Skip to main content
SpringerLink
Log in

Graph entropies-graph energies indices for quantifying network structural irregularity

  • Published:
The Journal of Supercomputing Aims and scope Submit manuscript

Abstract

Quantifying similarities/dissimilarities among different graph models and studying the irregularity (heterogeneity) of graphs in graphs and complex networks are one of the fundamental issues as well as a challenge of scientific and practical importance in many fields of science and engineering. This paper has been motivated by the necessity to establish novel and efficient entropy-based methods to quantify the structural irregularity properties of graphs, measure structural complexity, classify, and compare complex graphs and networks. In particular, we explore how criteria such as Shannon entropy, Von Newman, and generalized graph entropies, already defined for graphs, can be used to evaluate and measure irregularities in complex graphs and networks. To do so, we use some results obtained from graph spectral theory related to the construction of matrices obtained from graphs. We show how to use these irregularity indices based on graph entropies and demonstrate the usefulness and efficiency of each of these complexity measures on both synthetic networks and real-world data sets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

References

  1. Barabási A-L. (2022) Network science book, Boston, MA: Center for Complex Network, Northeastern University, Available online at: http://barabasi.com/networksciencebook

  2. Estrada E, Knight PA (2015) A first course in network theory. Oxford University Press, USA

    MATH  Google Scholar 

  3. Watts DJ, Strogatz SH (1998) Collective dynamics of small-world networks. Nature 393(6684):440–442

    MATH  Google Scholar 

  4. Barabási A-L, Albert R (1999) Emergence of scaling in random networks. Science 286(5439):509–512

    MathSciNet  MATH  Google Scholar 

  5. Harary F. (2018) Graph theory, Taylor & Francis

  6. Bondy JA, Murty USR (1976) Graph theory with applications, vol 290. Macmillan, London

    MATH  Google Scholar 

  7. Calderone A et al (2016) Comparing Alzheimers and Parkinsons diseases networks using graph communities structure. BMC Syst Biol 10:1–10

    Google Scholar 

  8. Carpi L et al (2012) Structural evolution of the Tropical Pacific climate network. Euro Phys J B 85:1434–6028

    Google Scholar 

  9. Dehmer M, Mowshowitz A (2011) A history of graph entropy measures. Inf Sci 181(1):57–78

    MathSciNet  MATH  Google Scholar 

  10. Li X, Qin Z, Wei M, Gutman I, Dehmer M (2015) Novel inequalities for generalized graph entropies–Graph energies and topological indices. Appl Math Comput 259:470–479

    MathSciNet  MATH  Google Scholar 

  11. Gutman I (2018) Topological indices and irregularity measures. J Bull 8:469–475

    MathSciNet  MATH  Google Scholar 

  12. Erdos P, Rényi A (1960) On the evolution of random Graphs. Publ Math Inst Hung Acad Sci 5(1):17–60

    MathSciNet  MATH  Google Scholar 

  13. Feldman D, Crutchfield J (1998) Measures of statistical complexity: Why? Phys Lett A 238:244–252

    MathSciNet  MATH  Google Scholar 

  14. Balaban AT (1982) Highly discriminating distance-based topological index. Chem Phys Lett 89:399–404

    MathSciNet  Google Scholar 

  15. Estrada E (2010) Quantifying network heterogeneity. Phys Rev E 82(6):066102

    Google Scholar 

  16. Li X, Shi Y (2008) A survey on the Randić index, MATCH: Comm. Math Comput Chem 59:127–156

    MathSciNet  MATH  Google Scholar 

  17. Maier M, Luxburg U, Hein M. (2010) Influence of graph construction on graph-based clustering measures. NIPS, pp. 1–9

  18. Han L, Escolano F, Hancock ER, Wilson RC (2012) Graph characterizations from von Neumann entropy. Pattern Recogn Lett 33(15):1958–1967

