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.

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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