- Physical Review Journals
All JournalsPhysics Magazine

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Insights on structure and influence from the adjacency and Laplacian eigenspectra of intersecting ring networks

Agathe Bouis*, Ruaridh A. Clark, and Malcolm Macdonald

  • Applied Space Technology Laboratory, Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow, United Kingdom

  • *Contact author: agathe.bouis@strath.ac.uk
PDF

Phys. Rev. E 113, 034313 – Published 24 March, 2026

DOI: https://doi.org/10.1103/4595-wyvd

Abstract

A network's community structure commonly impacts its functions. For instance, networks seeking synchronization will see this process follow the topology's hierarchical and community structuring. Herein, the interplay of network adjacency and Laplacian eigenspectra is shown to uncover hierarchical influence and community structure. Ring networks embedded with hubs of high connectivity are first analyzed to characterize the differing insights of the adjacency and Laplacian eigenspectra. This understanding is then transferred to the case study of satellite networks composed of intersecting rings. From these scenarios, it is found that where the adjacency reflects a network's structure, the Laplacian detects nodes' influence profiles. The adjacency identifies the number and relative sizes of salient network structures, for example hubs of high connectivity. In contrast, the Laplacian decomposes these structures in sets of nodes with equivalent influence profiles, where influence emerges due to variations in nodes' connectivity.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (60)

