Structural preferential attachment of community structure and its relation to Dunbar’s number

Structural preferential attachment of community structure and its relation to Dunbar’s number

The social activity of individuals within communities are limited by their ability to maintain stable relationships with their peers. From a network perspective, this observation translates into empirical limits (Dunbar’s number) on the maximal degrees that nodes can have within each of the communities to which they belong. It has been proposed that this constraint arises as a consequence of an individual’s limited cognition resources. We show that such group behaviour can also be understood as an emerging property of a simple system of two social mechanisms,
independent of the actual nature of the network’s nodes. Our idea is based on the simple assumption that each individual can, for every social group to which it belongs, develop connections and introduce new members. The resulting model accurately reproduces the limited internal degrees that are observed in real social networks. In fact, using our growth mechanism within a recently introduced structural preferential attachment (SPA) model [1], we reproduce with unprecedented accuracy the community structure, the degree distribution and the realistic internal structure of the communities of actual complex networks. This combined stochastic growth model yields an important additional insight into the community structure of networks: it suggests that vast, sparse, and therefore undetectable, communities are naturally occurring in social networks.

[1] Hébert-Dufresne, L., Allard, A., Marceau, V., Noël, P.-A., and Dubé , L.J., Structural Preferential Attachment: Network Organization beyond the Link. Phys. Rev. Lett., 107:158702, 2011.

6f39fdc3a5c2445c6e3b32a19df9e3bb?s=128

Jean-Gabriel Young

June 04, 2014
Tweet

Transcript

  1. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Structural preferential attachment of community structure and its relation to Dunbar’s number 2 4 6 8 10 12 14 16 0 20 40 60 80 100 Mean internal degree Community sizes Jean-Gabriel Young, Laurent H´ ebert-Dufresne Antoine Allard and Louis J. Dub´ e D´ epartement de Physique, de G´ enie Physique, et d’Optique Universit´ e Laval, Qu´ ebec, QC, Canada http://dynamica.phy.ulaval.ca Netsci 2014 – June 4th Structural preferential attachment of community structure Jean-Gabriel Young
  2. Community structure of real networks Local model Global model Applications,

    discussion & perspectives A model of the structure of communities based on empirical observations and its implications 2 4 6 8 10 12 14 16 0 20 40 60 80 100 Mean internal degree Community sizes Jean-Gabriel Young, Laurent H´ ebert-Dufresne Antoine Allard and Louis J. Dub´ e D´ epartement de Physique, de G´ enie Physique, et d’Optique Universit´ e Laval, Qu´ ebec, QC, Canada http://dynamica.phy.ulaval.ca Netsci 2014 – June 4th Structural preferential attachment of community structure Jean-Gabriel Young
  3. Community structure of real networks Local model Global model Applications,

    discussion & perspectives A model of the structure of communities based on empirical observations and its implications 2 4 6 8 10 12 14 16 0 20 40 60 80 100 Mean internal degree Community sizes Jean-Gabriel Young, Laurent H´ ebert-Dufresne Antoine Allard and Louis J. Dub´ e D´ epartement de Physique, de G´ enie Physique, et d’Optique Universit´ e Laval, Qu´ ebec, QC, Canada http://dynamica.phy.ulaval.ca Netsci 2014 – June 4th Structural preferential attachment of community structure Jean-Gabriel Young
  4. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Community structure vs. structure of communities Community structure Or how communities are organized? • Extensively studied. • Large number of detection algorithms. e.g.: S. Fortunato, Physics Report 486 (2010) or J. Xie et al., ACM-CS 45 (2013). Structural preferential attachment of community structure Jean-Gabriel Young
  5. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Community structure vs. structure of communities Community structure Or how communities are organized? • Extensively studied. • Large number of detection algorithms. e.g.: S. Fortunato, Physics Report 486 (2010) or J. Xie et al., ACM-CS 45 (2013). Structure of communities Or what are communities? How are they organized internally? • Multiple definitions inherited from detection algorithms. • Often modeled as Erd˝ os-R´ enyi (ER) graphs. e.g.: C. Seshadri et al., Phys. Rev. E 85 (2012). Structural preferential attachment of community structure Jean-Gabriel Young
  6. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Empirical observation. Detection algorithms • Yield community structures and structure for communities. • Some features are universal across algorithms. Structural preferential attachment of community structure Jean-Gabriel Young
  7. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Empirical observation. Detection algorithms • Yield community structures and structure for communities. • Some features are universal across algorithms. Questions: • Do these features fit within the framework of a ER structure of communities? • What mechanisms lead to such? Structural preferential attachment of community structure Jean-Gabriel Young
  8. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Our goal Introduce a minimal growth model based on empirical observations that reproduces the structure of communities and the community structure of real networks. Structural preferential attachment of community structure Jean-Gabriel Young
  9. Community structure of real networks Local model Global model Applications,

