DEGREE PROPERTIES IN HETEROGENEOUS NETWORKS
Diego Ruiz
Department of Mathematics
University of Cauca
Popayán - 190001, COLOMBIA

Abstract

In social networks it is usual to find nodes that tend to establish relationships based on common attributes like gender and age. To represent this behavior, past models consider a partition of the set of nodes in types and introduce an affinity level, representing the tendency of a node to connect with other nodes of the same type. The partition of the set of nodes in types give rise to a class of networks called heterogeneous. In this work we characterize mathematical expressions for the dynamics and convergence of the probability and complementary cumulative degree distribution functions in a model of heterogeneous networks. We show that the degree distribution of each type of nodes follows a power law characterizing its scaling exponent. Furthermore, using the stability in the sense of Lyapunov of the expected average degree for each type, we propose an approach to detect instants at which the formation of new edges does not follow the mechanisms of the proposed network.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 34
Issue: 6
Year: 2021

DOI: 10.12732/ijam.v34i6.10

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References

  1. [1] A. Abbasi, L. Hossain, L. Leydesdorff, Betweenness centrality as a driver of preferential attachment in the evolution of research collaboration networks, Journal of Informetrics, 6, No 3 (2012), 403-412.
  2. [2] A. Barab´asi, R. Albert, Emergence of scaling in random networks, Science, 286, No 5439 (1999), 509-512.
  3. [3] G. Bianconi, A. Barab´asi, Competition and multiscaling in evolving networks, Europhysics Letters, 54, No 3 (2001), 436-442.
  4. [4] R. Burt, Attachment, decay, and social network, Journal of Organizational Behavior, 22, No 6 (2001), 619-643.
  5. [5] G. Caldarelli, A. Capocci, P. De Los Rios, M. Mu˜noz, Scale-free networks from varying vertex intrinsic fitness, Physical Review Letters, 89, No 25 (2002), Art. 258702.
  6. [6] I. Fernandez, K. Passino, J. Finke, Dynamics of degree distributions of social network, In: Proceeding of the Conference on Decision and Control, Melbourne, Australia, (2017), 0743-1546.
  7. [7] S. Fortunato, A. Flammini, F. Menczer, Scale-free network growth by ranking, Physical Review Letters, 96, No 21 (2006), Art. 218701.
  8. [8] F. Karimi, M. G´enois, C. Wagner, P. Singer, M. Strohmaier, Homophily influences ranking of minorities in social networks, Scientific Reports, 8, No 11077 (2018), 1-12.
  9. [9] E. Lee, F. Karimi, C. Wagner, H. H. Jo, M. Strohmaier, M. Galesic, Homophily and minority-group size explain perception biases in social networks, Nature Human Behavior, 3, No 10 (2019), 1078-1087.
  10. [10] Y. Liu, B. Wei, Y. Du, F. Xiao, Y. Deng, Identifying influential spreaders by weight degree centrality in complex networks, Chaos, Solitons & Fractals, 86 (2016), 1-7.
  11. [11] A. Mantzaris, D. J. Highman, A model for dynamic communicators, European Journal of Applied Mathematics, 23, No 6 (2012), 659668
  12. [12] M. McPherson, L. Smith-Lovin, J. M. Cook, Birds of a feather: homophily in social networks, Annual Review of Sociology, 27, No 1 (2001), 415-444.
  13. [13] R. Pastor-Satorras, A. Vespignani, Epidemic spreading in scale-free networks, Physical Review Letters, 86, No 14 (2001), 3200-3203.
  14. [14] D. Ruiz, J. Campos, J. Finke, Dynamics in affinity-weighted preferential attachment networks, Journal of Statistical Physics, 181, No 2 (2020), 673-689.
  15. [15] D. Ruiz, J. Finke, Lyapunov-based anomaly detection in highly-clustered networks, Journal of Statistical Physics, 172, No 4 (2018), 1127-1146.
  16. [16] D. Ruiz, J. Finke, Lyapunov-based anomaly detection in preferential attachment networks, International Journal of Applied Mathematics and Computer Science, 29, No 2 (2019), 1-11.
  17. [17] M. Sun, Q. Liu, Analysis of dynamical behaviors for a computer virus spreading model, International Journal of Applied Mathematics, 28, No 3 (2015), 237-252; doi: 10.12732/ijam.v28i3.5.
  18. [18] Q. Wu, S. Chen, Mean field theory of epidemic spreading with effective contacts on networks, Chaos, Solitons & Fractals, 81 (2015), 359-364.