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Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis.

Code ↔ Paper

5 matches between paragraphs of the paper and lines of its authors' code, computed by the harvester (lexical-v1). Click a colored paragraph or line to see its counterpart.

The 5 matches
  1. [1] § Methods › Database and neural network architecture › Neural network DIMNN model ↔ scripts/neural_network_training_two_phases.py, lines 23–101 · score 0.71 · hidden layers, AdamW, activations, dropout, optimizer, residual
  2. [2] § Methods › Database and neural network architecture › Neural network DIMNN model ↔ dimnn/neural_network_training_clean.py, lines 64–168 · score 0.69 · hidden layers, AdamW, dropout, optimizer, residual, MLP
  3. [3] § Results › Dimensionality estimation using neural networks ↔ geometric-randomization/notebooks/network-properties-d-mercator.ipynb, lines 194–214 · score 0.65 · clustering spectrum, connected component, neighbor degree, Mercator, properties, networks
  4. [4] § Methods › Multidimensional geometric soft configuration model › Microcanonical formulation of model ↔ SD-model/notebooks/tutorial.ipynb, lines 37–54 · score 0.55 · power law, generated synthetic networks, exponent, nodes, model, dimension
  5. [5] § Methods › Multidimensional geometric soft configuration model › Microcanonical formulation of model ↔ geometric-randomization/generate_synthetic_networks.py, lines 18–61 · score 0.51 · generated synthetic networks, geometric randomization, realizations, dimension

Paper

Loaded from Europe PMC by your browser, not stored by OSCR: doi.org · Europe PMC

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The authors' code

Python · 105 lines · 4.6 KB · no license · 1 match

  1. import pandas as pd
  2. import numpy as np
  3. import matplotlib.pyplot as plt
  4. from sklearn.preprocessing import StandardScaler
  5. import keras
  6. import tensorflow as tf
  7. from keras.callbacks import EarlyStopping
  8. import seaborn as sns
  10. from residual_network import build_residual_mlp
  12. gpus = tf.config.list_physical_devices('GPU')
  13. if gpus:
  14. # Restrict TensorFlow to only use the first GPU
  15. try:
  16. tf.config.set_visible_devices(gpus[0], 'GPU')
  17. logical_gpus = tf.config.list_logical_devices('GPU')
  18. print(len(gpus), "Physical GPUs,", len(logical_gpus), "Logical GPU")
  19. except RuntimeError as e:
  20. # Visible devices must be set before GPUs have been initialized
  21. print(e)
  23. def main():
  24. keras.utils.set_random_seed(42)
  26. data = pd.read_csv('../data/data_for_training_large.csv')
  27. #data2 = pd.read_csv('./data/training_synthetic_networks.csv', index_col=0)
  29. #data = pd.concat([data, data2], ignore_index=True)
  31. # Possible feature vectors
  32. columns = ['num_nodes', 'avg_degree', 'Ct', 'Cs', 'Cp']
  33. columns = ['num_nodes', 'avg_degree', 'tp1_t', 'tp1_s', 'tp1_p']
  34. columns = ['num_nodes', 'avg_degree', 'Ct', 'Cs', 'Cp','tp1_t', 'tp1_s', 'tp1_p']
  35. y_train_phase_1 = data[['gamma', 'beta']]
  36. y_train_phase_2 = data['dim'] - 1 # Ensure 'dim' is zero-indexed
  37. X_train_phase_1 = data[columns]
  38. X_train_phase_2 = data[columns + ['gamma', 'beta']]
  40. # Number of classes of phase 2
  41. N = len(np.unique(y_train_phase_2))
  42. print(f"Number of classes: {N}")
  44. # For test
  45. X_test = pd.read_csv('../data/test_real_networks.csv', index_col=0)
  46. X_test = X_test[columns]
  48. # Normalization
  49. scaler_phase_1 = StandardScaler()
  50. scaler_phase_2 = StandardScaler()
  51. X_train_phase_1 = scaler_phase_1.fit_transform(X_train_phase_1)
  52. X_train_phase_2 = scaler_phase_2.fit_transform(X_train_phase_2)
  54. model_phase_1 = build_residual_mlp(input_dim=len(columns),
  55. hidden_layer_sizes=[32, 64, 64, 128, 256, 512, 1024, 1024, 512,
  56. 256, 128, 64, 64, 64, 32, 32, 16, 16, 16, 8, 8],
  57. output_dim=2, dropout_rate=0.5)
  59. model_phase_2 = build_residual_mlp(input_dim=len(columns) + 2,
  60. hidden_layer_sizes=[32, 64, 64, 128, 256, 512, 1024, 1024, 512,
  61. 256, 128, 64, 64, 64, 32, 32, 16, 16, 16, 8, 8],
  62. output_dim=N, dropout_rate=0.5, output_activation='linear')
  64. optimizer_phase_1 = keras.optimizers.AdamW(learning_rate=0.0005)
  66. model_phase_1.compile(optimizer=optimizer_phase_1, loss='mse', metrics=['mae'])
  67. early_stopping_phase_1 = EarlyStopping(monitor='val_loss', patience=20, restore_best_weights=True)
  68. history = model_phase_1.fit(X_train_phase_1, y_train_phase_1, epochs=200, batch_size=64, validation_split=0.2,
  69. callbacks=[early_stopping_phase_1], verbose=2)
  71. loss, mae = model_phase_1.evaluate(X_train_phase_1, y_train_phase_1)
  72. print(f"Train mae: {mae:.4f}")
  74. model_phase_1.save('base_NN_model_residual_phase_1.keras')
  76. # Train phase 2
  77. optimizer_phase_2 = keras.optimizers.AdamW(learning_rate=0.0005)
  78. model_phase_2.compile(optimizer=optimizer_phase_2, loss='sparse_categorical_crossentropy', metrics=['accuracy'])
  79. early_stopping_phase_2 = EarlyStopping(monitor='val_loss', patience=20, restore_best_weights=True)
  80. history = model_phase_2.fit(X_train_phase_2, y_train_phase_2, epochs=200, batch_size=64, validation_split=0.2,
  81. callbacks=[early_stopping_phase_2], verbose=2)
  82. loss, accuracy = model_phase_2.evaluate(X_train_phase_2, y_train_phase_2)
  83. print(f"Train accuracy: {accuracy:.4f}")
  85. model_phase_2.save('base_NN_model_residual_phase_2.keras')
  87. # Perform predictions
  88. X_test_phase_1 = scaler_phase_1.transform(X_test)
  89. predictions_phase_1 = model_phase_1.predict(X_test_phase_1)
  90. # Concatenate predictions with previous X_test on columns
  91. X_test_phase_2 = pd.concat([X_test, pd.DataFrame(predictions_phase_1, columns=['gamma', 'beta'])], axis=1)
  92. X_test_phase_2 = scaler_phase_2.transform(X_test_phase_2)
  93. predictions_phase_2 = model_phase_2.predict(X_test_phase_2)
  94. y_hat = np.argmax(predictions_phase_2, axis=1)
  95. y_hat = y_hat + 1
  97. test_networks = pd.read_csv('../data/test_real_networks.csv', index_col=0)
  98. for i, name in enumerate(test_networks['name']):
  99. print("Network "+name)
  100. print("Inferred dimension: "+str(y_hat[i]))
  101. print("Probability: "+format(predictions_phase_2[i][y_hat[i]-1], '.4f'))
  104. if __name__ == '__main__':
  105. main()