    Google Scholar 

  19. Estrada E, Vargas-Estrada E (2012) Distance-sum Heterogeneity in Graphs and Complex Networks. Appl Math Comput 218:10393–10405

    MathSciNet  MATH  Google Scholar 

  20. Olfati-Saber R, Fax JA, Murray RM (2007) Consensus and cooperation in networked multi-agent systems. Proc IEEE 95(1):215–233. https://doi.org/10.1109/JPROC.2006.887293

    Article  MATH  Google Scholar 

  21. Han L (2012) Graph Generative Models from Information Theory. University of York, UK

    Google Scholar 

  22. Collatz L, Sinogowitz U (1957) Spektren endlicher Grafen. Abh Math Sem Univ Hamburg 21:63–77

    MathSciNet  MATH  Google Scholar 

  23. Cvetković D, Rowlinson P (1988) On connected graphs with maximal index. Publ Inst Math (Beograd) 44:29–34

    MathSciNet  MATH  Google Scholar 

  24. Bell FK (1992) A note on the irregularity of a graph. Linear Algebra Appl 161:45–54

    MathSciNet  MATH  Google Scholar 

  25. Snijders TAB (1981) The degree variance: an index of graph heterogeneity. Social Networks 3(3):163–174

    MathSciNet  Google Scholar 

  26. Albertson MO (1997) The irregularity of a graph. Ars Comb 46:219–225

    MATH  Google Scholar 

  27. Fath-Tabar GH, Gutman I, Nasiri R, (2013) Extremely irregular trees, Bulletin (Académie serbe des sciences et des arts. Classe des sciences mathématiques et naturelles. Sciences mathématiques), pp.1–8

  28. Hansen P, Mélot H. (2005) Graphs and Discovery. In S. Fajtlowicz (Eds.), DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Providence, American Mathematical Society, Vol. 69, pp. 253–264

  29. Abdo H, Dimitrov D, Gutman I (2018) Graphs with maximal s irregularity. Discrete Appl Math 250:57–64

    MathSciNet  MATH  Google Scholar 

  30. Gutman I, Togan M, Yurttas A, Cevik AS, Cangul IN (2018) Inverse problem for sigma index. MATCH Commun Math Comput Chem 79:491–508

    MathSciNet  MATH  Google Scholar 

  31. Abdo H, Brandt S, Dimitrov D (2014) The total irregularity of a graph. Discrete Math Theory Comput Sci 16:201–206

    MathSciNet  MATH  Google Scholar 

  32. Nikiforov V (2006) Eigenvalues and degree deviation in graphs. Linear Algebra Appl. 414:347–360

    MathSciNet  MATH  Google Scholar 

  33. Shi L (2009) Bounds on Randic indices. Discr Appl Math 309:5238–5241

    MathSciNet  MATH  Google Scholar 

  34. Gutman I, Furtula B, Elphick C (2014) The new/old vertex-degree-based topological indices. MATCH Commun Math Comput Chem 72:617–632

    MathSciNet  MATH  Google Scholar 

  35. Goldberg F. (2014) Spectral radius minus average degree: a better bound, arXiv: 1407.4285v1 [math.co] 16 July

  36. Réti T, Tóth-Laufer E (2017) On the construction and comparison of graph irregularity indices. Kragujevac J Sci 39:53–75

    Google Scholar 

  37. Hamzeh A, Réti T (2014) An analogue of Zagreb index inequality obtained from graph irregularity measures. MATCH Commun Math Comput Chem 72(3):669–683

    MathSciNet  MATH  Google Scholar 

  38. Elphick C, Wocjan P (2014) New measures of graph irregularity. Electron J Graph Theory Appli 2(1):52–65

    MathSciNet  MATH  Google Scholar 

  39. Izumino S, Mori H, Seo Y (1998) On Ozeki’s inequality. J Inequalities Appli 2:235–253

    MathSciNet  MATH  Google Scholar 

  40. Favaron O, Mahéo M, Saclé JF (1993) Some eigenvalue properties in Graphs (conjectures of Graffiti-II. Discrete Math 111:197–220