    discussion & perspectives 1 Community structure of real networks 2 Local model 3 Global model 4 Applications, discussion & perspectives Structural preferential attachment of community structure Jean-Gabriel Young
  10. Community structure of real networks Local model Global model Applications,

    discussion & perspectives 1 Community structure of real networks 2 Local model 3 Global model 4 Applications, discussion & perspectives Structural preferential attachment of community structure Jean-Gabriel Young
  11. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Empirical observations 2 4 6 8 10 12 14 16 0 20 40 60 80 100 Mean internal degree Community sizes Cascading Clique Percolation [1] Link Clustering [2] OSLOM [3] Linegraph + Louvain [4] Greedy Clique Expansion [5] Dataset: arXiv cond-mat 2005, G. Palla et al., Nature 435 (2005). Algorithms: [1] J.-G. Young et al., arXiv:1211.1364 (2012). [2] Y.-Y. Ahn et al., Nature 466 (2010). [3] A. Lancichinetti et al., PLoS ONE 6 (2011). [4] T. Evans et al., Phys. Rev. E 80 (2009). [5] C. Lee et al., arXiv:1002.1827 (2010). Structural preferential attachment of community structure Jean-Gabriel Young
  12. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Empirical observations 2 4 6 8 10 12 14 16 0 20 40 60 80 100 Mean internal degree Community sizes Dense regime Cascading Clique Percolation [1] Link Clustering [2] OSLOM [3] Linegraph + Louvain [4] Greedy Clique Expansion [5] Dataset: arXiv cond-mat 2005, G. Palla et al., Nature 435 (2005). Algorithms: [1] J.-G. Young et al., arXiv:1211.1364 (2012). [2] Y.-Y. Ahn et al., Nature 466 (2010). [3] A. Lancichinetti et al., PLoS ONE 6 (2011). [4] T. Evans et al., Phys. Rev. E 80 (2009). [5] C. Lee et al., arXiv:1002.1827 (2010). Structural preferential attachment of community structure Jean-Gabriel Young
  13. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Empirical observations 2 4 6 8 10 12 14 16 0 20 40 60 80 100 Mean internal degree Community sizes Dense regime Sparse regime Cascading Clique Percolation [1] Link Clustering [2] OSLOM [3] Linegraph + Louvain [4] Greedy Clique Expansion [5] Dataset: arXiv cond-mat 2005, G. Palla et al., Nature 435 (2005). Algorithms: [1] J.-G. Young et al., arXiv:1211.1364 (2012). [2] Y.-Y. Ahn et al., Nature 466 (2010). [3] A. Lancichinetti et al., PLoS ONE 6 (2011). [4] T. Evans et al., Phys. Rev. E 80 (2009). [5] C. Lee et al., arXiv:1002.1827 (2010). Structural preferential attachment of community structure Jean-Gabriel Young
  14. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Empirical observations: modeling principles (a) Communities of size n 1 are not completely connected in general ∀ n. (b) Communities of size n ∼ 1 are not sparsely connected in general. (c) Communities are not split in multiple components in general. Structural preferential attachment of community structure Jean-Gabriel Young
  15. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Empirical observations: modeling principles Too dense (a) Communities of size n 1 are not completely connected in general ∀ n. (b) Communities of size n ∼ 1 are not sparsely connected in general. (c) Communities are not split in multiple components in general. Structural preferential attachment of community structure Jean-Gabriel Young
  16. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Empirical observations: modeling principles Too dense Too sparse (a) Communities of size n 1 are not completely connected in general ∀ n. (b) Communities of size n ∼ 1 are not sparsely connected in general. (c) Communities are not split in multiple components in general. Structural preferential attachment of community structure Jean-Gabriel Young
  17. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Empirical observations: modeling principles Too dense Too sparse Not connected (a) Communities of size n 1 are not completely connected in general ∀ n. (b) Communities of size n ∼ 1 are not sparsely connected in general. (c) Communities are not split in multiple components in general. Structural preferential attachment of community structure Jean-Gabriel Young
  18. Community structure of real networks Local model Global model Applications,