  1. M. A. Javed, M. S. Younis, S. Latif, J. Qadir, and A. Baig, Community detection in networks: A multidisciplinary review, J. Netw. Comput. Appl. 108, 87 (2018).
  2. S. Fortunato, Community detection in graphs, Phys. Rep. 486, 75 (2010).
  3. M.-Y. Zhou, Z. Zhuo, S.-m. Cai, and Z. Fu, Community structure revealed by phase locking, Chaos 24, 033128 (2014).
  4. M. E. J. Newman, Finding community structure in networks using the eigenvectors of matrices, Phys. Rev. E 74, 036104 (2006).
  5. R. Clark, G. Punzo, and M. Macdonald, Network communities of dynamical influence, Sci. Rep. 9, 17590 (2019).
  6. K. Klemm, M. Á. Serrano, V. M. Eguíluz, and M. S. Miguel, A measure of individual role in collective dynamics, Sci. Rep. 2, 292 (2012).
  7. U. Von Luxburg, A tutorial on spectral clustering, Stat. Comput. 17, 395 (2007).
  8. H.-W. Shen and X.-Q. Cheng, Spectral methods for the detection of network community structure: A comparative analysis, J. Stat. Mech. Theory Exp. (2010) P10020.
  9. M. E. J. Newman, Spectral methods for community detection and graph partitioning, Phys. Rev. E 88, 042822 (2013).
  10. C. Hoppen, D. P. Jacobs, and V. Trevisan, Locating Eigenvalues in Graphs (Springer, New York, 2012).
  11. D. Cvetković, P. Rowlinson, and S. Simić, An Introduction to the Theory of Graph Spectra of London Mathematical Society Student Texts (Cambridge University Press, Cambridge, 2009).
  12. P. Van Mieghem, D. Stevanović, F. Kuipers, C. Li, R. V. De Bovenkamp, D. Liu, and H. Wang, Decreasing the spectral radius of a graph by link removals, Phys. Rev. E 84, 016101 (2011).
  13. P. Van Mieghem, J. Omic, and R. Kooij, Virus spread in networks, IEEE/ACM Trans. Network. 17, 1 (2008).
  14. J. G. Restrepo, E. Ott, and B. R. Hunt, Onset of synchronization in large networks of coupled oscillators, Phys. Rev. E 71, 036151 (2005).
  15. P. Bonacich, Some unique properties of eigenvector centrality, Soc. Netw. 29, 555 (2007).
  16. A. Y. Ng, M. I. Jordan, and Y. Weiss, On spectral clustering: Analysis and an algorithm, in Proceedings of the 15th International Conference on Neural Information Processing Systems: Natural and Synthetic (NIPS'01) (MIT Press, Cambridge, MA, 2001), pp. 849–856.
  17. P. S. Bradley and U. M. Fayyad, Refining initial points for k-means clustering, in Proceedings of the Fifteenth International Conference on Machine Learning (ICML, San Francisco, CA, 1998), pp. 91–99.
  18. B. Luo, R. C. Wilson, and E. R. Hancock, Spectral embedding of graphs, Pattern Recognit. 36, 2213 (2003).
  19. F. M. Atay, T. Bıyıkoğlu, and J. Jost, Network synchronization: Spectral versus statistical properties, Physica D 224, 35 (2006).
  20. F. M. Atay, T. Biyikoglu, and J. Jost, Synchronization of networks with prescribed degree distributions, IEEE Trans. Circ. Syst. Regul. Pap. 53, 92 (2006).
  21. C. Grabow, S. Grosskinsky, J. Kurths, and M. Timme, Collective relaxation dynamics of small-world networks, Phys. Rev. E 91, 052815 (2015).
  22. A. Arenas, A. Díaz-Guilera, and C. J. Pérez-Vicente, Synchronization reveals topological scales in complex networks, Phys. Rev. Lett. 96, 114102 (2006).
  23. Z. Zhang, Y. Qi, S. Zhou, Y. Lin, and J. Guan, Recursive solutions for Laplacian spectra and eigenvectors of a class of growing treelike networks, Phys. Rev. E 80, 016104 (2009).
  24. A. Raza and M. M. Munir, Insights into network properties: Spectrum-based analysis with Laplacian and signless Laplacian spectra, Eur. Phys. J. Plus 138, 802 (2023).
  25. L. Donetti and M. A. Munoz, Detecting network communities: A new systematic and efficient algorithm, J. Stat. Mech. Theory Exp. (2004) P10012.
  26. F. Boesch and R. Tindell, Circulants and their connectivities, J. Graph Theory 8, 487 (1984).
  27. F. Göbel and E. Neutel, Cyclic graphs, Discr. Appl. Math. 99, 3 (2000).
  28. J.-C. Bermond, F. Comellas, and D. F. Hsu, Distributed loop computer-networks: A survey, J. Parallel Distrib. Comput. 24, 2 (1995).
  29. S. Chen and X. D. Jia, Undirected loop networks, Networks 23, 257 (1993).
  30. J.-C. Bermond, G. Illiades, and C. Peyrat, An optimization problem in distributed loop computer networks, Ann. N.Y. Acad. Sci. 555, 45 (1989).
  31. R. Beivide, C. Martínez, C. Izu, J. Gutierrez, J.-Á. Gregorio, and J. Miguel-Alonso, Chordal topologies for interconnection networks, in International Symposium on High Performance Computing (Springer, Berlin, 2003), pp. 385–392.
  32. A. L. Liestman, J. Opatrny, and M. Zaragoza, Network properties of double and triple fixed step graphs, Int. J. Found. Comput. Sci. 9, 57 (1998).
  33. J. Fabrega and M. Zaragozá, Fault-tolerant routings in double fixed-step networks, Discr. Appl. Math. 78, 61 (1997).
