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.

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
   

   View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

    View Slide

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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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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
    

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

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