    discussion & perspectives 1 Community structure of real networks 2 Local model 3 Global model 4 Applications, discussion & perspectives Structural preferential attachment of community structure Jean-Gabriel Young
  19. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  20. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  21. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  22. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  23. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  24. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  25. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  26. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  27. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  28. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  29. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  30. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  31. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  32. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  33. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  34. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  35. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  36. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  37. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  38. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  39. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  40. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  41. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  42. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  43. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  44. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  45. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  46. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  47. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Modeling the growth of a community Node recruiting event (rate ρn) Link birth event (rate ρ ) Structural preferential attachment of community structure Jean-Gabriel Young
  48. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Degree distribution 0.0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 14 pk k n = 4 n = 7 n = 10 n = 13 0.00 0.02 0.04 0.06 0.08 0.10 0 5 10 15 20 25 30 35 40 45 pk k n = 25 n = 30 n = 35 n = 40 0.00 0.01 0.02 0.03 0.04 0.05 0 20 40 60 80 pk k n = 125 n = 150 n = 200 n = 300 0 0.2 0.4 0.6 0.8 0 10 20 30 40 50 pk k n = 10 n = 20 n = 30 n = 50 0 0.02 0.04 0.06 0 50 100 150 200 250 pk k n = 100 n = 200 n = 300 n = 500 0e+00 2e-03 4e-03 6e-03 8e-03 1e-02 0 100 200 300 400 pk k n = 1000 n = 1300 n = 1700 n = 2000 Internal degree distributions with rates ratios r := ρ /ρn of r = 9 (top) and r = 49 (bottom) Structural preferential attachment of community structure Jean-Gabriel Young
  49. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Degree distribution 0.0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 14 pk k n = 4 n = 7 n = 10 n = 13 0.00 0.02 0.04 0.06 0.08 0.10 0 5 10 15 20 25 30 35 40 45 pk k n = 25 n = 30 n = 35 n = 40 0.00 0.01 0.02 0.03 0.04 0.05 0 20 40 60 80 pk k n = 125 n = 150 n = 200 n = 300 0 0.2 0.4 0.6 0.8 0 10 20 30 40 50 pk k n = 10 n = 20 n = 30 n = 50 0 0.02 0.04 0.06 0 50 100 150 200 250 pk k n = 100 n = 200 n = 300 n = 500 0e+00 2e-03 4e-03 6e-03 8e-03 1e-02 0 100 200 300 400 pk k n = 1000 n = 1300 n = 1700 n = 2000 0.0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 14 pk k n = 4 n = 7 n = 10 n = 13 Internal degree distributions with rates ratios r := ρ /ρn of r = 9 (top) and r = 49 (bottom) Structural preferential attachment of community structure Jean-Gabriel Young