    MathSciNet  MATH  Google Scholar 

  41. Newman MEJ (xxxx) Random graphs as models of networks, Sante Fe Institute, Working Paper 2002–02–005.

  42. Ilić A, Stevanović D (2009) On comparing Zagreb indices. MATCH Commun Math Comput Chem 62:681–687

    MathSciNet  MATH  Google Scholar 

  43. Ilić A, Zhou B (2012) On reformulated Zagreb indices. Discr Appl Math 160:204–209

    MathSciNet  MATH  Google Scholar 

  44. Hao J (2011) Theorems about Zagreb indices and modified Zagreb indices. MATCH Commun Math Comput Chem 65:659–670

    MathSciNet  MATH  Google Scholar 

  45. Zimmermann MG, Eguiluz VM, San Miguel M (2004) Coevolution of dynamical states and interactions in dynamic networks. Phys Rev E 69:065102

    Google Scholar 

  46. Smith KM, Escudero J (2020) Normalised degree variance. Appl Netw Science 5(1):1–22

    Google Scholar 

  47. Safaei F, Tabrizchi S, Rasanan AH, Zare M (2019) An energy-based heterogeneity measure for quantifying structural irregularity in complex networks. J Comput Sci 36:101011

    MathSciNet  Google Scholar 

  48. Gutman I (1978) The energy of a graph. Ber Math-Statist Sekt Forschungsz Graz 103:1–22

    MATH  Google Scholar 

  49. Estrada E, Benzi M (2017) What is the meaning of the graph energy after all? Discret Appl Math 230:71–77

    MathSciNet  MATH  Google Scholar 

  50. Safaei F, Kashkooei Jahromi F, Fathi S (2019) A method for computing local contributions to graph energy based on Estrada-Benzi approach. Discrete Appl Math 260:214–226

    MathSciNet  MATH  Google Scholar 

  51. Safaei F, Kashkooei Jahromi F, Fathi S (2021) Graphlets importance ranking in complex networks based on the spectral energy contribution. Int J Comput Math: Comput Syst Theory 6(1):21–36

    MathSciNet  Google Scholar 

  52. Gutman I, Zhou B (2006) Laplacian energy of a graph. Linear Algebra Appl 414(1):29–37

    MathSciNet  MATH  Google Scholar 

  53. Gutman I et al (2008) Relation between Energy and Laplacian Energy. MATCH Commun Math Comput Chem 59:343–354

    MathSciNet  MATH  Google Scholar 

  54. McClelland BJ (1971) Properties of the latent roots of a matrix: The estimation of π-electron energies. J Chem Phys 54:640–643

    Google Scholar 

  55. Dehmer M, Li X, Shi Y (2015) Connections between generalized graph entropies and graph energy. Complexity 21(1):35–41

    MathSciNet  Google Scholar 

  56. Dehmer M (2008) Information processing in complex networks: graph entropy and information functionals. Appl Math Comput 201:82–94

    MathSciNet  MATH  Google Scholar 

  57. Shannon CE, Weaver W (1949) The Mathematical Theory of Communication. University of Illinois Press, Urbana

    MATH  Google Scholar 

  58. Dehmer M, Mowshowitz A (2011) Generalized graph entropies. Complexity 17:45–50

    MathSciNet  MATH  Google Scholar 

  59. Rényi P. (1961) On measures of information and entropy. In: Proceedings of the 4th Berkeley symposium on mathematics, statistics and probability, Vol. 1, University of California Press: Berkeley, CA, pp 547–561

  60. Daròczy Z, Jarai A (1979) On the measurable solutions of functional equation arising in information theory. Acta Math Acad SciHungar 34:105–116

    MathSciNet  MATH  Google Scholar 

  61. Faloutsos M, Faloutsos P, Faloutsos C (1999) On power-law relationships of the internet topology. Comp Comm Rev 29:251–262

    MATH  Google Scholar 

  62. Dorogovtsev SN, Mendes JFF, Samukhin AN (2003) Metric structure of random networks. Nucl Phys B 653:307–338

    MathSciNet  MATH  Google Scholar 

  63. Malarz K, Karpinska J, Kardas A, Kulalowski K (xxxx) Node-node distance distribution for growing networks, arXiv:cond-mat/0309255v2.

