IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v156y2022ics0960077922000285.html

A graph complexity measure based on the spectral analysis of the Laplace operator

Author

Listed:
  • Mateos, Diego M.
  • Morana, Federico
  • Aimar, Hugo

Abstract

In this work we introduce a concept of complexity for undirected graphs in terms of the spectral analysis of the Laplacian operator defined by the incidence matrix of the graph. Precisely, we compute the norm of the vector of eigenvalues of both the graph and its complement and take their product. Doing so, we obtain a quantity that satisfies two basic properties that are the expected for a measure of complexity. First, complexity of fully connected and fully disconnected graphs vanish. Second, complexity of complementary graphs coincide. This notion of complexity allows us to distinguish different kinds of graphs by placing them in a “croissant-shaped” region of the plane link density - complexity, highlighting some features like connectivity, concentration, uniformity or regularity and existence of clique-like clusters. Indeed, considering graphs with a fixed number of nodes, by plotting the link density versus the complexity we find that graphs generated by different methods take place at different regions of the plane. We consider some of the paradigmatic randomly generated graphs, in particular the Erdös-Rényi, the Watts-Strogatz and the Barabási-Albert models. Also, we place some particular, let us say deterministic, well known hand-crafted graphs, to wit, lattices, stars, hyper-concentrated and cliques-containing graphs. It is worthy noticing that these deterministic classical models of graphs depict the boundary of the croissant-shaped region. Finally, as an application to graphs generated by real measurements, we consider the brain connectivity graphs from two epileptic patients obtained from magnetoencephalography (MEG) recording, both in a baseline period and in ictal periods (epileptic seizures). In this case, our definition of complexity could be used as a tool for discerning between states, by the analysis of differences at distinct frequencies of the MEG recording.

Suggested Citation

  • Mateos, Diego M. & Morana, Federico & Aimar, Hugo, 2022. "A graph complexity measure based on the spectral analysis of the Laplace operator," Chaos, Solitons & Fractals, Elsevier, vol. 156(C).
    • Handle: RePEc:eee:chsofr:v:156:y:2022:i:c:s0960077922000285
    DOI: 10.1016/j.chaos.2022.111817
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077922000285
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2022.111817?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to

    for a different version of it.

    References listed on IDEAS

    as
    1. Anna D. Broido & Aaron Clauset, 2019. "Scale-free networks are rare," Nature Communications, Nature, vol. 10(1), pages 1-10, December.
    2. Passerini, Filippo & Severini, Simone, 2008. "The von Neumann entropy of networks," MPRA Paper 12538, University Library of Munich, Germany.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Goldrosen, Nicholas, 2024. "Is corrections officers' use of illegal force networked? Network structure, brokerage, and key players in the New York City Department of Correction," Journal of Criminal Justice, Elsevier, vol. 92(C).
    2. Vinayak, & Raghuvanshi, Adarsh & kshitij, Avinash, 2023. "Signatures of capacity development through research collaborations in artificial intelligence and machine learning," Journal of Informetrics, Elsevier, vol. 17(1).
    3. Kaushik Gala & Andreas Schwab, 2026. "A Methodological Guide for Quantitative Analysis of Star Performance in Entrepreneurship," Entrepreneurship Theory and Practice, , vol. 50(3), pages 931-973, May.
    4. Martins, Francisco Leonardo Bezerra & do Nascimento, José Cláudio, 2022. "Power law dynamics in genealogical graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 596(C).
    5. Fleming, Sean W., 2021. "Scale-free networks, 1/f dynamics, and nonlinear conflict size scaling from an agent-based simulation model of societal-scale bilateral conflict and cooperation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 567(C).
    6. Nicolas Victor Noak & Lance Christian, 2025. "Exploring spatial network structures in entrepreneurial ecosystems: a network and clustering analysis of global venture funding flows," Small Business Economics, Springer, vol. 65(3), pages 1483-1500, October.
    7. Caravelli, Francesco, 2015. "Ranking nodes according to their path-complexity," Chaos, Solitons & Fractals, Elsevier, vol. 73(C), pages 90-97.
    8. Shang, Yilun, 2021. "Generalized k-cores of networks under attack with limited knowledge," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    9. Amit Kumar Srivastava & Rajhans Mishra, 2025. "What is my privacy score? Measuring users’ privacy on social networking websites," Electronic Commerce Research, Springer, vol. 25(5), pages 3427-3450, October.
    10. Cádiz, Macarena & Gallo-Méndez, Iván & Moya, Pablo S. & Pastén, Denisse, 2025. "Analysis of persistent and antipersistent time series with the Visibility Graph method," Chaos, Solitons & Fractals, Elsevier, vol. 201(P2).
    11. Wang, Ziyulong & Huang, Ketong & Massobrio, Renzo & Bombelli, Alessandro & Cats, Oded, 2024. "Quantification and comparison of hierarchy in Public Transport Networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 634(C).
    12. D'Acci, Luca S., 2023. "Is housing price distribution across cities, scale invariant? Fractal distribution of settlements' house prices as signature of self-organized complexity," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    13. Oscar Contreras-Velasco & Nathan P. Jones & Daniel Weisz Argomedo & John P. Sullivan & Chris Callaghan, 2025. "Uncovering hidden alliances in organized crime networks with machine learning: from node similarity to graph neural networks," Journal of Computational Social Science, Springer, vol. 8(4), pages 1-25, November.
    14. Liu, Huixia & Lu, Lulu & Zhu, Yuan & Wei, Zhouchao & Yi, Ming, 2022. "Stochastic resonance: The response to envelope modulation signal for neural networks with different topologies," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 607(C).
    15. Meng, Xiangyi & Zhou, Bin, 2023. "Scale-free networks beyond power-law degree distribution," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    16. Partida, Alberto & Gerassis, Saki & Criado, Regino & Romance, Miguel & Giráldez, Eduardo & Taboada, Javier, 2022. "The chaotic, self-similar and hierarchical patterns in Bitcoin and Ethereum price series," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    17. Schulz, Jan & Weber, Jan David, 2025. "Power laws in socio-economics," BERG Working Paper Series 203, Bamberg University, Bamberg Economic Research Group.
    18. Coulibaly, Saliya & Bessin, Florent & Clerc, Marcel G. & Mussot, Arnaud, 2022. "Precursors-driven machine learning prediction of chaotic extreme pulses in Kerr resonators," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    19. Mohammad Ghaderi, 2020. "Public health interventions in the face of pandemics: network structure, social distancing, and heterogeneity," Economics Working Papers 1732, Department of Economics and Business, Universitat Pompeu Fabra.
    20. Jianjia Wang & Chenyue Lin & Yilei Wang, 2019. "Thermodynamic Entropy in Quantum Statistics for Stock Market Networks," Complexity, Hindawi, vol. 2019, pages 1-11, April.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:156:y:2022:i:c:s0960077922000285. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.