  50. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Degree distribution 0.0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 14 pk k n = 4 n = 7 n = 10 n = 13 0.00 0.02 0.04 0.06 0.08 0.10 0 5 10 15 20 25 30 35 40 45 pk k n = 25 n = 30 n = 35 n = 40 0.00 0.01 0.02 0.03 0.04 0.05 0 20 40 60 80 pk k n = 125 n = 150 n = 200 n = 300 0 0.2 0.4 0.6 0.8 0 10 20 30 40 50 pk k n = 10 n = 20 n = 30 n = 50 0 0.02 0.04 0.06 0 50 100 150 200 250 pk k n = 100 n = 200 n = 300 n = 500 0e+00 2e-03 4e-03 6e-03 8e-03 1e-02 0 100 200 300 400 pk k n = 1000 n = 1300 n = 1700 n = 2000 0.00 0.02 0.04 0.06 0.08 0.10 0 5 10 15 20 25 30 35 40 45 pk k n = 25 n = 30 n = 35 n = 40 Internal degree distributions with rates ratios r := ρ /ρn of r = 9 (top) and r = 49 (bottom) Structural preferential attachment of community structure Jean-Gabriel Young
  51. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Degree distribution 0.0 0.2 0.4 0.6 0.8 0 2 4 6 8 10 12 14 pk k n = 4 n = 7 n = 10 n = 13 0.00 0.02 0.04 0.06 0.08 0.10 0 5 10 15 20 25 30 35 40 45 pk k n = 25 n = 30 n = 35 n = 40 0.00 0.01 0.02 0.03 0.04 0.05 0 20 40 60 80 pk k n = 125 n = 150 n = 200 n = 300 0 0.2 0.4 0.6 0.8 0 10 20 30 40 50 pk k n = 10 n = 20 n = 30 n = 50 0 0.02 0.04 0.06 0 50 100 150 200 250 pk k n = 100 n = 200 n = 300 n = 500 0e+00 2e-03 4e-03 6e-03 8e-03 1e-02 0 100 200 300 400 pk k n = 1000 n = 1300 n = 1700 n = 2000 0.00 0.01 0.02 0.03 0.04 0.05 0 20 40 60 80 pk k n = 125 n = 150 n = 200 n = 300 Internal degree distributions with rates ratios r := ρ /ρn of r = 9 (top) and r = 49 (bottom) Structural preferential attachment of community structure Jean-Gabriel Young
  52. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Results: Internal structure 0.00 0.25 0.50 0.75 1.00 0 5 10 15 pk k Observed internal degree distribution for small size communities of a Sexual Network. Dataset: L.E.C. Da Rocha et al., PLoS Comput. Biol. 7 (2011). Algorithm: Y.-Y. Ahn et al., Nature 466 (2010). Structural preferential attachment of community structure Jean-Gabriel Young
  53. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Results: Internal structure 0.00 0.02 0.04 0.06 0.08 20 40 60 80 100 120 pk k Observed internal degree distribution for medium size communities of a Sexual Network. Dataset: L.E.C. Da Rocha et al., PLoS Comput. Biol. 7 (2011). Algorithm: Y.-Y. Ahn et al., Nature 466 (2010). Structural preferential attachment of community structure Jean-Gabriel Young
  54. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Results: Internal structure 0.00 0.05 0.10 0.15 0.20 0 10 20 30 40 50 pk k Observed internal degree distribution for large size communities of the arXiv. Dataset: G. Palla et al., NJP 10 (2008). Algorithm:T. Evans et al., Phys. Rev. E 80 (2009). Structural preferential attachment of community structure Jean-Gabriel Young
  55. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Average quantities Assuming uncorrelated links in the sparse regime, we may solve for the average number of links L: dL dn = 1 + r 1 − L Lmax(n) n−1 1 + r =⇒ k max = 2 (1 + r) Structural preferential attachment of community structure Jean-Gabriel Young
  56. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Average quantities Assuming uncorrelated links in the sparse regime, we may solve for the average number of links L: dL dn = 1 + r 1 − L Lmax(n) n−1 1 + r =⇒ k max = 2 (1 + r) This simple relation is used to fit the model to data 2 4 6 8 10 12 14 16 0 20 40 60 80 100 Mean internal degree Community sizes (a) arXiv cond-mat 2005 2 3 4 5 6 7 8 9 10 0 20 40 60 80 100 Mean internal degree Community sizes (b) MathSci 2008 Structural preferential attachment of community structure Jean-Gabriel Young
  57. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Average quantities Assuming uncorrelated links in the sparse regime, we may solve for the average number of links L: dL dn = 1 + r 1 − L Lmax(n) n−1 1 + r =⇒ k max = 2 (1 + r) This simple relation is used to fit the model to data 2 4 6 8 10 12 14 16 0 20 40 60 80 100 Mean internal degree Community sizes (a) arXiv cond-mat 2005 2 3 4 5 6 7 8 9 10 0 20 40 60 80 100 Mean internal degree Community sizes (b) MathSci 2008 Structural preferential attachment of community structure Jean-Gabriel Young
  58. Community structure of real networks Local model Global model Applications,

