IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v156y2022ics0960077922000285.html
   My bibliography  Save this article

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 search 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. 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).
    2. 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).
    3. 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).
    4. Meng, Xiangyi & Zhou, Bin, 2023. "Scale-free networks beyond power-law degree distribution," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    5. 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).
    6. Rumeng Zhang & Lihong Li, 2023. "Research on Evolutionary Game and Simulation of Information Sharing in Prefabricated Building Supply Chain," Sustainability, MDPI, vol. 15(13), pages 1-24, June.
    7. Mohammad Ghaderi, 2020. "Public Health Interventions in the Face of Pandemics: Network Structure, Social Distancing, and Heterogeneity," Working Papers 1193, Barcelona School of Economics.
    8. Michele Battisti & Andrea Mario Lavezzi & Roberto Musotto, 2022. "Organizing Crime: an Empirical Analysis of the Sicilian Mafia," Papers 2205.02310, arXiv.org.
    9. Jiang, Xiong-Fei & Xiong, Long & Bai, Ling & Lin, Jie & Zhang, Jing-Feng & Yan, Kun & Zhu, Jia-Zhen & Zheng, Bo & Zheng, Jian-Jun, 2022. "Structure and dynamics of human complication-disease network," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    10. Luka Naglić & Lovro Šubelj, 2019. "War pact model of shrinking networks," PLOS ONE, Public Library of Science, vol. 14(10), pages 1-14, October.
    11. Kim, Seyun & Yoon, Yoonjin, 2019. "On node criticality of the Northeast Asian air route network," Journal of Air Transport Management, Elsevier, vol. 80(C), pages 1-1.
    12. Ghaderi, Mohammad, 2022. "Public health interventions in the face of pandemics: Network structure, social distancing, and heterogeneity," European Journal of Operational Research, Elsevier, vol. 298(3), pages 1016-1031.
    13. Kelty-Stephen, Damian G. & Furmanek, Mariusz P. & Mangalam, Madhur, 2021. "Multifractality distinguishes reactive from proactive cascades in postural control," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    14. Valero, Jordi & Pérez-Casany, Marta & Duarte-López, Ariel, 2022. "The Zipf-Polylog distribution: Modeling human interactions through social networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 603(C).
    15. Huang, Chuangxia & Deng, Yunke & Yang, Xiaoguang & Cao, Jinde & Yang, Xin, 2021. "A network perspective of comovement and structural change: Evidence from the Chinese stock market," International Review of Financial Analysis, Elsevier, vol. 76(C).
    16. 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).
    17. Caravelli, Francesco, 2015. "Ranking nodes according to their path-complexity," Chaos, Solitons & Fractals, Elsevier, vol. 73(C), pages 90-97.
    18. Shang, Yilun, 2021. "Generalized k-cores of networks under attack with limited knowledge," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    19. 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).
    20. 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).

    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.