neural_network_training_two_phases.py at commit 870c695, no license · at the source

Overview

Authors: Aina Ferrà Marcús1, Robert Jankowski2,3,4, Meritxell Vila-Miñana5, Carles Casacuberta1, M Ángeles Serrano2,3,6
ORCID iDs: Meritxell Vila-Miñana, Carles Casacuberta, M Ángeles Serrano
  1. Departament de Matèmatiques i Informàtica, Universitat de Barcelona, Barcelona, Spain
  2. Departament de Física de la Matèria Condensada, Universitat de Barcelona, Barcelona, Spain
  3. Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Spain
  4. Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Delft, Netherlands
  5. Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University, Bloomington, IN USA
  6. ICREA, Barcelona, Spain
Institutions: Universitat de Barcelona (Spain); Delft University of Technology (Netherlands); Indiana University Bloomington (United States); Indiana University (United States); Institució Catalana de Recerca i Estudis Avançats (Spain)
Journal: Nature communications, volume 17, issue 1, article 6105
Dates: received 16 September 2025; accepted 22 April 2026; published online 6 May 2026
Type: Research article · Language: English
License: CC BY
Identifiers: DOI 10.1038/s41467-026-72687-z · PMID 42091870 · PMCID PMC13358065 · OpenAlex W4417065225
Open access: gold, a free copy (OpenAlex)
Preprint: arxiv:2509.08350
Status: code verified
Categories: systems (subfield)
Methods: Graphs, Machine learning, Spectral & time-frequency
Keywords: Applied mathematics, Complex networks
Topic: Topological and Geometric Data Analysis (Computational Theory and Mathematics, Computer Science), according to OpenAlex
Funding: Agencia Estatal de Investigación de España, AEI
Citations: cited by 1 paper (Europe PMC); 73 references in the paper

Abstract

Many complex networks, ranging from social to biological systems, exhibit structural patterns consistent with an underlying hyperbolic geometry. Revealing the dimensionality of this latent space can disentangle the structural complexity of communities, impact efficient network navigation, and fundamentally shape connectivity and system behavior. We introduce a topological data analysis weighting scheme for graphs based on chordless cycles to estimate network dimensionality in a data-driven way. We further show that the resulting descriptors can effectively estimate network dimensionality using a neural network architecture trained on a synthetic graph database constructed for this purpose, which requires no retraining to transfer effectively to real-world networks. Thus, by combining cycle-aware filtrations, algebraic topology, and machine learning, our approach provides a robust and effective method for uncovering the hidden geometry of complex networks and guiding accurate modeling and low-dimensional embedding.

