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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Community structure of real networks Local model Global model Applications, discussion & perspectives Acknowledgements Structural preferential attachment of community structure Jean-Gabriel Young