    discussion & perspectives 1 Community structure of real networks 2 Local model 3 Global model 4 Applications, discussion & perspectives Structural preferential attachment of community structure Jean-Gabriel Young
  59. Community structure of real networks Local model Global model Applications,

    discussion & perspectives From a local model to a global model An important feature of the local model The growth rate of a community is proportional to its size: dn dt = nρn Structural preferential attachment of community structure Jean-Gabriel Young
  60. Community structure of real networks Local model Global model Applications,

    discussion & perspectives From a local model to a global model An important feature of the local model The growth rate of a community is proportional to its size: dn dt = nρn Structural Preferential Attachment (SPA) [1] • Communities grow preferentially to their size. • Nodes join communities preferentially to their memberships. • Parametrized by only 2 probabilities p, q that dictate how fast new communities/nodes are introduced. [1] L. H´ ebert-Dufresne et al., Phys. Rev. Lett. 107 (2011) Structural preferential attachment of community structure Jean-Gabriel Young
  61. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Structural Preferential Attachment (SPA) One time step in the life of a growing network. Old node joins Old structure New node joins New structure Old node joins New structure New node joins Old structure L. H´ ebert-Dufresne et al., Phys. Rev. Lett. 107 (2011) Structural preferential attachment of community structure Jean-Gabriel Young
  62. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Structural Preferential Attachment (SPA) One time step in the life of a growing network. Old node joins Old structure New node joins New structure Old node joins New structure New node joins Old structure L. H´ ebert-Dufresne et al., Phys. Rev. Lett. 107 (2011) Structural preferential attachment of community structure Jean-Gabriel Young
  63. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Structural Preferential Attachment (SPA) One time step in the life of a growing network. Old node joins Old structure New node joins New structure Old node joins New structure New node joins Old structure L. H´ ebert-Dufresne et al., Phys. Rev. Lett. 107 (2011) Structural preferential attachment of community structure Jean-Gabriel Young
  64. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Structural Preferential Attachment (SPA) One time step in the life of a growing network. Old node joins Old structure New node joins New structure Old node joins New structure New node joins Old structure L. H´ ebert-Dufresne et al., Phys. Rev. Lett. 107 (2011) Structural preferential attachment of community structure Jean-Gabriel Young
  65. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Structural Preferential Attachment (SPA) One time step in the life of a growing network. Old node joins Old structure New node joins New structure Old node joins New structure New node joins Old structure L. H´ ebert-Dufresne et al., Phys. Rev. Lett. 107 (2011) Structural preferential attachment of community structure Jean-Gabriel Young
  66. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Structural Preferential Attachment (SPA) Varying p and q, plethora of structures can be obtained L. H´ ebert-Dufresne et al., Phys. Rev. Lett. 107 (2011) Structural preferential attachment of community structure Jean-Gabriel Young
  67. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Results (1 of 2): Global distributions 4 8 12 16 0 20 40 60 80 100 k n n r = 6.5 5 10 15 20 0 20 40 60 80 100 k n n r = 1.6 10−5 10−4 10−3 10−2 10−1 100 100 101 102 Distribution Quantity Membership Size (a) arXiv with CCP 10−3 10−2 10−1 100 100 101 102 Distribution Quantity Membership Size (b) Enron with GCE CCP: Cascading Clique Percolation. J.-G. Young et al., arXiv:1211.1364 (2012). GCE: Greedy Clique Expansion. C. Lee et al., arXiv:1002.1827 (2010). Structural preferential attachment of community structure Jean-Gabriel Young
  68. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Results (1 of 2): Global distributions 4 8 12 16 0 20 40 60 80 100 k n n r = 6.5 5 10 15 20 0 20 40 60 80 100 k n n r = 1.6 10−5 10−4 10−3 10−2 10−1 100 100 101 102 Distribution Quantity Degree Membership Size (a) arXiv with CCP 10−3 10−2 10−1 100 100 101 102 Distribution Quantity Degree Membership Size (b) Enron with GCE CCP: Cascading Clique Percolation. J.-G. Young et al., arXiv:1211.1364 (2012). GCE: Greedy Clique Expansion. C. Lee et al., arXiv:1002.1827 (2010). Structural preferential attachment of community structure Jean-Gabriel Young
  69. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Results (2 of 2): Structural correlations m n k(n,m) ____ k(n,1) 2 4 6 8 10 12 4 6 8 10 12 4 6 8 10 12 2 4 6 8 10 12 arXiv LCA 2 1 0 Structural correlations: normalized average internal degree as a function of memberships and community sizes k(m, n). arXiv cond-mat 2005 co-authorship network, for a community structure detected by LCA [1]. [1] LCA: Link Clustering Algorithm. Y.-Y. Ahn et al., Nature 466 (2010) Structural preferential attachment of community structure Jean-Gabriel Young
  70. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Results (2 of 2): Structural correlations 2 1 0 m n k(n,m) ____ k(n,1) 2 4 6 8 10 12 4 6 8 10 12 4 6 8 10 12 2 4 6 8 10 12 4 6 8 10 12 2 4 6 8 10 12 4 6 8 10 12 2 4 6 8 10 12 4 6 8 10 12 2 4 6 8 10 12 4 6 8 10 12 2 4 6 8 10 12 4 6 8 10 12 2 4 6 8 10 12 4 6 8 10 12 2 4 6 8 10 12 Detected Modeled arXiv Enron arXiv Enron OSLOM LCA Structural correlations: normalized average internal degree as a function of memberships and community sizes k(m, n). Enron email exchange and arXiv cond-mat 2005 co-authorship networks, for a community structure detected by LCA [1] and OSLOM [2]. [1] LCA: Link Clustering Algorithm. Y.-Y. Ahn et al., Nature 466 (2010) [2] OSLOM: Order statistics local optimization method. A. Lancichinetti et al., PLoS ONE 6 (2011) Structural preferential attachment of community structure Jean-Gabriel Young
  71. Community structure of real networks Local model Global model Applications,