Reproduced under the paper's license (CC BY), from the paper cited above.

Repository

Its files are read in the Code ↔ Paper reader above, with 5 matches between paragraphs and lines of code.

networkgeometry/detecting-dimensionality-TDA-DimNN

License: none: the authors keep all their rights
State: the link answers, verified on 28 September 2026
Evidence: files inventoried
Commit: 870c69581f70a1503691efa8713f09b152136bd3, 8 April 2026
Languages: Python (30), Jupyter (7), C++ (5), C/C++ (3)
Size: 95 files, 45 scripts
Software Heritage: not archived
Found in: “Code availability”
Holds: README, environment (requirements.txt), 7 notebooks
Not found: license file, CITATION.cff, tests, continuous integration, documentation
Tools: pandas (21 files), NumPy (20 files), NetworkX (11 files), TensorFlow (10 files), Keras (9 files), Matplotlib (9 files), scikit-learn (8 files), seaborn (8 files), Numba (2 files), SciPy (2 files)
Availability: 1 check, the latest on 28 September 2026: the link answers
  • 28 September 2026: the link answers
46 files
At the source: github.com/networkgeometry/detecting-dimensionality-TDA-DimNN

Code availability

The open-source code generated in this work, along with the code to reproduce the figures, is available on GitHub at https://github.com/networkgeometry/detecting-dimensionality-TDA-DimNN.

Reproduced under the paper's license (CC BY), from the paper cited above.

Tracing map

Proposed by the machine: these links were found in the paper and verified at the source, without human review. The map will receive a Zenodo DOI once one of the paper's authors has validated it with their ORCID.

What the map holds:

  • 1 repository of the authors' code, each at its verified commit, with its license and how the link was found in the paper;
  • 45 scripts, each with its path and the digest of its content;
  • 5 matches between paragraphs of the paper and lines of the code (method lexical-v1);
  • neither the text of the paper nor the code itself.

Its JSON (tracing-map.json) is deposited on Zenodo with its DOI once the map is validated.

Data

Datasets cited

Data availability

The real network datasets used in this study are available from the sources referenced in the manuscript and the Supplementary Information. The SYNNET dataset of 792 000 synthetic networks generated with the SD model used for neural network training, along with the neural network checkpoints, is available on Zenodo73.

Reproduced under the paper's license (CC BY), from the paper cited above.

Versions

The history of this record: each version stored by the harvester or made by a correction of its authors or of the maintainers of its code, and what changed in its facts. The texts of the paper (its abstract, its availability statements) are not part of it; versions that changed only those are not listed.

Version 1, 28 September 2026: the first record

Recorded: type, language, journal, volume, issue, pages, dates, 5 authors, 2 keywords, 1 funder, 29 references.

Cite

This paper

Ferrà Marcús, A., Jankowski, R., Vila-Miñana, M., Casacuberta, C., & Serrano, M. Á. (2026). Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis. Nature communications, 17(1), 6105. https://doi.org/10.1038/s41467-026-72687-z

BibTeX

@article{ferramarcus2026chordless,
author = {Ferrà Marcús, Aina and Jankowski, Robert and Vila-Miñana, Meritxell and Casacuberta, Carles and Serrano, M Ángeles},
title = {{Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis}},
journal = {Nature communications},
year = {2026},
month = may,
volume = {17},
number = {1},
pages = {6105},
publisher = {Nature Publishing Group},
issn = {2041-1723},
doi = {10.1038/s41467-026-72687-z},
url = {https://doi.org/10.1038/s41467-026-72687-z},
pmid = {42091870},
pmcid = {PMC13358065}
}

RIS

TY - JOUR
AU - Ferrà Marcús, Aina
AU - Jankowski, Robert
AU - Vila-Miñana, Meritxell
AU - Casacuberta, Carles
AU - Serrano, M Ángeles
TI - Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis
T2 - Nature communications
J2 - Nat Commun
PY - 2026
DA - 2026/05/06
VL - 17
IS - 1
SP - 6105
SN - 2041-1723
PB - Nature Publishing Group
DO - 10.1038/s41467-026-72687-z
UR - https://doi.org/10.1038/s41467-026-72687-z
LA - en
ER -

CSL-JSON

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