  64. Blondell VD, Guillaume J-L, Hendrickx JM, Jungers RM (2007) Distance distribution in random graphs and application to network exploration. Phys Rev E 76:066101

    MathSciNet  Google Scholar 

  65. Wiener H (1947) Structural determination of paraffin boiling points. J Amer Chem Soc 69:17–20

    Google Scholar 

  66. Freeman LC (1979) Centrality in networks: I Conceptual clarification. Social Networks 1:215–239

    Google Scholar 

  67. Dangalchev C (2006) Residual closeness in networks. Physica A 365(2):556–564

    Google Scholar 

  68. Alikhani S, Ghanbari N. (2014) On the Randic characteristic polynomial of specific graphs, In: The First Conference on Computational Group Theory, Computational Number Theory and Applications, University of Kashan, pp. 26–28, Dec. 17–19, 2014, pp. 11–15

  69. Passerini F, Severini S. (2008) The von Neumann entropy of networks. arXiv preprint arXiv:0812.2597

  70. Anand K, Bianconi G, Severini S (2011) Shannon and von Neumann entropy of random networks with heterogeneous expected degree. Phys Rev E 83(3):036109

    MathSciNet  Google Scholar 

  71. Safaei F, Yeganloo H, Akbar R (2020) Robustness on topology reconfiguration of complex networks: an entropic approach. Math Comput Simul 170:379–409

    MathSciNet  MATH  Google Scholar 

  72. Ellens W, Spieksma FM, Van Mieghem P, Jamakovic A, Kooij RE (2011) Effective graph resistance. Linear Algebra Appl 435(10):2491–2506

    MathSciNet  MATH  Google Scholar 

  73. http://www-personal.umich.edu/~mejn/netdata/ , Available online at 19 December 2021

  74. Zare M, Rezvani Z, Benasich AA (2016) Automatic classification of 6-month-old infants at familial risk for language-based learning disorder using a support vector machine. Clin Neurophysiol 127:2695–2703

    Google Scholar 

  75. http://cosinproject.eu/extra/data/foodwebs/WEB.html , Available online at 19 December 2021

  76. Estrada E (2019) Degree heterogeneity of graphs and networks. I. Interpretation and the “heterogeneity paradox.” J Interdiscip Math 22(4):503–529

    Google Scholar 

  77. Estrada E (2019) Degree heterogeneity of graphs and networks. II. Comparison with other indices. J Interdiscip Math 22(5):711–735

    Google Scholar 

  78. Safaei F, Babaei A, Moudi M (2020) Optimally connected hybrid complex networks with Windmill Graphs Backbone. J Syst Sci Complexity 33:903–929

    MATH  Google Scholar 

  79. Abdo H, Dimitrov D, Gutman I (2019) Graph irregularity and its measures. Appl Math Comput 357:317–324

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran

    M. M. Emadi Kouchak & M. Reshadi

  2. Faculty of Computer Science and Engineering, Shahid Beheshti University, Tehran, Iran

    F. Safaei

Authors
  1. M. M. Emadi Kouchak
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Safaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Reshadi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Safaei.

Ethics declarations

Conflict of interests

The authors declare that they have no conflicts of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Emadi Kouchak, M.M., Safaei, F. & Reshadi, M. Graph entropies-graph energies indices for quantifying network structural irregularity. J Supercomput 79, 1705–1749 (2023). https://doi.org/10.1007/s11227-022-04724-9

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11227-022-04724-9

Keywords

Search

Navigation