    discussion & perspectives 1 Community structure of real networks 2 Local model 3 Global model 4 Applications, discussion & perspectives Structural preferential attachment of community structure Jean-Gabriel Young
  72. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Dunbar’s number or the social brain Dunbar: Maximal clique size is related to the organization of the brain. Our model: The opposition of linear densification and ex- ponential growth is a sufficient condition. → Implemented through 2 competing processes: 1 Bonding. 2 ’Recruiting’. 2 4 6 8 10 12 14 16 0 20 40 60 80 100 Mean internal degree Community sizes R.I.M. Dunbar, Evolutionary Anthropology 6 (1998) Structural preferential attachment of community structure Jean-Gabriel Young
  73. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Future directions Benchmarking • Complex structural properties are emergent features of the SPA model. • Low dimensionality parameter space (3 parameters:p,q,r). • Natural resolution limit. Hierarchical generalization • SPA uses only 2 level of organizations. • A straightforward generalization to d level has been proposed. • This growth model could be used at the lower level of organization. Structural preferential attachment of community structure Jean-Gabriel Young
  74. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Acknowledgements dynamica.phy.ulaval.ca • Laurent H´ ebert-Dufresne • Antoine Allard • Prof. Louis J. Dub´ e Structural preferential attachment of community structure Jean-Gabriel Young
  75. Community structure of real networks Local model Global model Applications,

    discussion & perspectives Acknowledgements Structural preferential attachment of community structure Jean-Gabriel Young