  34. E. A. Monakhova, A survey on undirected circulant graphs, Discr. Math. Algor. Appl. 4, 1250002 (2012).
  35. X. Zhao, J. Liu, G. Chen, L. Chai, and D. Wang, Dynamics and synchronization of complex-valued ring networks, Int. J. Bifurcat. Chaos 32, 2230011 (2022).
  36. C. Nathe, K. Huang, M. Lodi, M. Storace, and F. Sorrentino, Delays induced cluster synchronization in chaotic networks, Chaos 30, 121105 (2020).
  37. M. J. Panaggio and D. M. Abrams, Chimera states: Coexistence of coherence and incoherence in networks of coupled oscillators, Nonlinearity 28, R67 (2015).
  38. S. He, K. Sun, and H. Wang, Synchronization of fractional-order time delayed chaotic systems with ring connection, Eur. Phys. J.: Spec. Top. 225, 97 (2016).
  39. E. A. Martens, E. Barreto, S. H. Strogatz, E. Ott, P. So, and T. M. Antonsen, Exact results for the Kuramoto model with a bimodal frequency distribution, Phys. Rev. E 79, 026204 (2009).
  40. D. A. Wiley, S. H. Strogatz, and M. Girvan, The size of the sync basin, Chaos 16, 015103 (2006).
  41. Y. Hauri, D. Bhattacherjee, M. Grossmann, and A. Singla, “Internet from Space” without Inter-satellite Links, in Proceedings of the 19th ACM Workshop on Hot Topics in Networks (ACM, New York, 2020), pp. 205–211.
  42. R. A. Clark, C. N. McGrath, and M. Macdonald, Dynamical influence driven space system design, in International Conference on Complex Networks and Their Applications (Springer, Berlin, 2021), pp. 27–38.
  43. J. G. Walker, Continuous whole-earth coverage by circular-orbit satellite patterns, NASA Technical Reports Server 78, 11169 (1977).
  44. M. Albulet, Spacex non-geostationary satellite system: Attachment a technical information to supplement schedules, U.S. Federal Communications Commission, Washington, DC, Report No. SAT-OA-20161115-00118 (2018).
  45. C. Bassa, O. Hainaut, and D. Galadí-Enríquez, Analytical simulations of the effect of satellite constellations on optical and near-infrared observations, Astron. Astrophys. 657, A75 (2022).
  46. N. M. Choo, B. J. Lunday, D. K. Ahner, N. B. Gaw, B. D. Little, and B. K. McBee, Investigating regional communication network robustness of an asymmetric string-of-pearls satellite constellation design framework, J. Spacecraft Rockets 61, 1 (2024).
  47. J. Lutzeyer, Network representation matrices and their eigen- properties: A comparative study, Ph.D. thesis, Imperial College, London, 2019.
  48. A. Bouis, R. A. Clark, and M. Macdonald, An autonomous distributed timing signal in-space as alternative to GNSS time Synchronisation, in 75th International Astronautical Congress (International Astronautical Federation (IAF), Milan, 2024).
  49. L. M. Marrero, et al., Architectures and synchronization techniques for distributed satellite systems: A survey, IEEE Access 10, 45375 (2022).
  50. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer Berlin Heidelberg, Berlin, Heidelberg, 1984), pp. 111–140.
  51. A. Bouis, E. Steinmann, J. Gribben, R. A. Clark, P. Reiss, and M. Macdonald, Decentralised asynchronous resilient time synchronization for satellite networks, in AIAA Aviation Forum and Ascend 2025 (AIAA, Las Vegas, NV, 2025), p. 4091.
  52. R. A. Clark, F. Arrigo, A. Bouis, and M. Macdonald, A local eigenvector centrality, arXiv:2511.03608.
  53. L. J. Grady and J. R. Polimeni, Discrete Calculus: Applied Analysis on Graphs for Computational Science (Springer, London, 2010), Vol. 3.
  54. A. Ortega, Introduction to Graph Signal Processing (Cambridge University Press, Cambridge, 2022).
  55. R. M. Gray, Toeplitz and circulant matrices: A review, Found. Trends Commun. Inf. Theory 2, 155 (2006).
  56. C.-J. Wang, Structural properties of a low earth orbit satellite constellation—The Walker delta network, in Proceedings of the IEEE Military Communications Conference (MILCOM'93), Vol. 3 (IEEE, Los Alamitos, CA, 1993), pp. 968–972.
  57. M. Macdonald and V. Badescu, The International Handbook of Space Technology (Springer, Berlin, Heidelberg, 2014).
  58. D. A. Vallado, Fundamentals of Astrodynamics and Applications. (Springer Science & Business Media, New York, 2001).
  59. C. Sarkar and S. Jalan, Spectral properties of complex networks, Chaos 28, 102101 (2018).
  60. A. Bouis, Code for Insights on structure and influence from the adjacency and Laplacian eigenspectra of intersecting ring networks, Zenodo (2025), doi:10.5281/zenodo.17661737.

Outline

Information

Phys. Rev. E 113, 034313– Published 24 March, 2026
  • Received 26 May 2025
  • Accepted 23 February 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation