Popularity versus similarity in growing networks

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Abstract

The principle1 that ‘popularity is attractive’ underlies preferential attachment2, which is a common explanation for the emergence of scaling in growing networks. If new connections are made preferentially to more popular nodes, then the resulting distribution of the number of connections possessed by nodes follows power laws3,4, as observed in many real networks5,6. Preferential attachment has been directly validated for some real networks (including the Internet7,8), and can be a consequence of different underlying processes based on node fitness, ranking, optimization, random walks or duplication9,10,11,12,13,14,15,16. Here we show that popularity is just one dimension of attractiveness; another dimension is similarity17,18,19,20,21,22,23,24. We develop a framework in which new connections optimize certain trade-offs between popularity and similarity, instead of simply preferring popular nodes. The framework has a geometric interpretation in which popularity preference emerges from local optimization. As opposed to preferential attachment, our optimization framework accurately describes the large-scale evolution of technological (the Internet), social (trust relationships between people) and biological (Escherichia coli metabolic) networks, predicting the probability of new links with high precision. The framework that we have developed can thus be used for predicting new links in evolving networks, and provides a different perspective on preferential attachment as an emergent phenomenon.

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Figure 1: Geometric interpretation of popularity × similarity optimization.
Figure 2: Emergence of PA from popularity × similarity optimization.
Figure 3: Popularity × similarity optimization for three different networks.

References

  1. 1

    Dorogovtsev, S., Mendes, J. & Samukhin, A. WWW and Internet models from 1955 till our days and the “popularity is attractive” principle. Preprint at http://arXiv.org/abs/cond-mat/0009090 (2000)

  2. 2

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

  3. 3

    Krapivsky, P. L., Redner, S. & Leyvraz, F. Connectivity of growing random networks. Phys. Rev. Lett. 85, 4629–4632 (2000)

  4. 4

    Dorogovtsev, S. N., Mendes, J. F. F. & Samukhin, A. N. Structure of growing networks with preferential linking. Phys. Rev. Lett. 85, 4633–4636 (2000)

  5. 5

    Dorogovtsev, S. N. Lectures on Complex Networks (Oxford Univ. Press, 2010)

  6. 6

    Newman, M. E. J. Networks: An Introduction (Oxford Univ. Press, 2010)

  7. 7

    Pastor-Satorras, R., Vázquez, A. & Vespignani, A. Dynamical and correlation properties of the internet. Phys. Rev. Lett. 87, 258701 (2001)

  8. 8

    Jeong, H., Néda, Z. & Barabási, A. L. Measuring preferential attachment in evolving networks. Europhys. Lett. 61, 567–572 (2003)

  9. 9

    Dorogovtsev, S. N., Mendes, J. & Samukhin, A. Size-dependent degree distribution of a scale-free growing network. Phys. Rev. E 63, 062101 (2001)

  10. 10

    Bianconi, G. & Barabási, A.-L. Bose-Einstein Condensation in complex networks. Phys. Rev. Lett. 86, 5632–5635 (2001)

  11. 11

    Caldarelli, G., Capocci, A. & Rios, P. D. L. &. Muñoz, M. A. Scale-free networks from varying vertex intrinsic fitness. Phys. Rev. Lett. 89, 258702 (2002)

  12. 12

    Vázquez, A. Growing network with local rules: preferential attachment, clustering hierarchy, and degree correlations. Phys. Rev. E 67, 056104 (2003)

  13. 13

    Pastor-Satorras, R., Smith, E. & Sole, R. V. Evolving protein interaction networks through gene duplication. J. Theor. Biol. 222, 199–210 (2003)

  14. 14

    Fortunato, S., Flammini, A. & Menczer, F. Scale-free network growth by ranking. Phys. Rev. Lett. 96, 218701 (2006)

  15. 15

    D'Souza, R. M., Borgs, C., Chayes, J. T., Berger, N. & Kleinberg, R. D. Emergence of tempered preferential attachment from optimization. Proc. Natl Acad. Sci. USA 104, 6112–6117 (2007)

  16. 16

    Motter, A. E. & Toroczkai, Z. Introduction: optimization in networks. Chaos 17, 026101 (2007)

  17. 17

    McPherson, M., Smith-Lovin, L. & Cook, J. M. Birds of a feather: homophily in social networks. Annu. Rev. Sociol. 27, 415–444 (2001)

  18. 18

    Simşek, O. & Jensen, D. Navigating networks by using homophily and degree. Proc. Natl Acad. Sci. USA 105, 12758–12762 (2008)

  19. 19

    Redner, S. How popular is your paper? An empirical study of the citation distribution. Eur. Phys. J. B 4, 131–134 (1998)

  20. 20

    Watts, D. J., Dodds, P. S. & Newman, M. E. J. Identity and search in social networks. Science 296, 1302–1305 (2002)

  21. 21

    Börner, K., Maru, J. T. & Goldstone, R. L. The simultaneous evolution of author and paper networks. Proc. Natl Acad. Sci. USA 101, 5266–5273 (2004)

  22. 22

    Crandall, D., Cosley, D., Huttenlocher, D., Kleinberg, J. & Suri, S. in Proc. 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD 2008) (eds Li, Y., Liu, B. & Sarawagi, S. ) 160–168 (ACM, 2008)

  23. 23

    Menczer, F. Growing and navigating the small world Web by local content. Proc. Natl Acad. Sci. USA 99, 14014–14019 (2002)

  24. 24

    Menczer, F. Evolution of document networks. Proc. Natl Acad. Sci. USA 101, 5261–5265 (2004)

  25. 25

    Bonahon, F. Low-Dimensional Geometry (AMS, 2009)

  26. 26

    Bollobás, B. & Riordan, O. in Handbook of Graphs and Networks (eds Bornholdt, S. & Schuster, H. G. ) Ch. 1 1–34 (Wiley-VCH, 2003)

  27. 27

    Adamic, L. A. & Huberman, B. A. Power-law distribution of the World Wide Web. Science 287, 2115 (2000)

  28. 28

    van Raan, A. F. J. On growth, ageing, and fractal differentiation of science. Scientometrics 47, 347–362 (2000)

  29. 29

    Clauset, A., Moore, C. & Newman, M. E. J. Hierarchical structure and the prediction of missing links in networks. Nature 453, 98–101 (2008)

  30. 30

    Menon, A. K. & Elkan, C. in Machine Learning and Knowledge Discovery in Databases (ECML) (eds Gunopulos, D., Hofmann, T., Malerba, D. & Vazirgiannis, M. ) 437–452 (Lecture Notes in Computer Science, Vol. 6912, Springer, 2011)

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Acknowledgements

We thank C. Elkan, G. Bianconi, P. Krapivsky, S. Redner, S. Havlin, E. Stanley and A.-L. Barabási for discussions and suggestions. This work was supported by a Marie Curie International Reintegration Grant within the 7th European Community Framework Programme; MICINN Projects FIS2010-21781-C02-02 and BFU2010-21847-C02-02; Generalitat de Catalunya grant 2009SGR838; the Ramón y Cajal programme of the Spanish Ministry of Science; ICREA Academia prize 2010, funded by the Generalitat de Catalunya; NSF grants CNS-0964236, CNS-1039646, CNS-0722070; DHS grant N66001-08-C-2029; DARPA grant HR0011-12-1-0012; and Cisco Systems.

Author information

F.P. and D.K. planned research, performed research and wrote the paper; M.K., M.A.S. and M.B. planned and performed research. All authors discussed the results and reviewed the manuscript.

Correspondence to Fragkiskos Papadopoulos or Dmitri Krioukov.

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The authors declare no competing financial interests.

Supplementary information

Supplementary Information

This file contains: (i) Supplementary Methods, including details on the real-world network data used in the main text to validate the popularity×similarity optimization approach, and on the network mapping method used to infer the popularity and similarity coordinates; (ii) Supplementary Notes including the technical details of the popularity×similarity model, comparisons between the properties of real-world and modelled networks, and discussion of related work; (iii) Supplementary Figures S1-S16 with legends; and (iv) additional references. (PDF 1064 kb)

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Papadopoulos, F., Kitsak, M., Serrano, M. et al. Popularity versus similarity in growing networks. Nature 489, 537–540 (2012) doi:10.1038/nature11459

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