =

    What Adaptive Neuronal Networks Teach Us about Power Grids , author =. Physical Review E , volume =. doi:10.1103/PhysRevE.103.042315 , urldate =

    work page doi:10.1103/physreve.103.042315

  59. [59]

    Effective

    Besard, Tim and Foket, Christophe and De Sutter, Bjorn , year = 2019, month = apr, journal =. Effective. doi:10.1109/TPDS.2018.2872064 , abstract =

    work page doi:10.1109/tpds.2018.2872064 2019

  60. [60]

    Bezanson , author A

    Bezanson, Jeff and Edelman, Alan and Karpinski, Stefan and Shah, Viral B. , year = 2017, journal =. Julia:. doi:10.1137/141000671 , abstract =

    work page doi:10.1137/141000671 2017

  61. [61]

    Bianchi, Filippo Maria and Grattarola, Daniele and Livi, Lorenzo and Alippi, Cesare , year = 2021, month = jan, journal =. Graph. doi:10.1109/TPAMI.2021.3054830 , urldate =

    work page doi:10.1109/tpami.2021.3054830 2021

  62. [62]

    Pyramidal

    Bianchi, Filippo Maria and Gallicchio, Claudio and Micheli, Alessio , year = 2021, month = apr, number =. Pyramidal. doi:10.48550/arXiv.2104.04710 , urldate =. arXiv , langid =:2104.04710 , primaryclass =

    work page doi:10.48550/arxiv.2104.04710 2021

  63. [63]

    The Topological

    Bianconi, Ginestra , year = 2021, month = sep, journal =. The Topological. doi:10.1088/2632-072X/ac19be , urldate =

    work page doi:10.1088/2632-072x/ac19be 2021

  64. [64]

    and Schaub, Michael T

    Bick, Christian and Gross, Elizabeth and Harrington, Heather A. and Schaub, Michael T. , year = 2023, month = aug, journal =. What. doi:10.1137/21M1414024 , urldate =

    work page doi:10.1137/21m1414024 2023

  65. [65]

    Wicked Facets of the

    Biehl, Juliane and Missbach, Leonard and Riedel, Franziska and Stemmle, Ruben and J. Wicked Facets of the. International Journal of Sustainable Energy , volume =. doi:10.1080/14786451.2023.2244602 , urldate =

    work page doi:10.1080/14786451.2023.2244602 2023

  66. [66]

    and Ceriotti, Michele , year = 2023, month = mar, number =

    Bigi, Filippo and Pozdnyakov, Sergey N. and Ceriotti, Michele , year = 2023, month = mar, number =. Wigner Kernels: Body-Ordered Equivariant Machine Learning without a Basis , shorttitle =. arXiv , langid =:2303.04124 , primaryclass =

    work page arXiv 2023

  67. [67]

    Reliability

    Billinton, Roy and Li, Wenyuan , year = 1994, publisher =. Reliability. doi:10.1007/978-1-4899-1346-3 , urldate =

    work page doi:10.1007/978-1-4899-1346-3 1994

  68. [68]

    Birchfield, Adam , year = 2021, url =

    work page 2021

  69. [69]

    and Xu, Ti and Gegner, Kathleen M

    Birchfield, Adam B. and Xu, Ti and Gegner, Kathleen M. and Shetye, Komal S. and Overbye, Thomas J. , year = 2017, month = jul, journal =. Grid. doi:10.1109/TPWRS.2016.2616385 , abstract =

    work page doi:10.1109/tpwrs.2016.2616385 2017

  70. [70]

    Time and

    Blakely, Derrick and Lanchantin, Jack and Qi, Yanjun , year = 2021, url =. Time and

    work page 2021

  71. [71]

    Complex Networks:

    Boccaletti, S and Latora, V and Moreno, Y and Chavez, M and Hwang, D , year = 2006, month = feb, journal =. Complex Networks:. doi:10.1016/j.physrep.2005.10.009 , urldate =

    work page doi:10.1016/j.physrep.2005.10.009 2006

  72. [72]

    Bodnar, Cristian and Giovanni, Francesco Di , abstract =. Neural

    work page

  73. [73]

    Bodnar, Cristian and Di Giovanni, Francesco and Chamberlain, Benjamin and Li. Neural. Advances in Neural Information Processing Systems , volume =

    work page

  74. [74]

    Bodnar, Cristian and Frasca, Fabrizio and Wang, Yuguang and Otter, Nina and Montufar, Guido F. and Li. Weisfeiler and. Proceedings of the 38th

    work page

  75. [75]

    Weisfeiler and

    Bodnar, Cristian and Frasca, Fabrizio and Wang, Yu Guang and Otter, Nina and Mont. Weisfeiler and. doi:10.48550/arXiv.2103.03212 , urldate =. arXiv , keywords =:2103.03212 , primaryclass =

    work page doi:10.48550/arxiv.2103.03212

  76. [76]

    Bolz, Valentin and Rue. Power. 2019 18th. doi:10.1109/ICMLA.2019.00274 , abstract =

    work page doi:10.1109/icmla.2019.00274 2019

  77. [78]

    Probabilistic

    Borkowska, Barbara , year = 1974, month = may, journal =. Probabilistic. doi:10.1109/TPAS.1974.293973 , urldate =

    work page doi:10.1109/tpas.1974.293973 1974

  78. [79]

    Physical Review Research , volume =

    Delay Master Stability of Inertial Oscillator Networks , author =. Physical Review Research , volume =. doi:10.1103/PhysRevResearch.2.023409 , urldate =

    work page doi:10.1103/physrevresearch.2.023409

  79. [80]

    Dynamic Stability of Electric Power Grids:

    B. Dynamic Stability of Electric Power Grids:. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume =. doi:10.1063/5.0082712 , urldate =

    work page doi:10.1063/5.0082712

  80. [81]

    Stability

    B. Stability. IEEE Access , volume =. doi:10.1109/ACCESS.2023.3320944 , urldate =

    work page doi:10.1109/access.2023.3320944 2023

Showing first 80 references.