Skip to main content
SpringerLink
Log in

The Structure Entropy of Social Networks

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

Micro triadic structure is an important motif and serves the building block of complex networks. In this paper, the authors define structure entropy for a social network and explain this concept by using the coded triads proposed by Davis and Leinhardt in 1972. The proposed structure entropy serves as a new macro-evolution index to measure the network’s stability at a given timestamp. Empirical analysis of real-world network structure entropy discloses rich information on the mechanism that yields given triadic motifs frequency distribution. This paper illustrates the intrinsic link between the micro dyadic/triadic motifs and network structure entropy. Importantly, the authors find that the high proportion of reciprocity and transitivity results in the emergence of hierarchy, order, and cooperation of online social networks.

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

Similar content being viewed by others

References

  1. Barabasi A L and Albert R, Emergence of scaling in random networks, Science, 1999, 286: 509–512.

    Article  MathSciNet  Google Scholar 

  2. Watts D J and Strogatz S H, Collective dynamics of small-world networks, Nature, 1998, 393: 440–442.

    Article  Google Scholar 

  3. Zlatic V, Bozicevic M, Stefancic H, et al., Wikipedias: Collaborative web-based encyclopedias as complex networks, Physical Review E, 2006, 74: 016115.

    Article  Google Scholar 

  4. Homans G C, The Human Group, Harcourt Brace, New York, 1950.

    Google Scholar 

  5. Lin N, Social Capital: A Theory of Social Structure and Action, Cambridge University, New York, 2001.

    Book  Google Scholar 

  6. Holland P W and Leinhardt S, Transitivity in structure models of small groups, Editor(s) by Samuel Leinhardts, Social Networks, Academic Press, New York, 1977.

    Google Scholar 

  7. Hua L A, Cqa B, Yn A, et al., The evolution of structural balance in time-varying signed networks, Future Generation Computer Systems, 2020, 102: 403–408.

    Article  Google Scholar 

  8. Burt R S, Structural Holes: The Social Structure of Competition, Harvard University Press, Cambridge, Massachusetts, 1992.

    Book  Google Scholar 

  9. Granovetter M, Economic action and social structure: The problem of embeddedness, American Journal of Sociology, 1985, 91: 481–510.

    Article  Google Scholar 

  10. Benson A R, Gleich D F, and Leskovec J, Higher-order organization of complex networks, Science, 2016, 353: 163–166.

    Article  Google Scholar 

  11. Thomas J F, Social activity and social structure: A contribution to the theory of social systems, Cybernetics and Systems, 1981, 12: 53–81.

    Article  MathSciNet  Google Scholar 

  12. Harary F, Norman R Z, and Cartwright D, Structure Models: An Introduction to the Theory of Directed Graphs, Wiley, New York, 1965.

    Google Scholar 

  13. Davis J A, Clustering and structural balance in graphs, Human Relations, 1967, 20: 181–187.

    Article  Google Scholar 

  14. Mitchell J C, Social Networks in Urban Situations, Manchester University Press, Manchester, 1969.

    Google Scholar 

  15. White H C, Boorman S A, and Breiger R L, Social structure from multiple networks I: Block-models of roles and positions, American Journal of Sociology, 1976, 81: 730–779.

    Article  Google Scholar 

  16. Jaynes E T, Information theory and statistical mechanics, The Physical Review, 1957, 106: 620–630.

    Article  MathSciNet  Google Scholar 

  17. Albert R and Barabasi A L, Statistical mechanics of complex networks, Reviews of Modern Physics, 2002, 74: 47–97.

    Article  MathSciNet  Google Scholar 

  18. Dinga E, Tanasescu C R, and Ionescu G M, Social entropy and normative network, Entropy, 2020, 22: 1051.

    Article  MathSciNet  Google Scholar 

  19. Matei S A and Bruno R J, Pareto’s 80/20 law and social differentiation: A social entropy perspective, Public Relations Review, 2015, 41: 178–186.

    Article  Google Scholar 

  20. Mowshowitz A and Dehmer M, Entropy and the complexity of graphs revisited, Entropy, 1967, 14: 559–570.

    Article  MathSciNet  Google Scholar 

  21. Bianconi G, The entropy of randomized network ensembles, Europhys. Lett., 2008, 81: 28005.

    Article  MathSciNet  Google Scholar 

  22. Minnhagen P and Bernhardsson S, Optimization and scale-freeness for complex networks, Chaos, 2007, 17: 026117.

    Article  Google Scholar 

  23. Park J and Newman M E J, Statistical mechanics of networks, Phys. Rev. E, 2004, 70: 066117.

    Article  MathSciNet  Google Scholar 

  24. Paranjape A, Benson A R, and Leskovec J, Motifs in temporal networks, Proceedings of the 10th ACM International Conference on Web Search and Data Mining, New York, 2017, 601–610.

  25. Kumar S, Spezzano F, Subrahmanianet V S, et al., Edge weight prediction in weighted signed networks, IEEE International Conference on Data Mining (ICDM), Barcelona, 2016, 221–230.

  26. Kumar S, Hamilton W L, Leskovec J, et al., Community interaction and conflict on the web, Proceedings of the 2018 World Wide Web Conference, Lyon, 2018, 933–943.

  27. Hogg T and Lerman K, Social dynamics of digg, EPJ Data Science, 2012, 1: 1–26.

    Article  Google Scholar 

  28. Leskovec J, Kleinberg J, and Faloutsos C. Graphs over time: Densification laws, shrinking diameters and possible explanations, Proceedings of the 11th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Chicago, 2005, 177–187.

  29. Boyd R, Gintis H, Bowles S, et al., Evolution of altruistic punishment, Proceedings of the National Academy of Sciences, 2003, 100: 3531–3535.

    Article  Google Scholar 

  30. Szolnoki A and Perc M, Correlation of positive and negative reciprocity fails to confer an evolutionary advantage: Phase transitions to elementary strategies, Physical Review X, 2013, 3: 041021.

    Article  Google Scholar 

  31. Frank O and Strauss D, Markov graphs, Journal of the American Statistical Association, 1986, 81: 832–842.

    Article  MathSciNet  Google Scholar 

  32. Wasserman S S and Pattison P E, Logit models and logistic regressions for social networks I: An introduction to Markov graphs and p*, Psychometrika, 1996, 61: 401–425.

    Article  MathSciNet  Google Scholar 

  33. Browne W J and Draper D, Implementation and performance issues in the Bayesian and likelihood fitting of multilevel models, Comput. Stat., 2000, 15: 391–420.

    Article  Google Scholar 

  34. Snijders T A, Markov chain monte carlo estimation of exponential random graph models, Journal of Social Structure, 2002, 3: 1–40.

    Google Scholar 

  35. Davis J A and Leinhardt S, The structure of positive interpersonal relations in small groups, Sociological Theories in Progress, 1972, 54: 218–251.

    Google Scholar 

  36. Milo R, ShenOrr S, Itzkovitz S, et al., Network motifs: Simple building blocks of complex networks, Science, 2002, 298: 824–827.

    Article  Google Scholar 

  37. Jazayeri A and Yang C C, Motif discovery algorithms in static and temporal networks: A survey, Journal of Complex Networks, 2020, 8(4): cnaa031.

    Article  MathSciNet  Google Scholar 

  38. Grasso R, Micale G, Ferro A, et al., MODIT: Motif discovery in temporal networks, Frontiers in Big Data, 2022, 4: 806014.

    Article  Google Scholar 

  39. Xie W J, Li M X, Jiang Z Q, et al., Triadic motifs in the dependence networks of virtual societies, Scientific Reports, 2014, 4: 5244.

    Article  Google Scholar 

  40. Winkler M and Reichardt J, Motifs in triadic random graphs based on Steiner triple systems, Physical Review E, 2013, 88(2): 022805.

    Article  Google Scholar 

  41. Wasserman S and Faust K, Social Network Analysis: Methods and Applications, Cambridge University, New York, 1994.

    Book  Google Scholar 

  42. Dehmer M and Mowshowitz A, Generalized graph entropies, Complexity, 2011, 17: 45–50.

    Article  MathSciNet  Google Scholar 

  43. Dehmer M and Mowshowitz A, A history of graph entropy measures, Information Sciences, 2011, 1: 57–78.

    Article  MathSciNet  Google Scholar 

  44. Jones C and Wiesner K, Clarifying how degree entropies and degree-degree correlations relate to network robustness, Entropy, 2022, 24: 1182.

    Article  MathSciNet  Google Scholar 

  45. Heider F, Attitudes and cognitive organization, Journal of Psychology, 1946, 21: 107–112.

    Article  Google Scholar 

  46. Rapoport A, Mathematical models of social interaction, Eds. by Luce R D, Bush R R, Galanteret E, et al., Handbook of Mathematical Psychology, John Wiley, New York, 1963.

    Google Scholar 

  47. Holland P W and Leinhardt S, Transitivity in structural models of small groups, Comparative Group Studies, 1971, 2: 107–124.

    Article  Google Scholar 

  48. Trivers R L, The evolution of reciprocal altruism, Quarterly Review of Biology, 1971, 46: 35–57.

    Article  Google Scholar 

  49. Leskovec J, Huthenlocher D, and Kleinberg J, Signed networks in social media, Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, Atlanta, 2010, 1361–1370.

  50. Elmore F R, Patterns of sociometric choices: Transitivity reconsidered, Social Psychology Quarterly, 1982, 45: 77–85.

    Article  Google Scholar 

  51. Goodreau S M, Kitts J A, and Morris M, Birds of a feather, or friend of a friend? Using exponential random graph models to investigate adolescent social networks, Demography, 2009, 46: 103–125.

    Article  Google Scholar 

  52. Gintis H, Strong reciprocity and human sociality, Journal of Theoretical Biology, 2000, 206: 169–179.

    Article  Google Scholar 

  53. Mowshowitz A, Entropy and the complexity of the graphs I: An index of the relative complexity of a graph, Bull. Math. Biophys., 1968, 30: 175–204.

    Article  MathSciNet  Google Scholar 

  54. Bonchev D, Information indices for atoms and molecules, MATCH-Commumications in Mathematical and in Computer Chemistry, 1979, 7: 65–113.

    Google Scholar 

  55. Bonchev D, Information Theoretic Indices for Characterization of Chemical Structures, Research Studies Press, Chichester, 1983.

    Google Scholar 

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

    MathSciNet  Google Scholar 

  57. Eckmann J P, Moses E, and Sergi D, Entropy of dialogues creates coherent structures in e-mail traffic, Proceedings of the National Academy of Sciences, 2004, 101: 14333.

    Article  MathSciNet  Google Scholar 

  58. Song C, Qu Z, Blumm N, et al., Limits of predictability in human mobility, Science, 2010, 327: 1018–1021.

    Article  MathSciNet  Google Scholar 

  59. Zhang Z, Gao Y, and Li Z L, Consensus reaching for social network group decision making by considering leadership and bounded confidence, Knowledge-Based Systems, 2020, 204: 106240.

    Article  Google Scholar 

  60. Gao Y and Zhang Z, Consensus reaching with non-cooperative behavior management for personalized individual semantics-based social network group decision making, Journal of the Operational Research Society, 2022, 73: 2518–2535.

    Article  Google Scholar 

  61. Dong Q X, Sheng Q, Martinez L, et al., An adaptive group decision making framework: Individual and local world opinions based opinion dynamics, Information Fusion, 2022, 78: 218–231.

    Article  Google Scholar 

  62. Li Z P and Tang X J, Stimuli strategy and learning dynamics promote the wisdom of crowds, Eur. Phys. J. B, 2021, 94: 248.

    Article  Google Scholar 

  63. Dionne S D, Sayama H, Yammarino F J, Diversity and social network structure in collective decision making: Evolutionary perspectives with agent-based simulations, Complexity, 2019, 2019: https://doi.org/10.1155/2019/7591072.

  64. Zhang W, Yan S S, Li J, et al., Credit risk prediction of SMEs in supply chain finance by fusing demographic and behavioral data, Transportation Research Part E, 2022, 158: 102611.

    Article  Google Scholar 

  65. Li Z P and Tang X J, The emergence of triads on signed social network, Eur. Phys. J. Plus, 2022, 137: 381.

    Article  Google Scholar 

Download references

Author information

Author notes

  1. These authors contributed equally: LI Zhenpeng and TANG Xijin.

Authors and Affiliations

  1. School of Electronics and Information Engineering, Taizhou University, Linhai, 317000, China

    Zhenpeng Li

  2. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Zhihua Yan & Xijin Tang

  3. Yunnan Key Laboratory of Smart City in Cyberspace Security, Yuxi Normal University, Yuxi, 653100, China

    Jian Yang

  4. University of Chinese Academy of Sciences, Beijing, 100039, China

    Xijin Tang

Authors
  1. Zhenpeng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhihua Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jian Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Xijin Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhenpeng Li.

Ethics declarations

TANG Xijin is an editorial board member for Journal of Systems Science & Complexity and was not involved in the editorial review or the decision to publish this article. All authors declare that there are no competing interests.

Additional information

This research was supported by the Natural Science Foundation of China under Grant Nos. 71661001 and 71971190, and the project of Yunnan Key Laboratory of Smart City and Cyberspace Security under Grant No. 202105AG070010.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Z., Yan, Z., Yang, J. et al. The Structure Entropy of Social Networks. J Syst Sci Complex 37, 1147–1162 (2024). https://doi.org/10.1007/s11424-024-2484-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-024-2484-x

Keywords

Search

Navigation