Model of Complex Networks based on Citation Dynamics Lovro Subelj - - PowerPoint PPT Presentation

Model of Complex Networks based on Citation Dynamics

Lovro ˇ Subelj & Marko Bajec

University of Ljubljana Faculty of Computer and Information Science

LSNA ’13

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Introduction

Introduction

Real-world networks are scale-free, small-world etc. Social networks are degree assortative. (Newman and Park, 2003) ֒ → Properties captured by many models in the literature. However, non-social networks are degree disassortative!

Figure: Part of Cora citation network with highlighted hubs.

For simplicity, we consider only undirected networks.

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Models of complex networks

Forest Fire model (Leskovec et al., 2007)

Let p be the burning probability.

1 i chooses an ambassador a and links to it; 2 i selects xp ∼ G(

p 1−p) neighbors a1, . . . , axp and links to them;

3 a1, . . . , axp are taken as the ambassadors of i. y z a x i w v y z a x i w v

Networks are scale-free, small-world, degree assortative etc. Natural interpretation for citation networks!

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Models of complex networks

Author citation dynamics

1 author chooses a paper (i.e., ambassador) and cites it; 2 author selects some of its references and cites them; 3 the latter are taken as the ambassadors. y z a x i w v y z a x i w v

Assumption → authors read all papers they cite (and vice-versa). Only ≈ 20% of cited papers are read. (Simkin and Roychowdhury, 2003) Authors read or cite papers due to two (independent) processes!

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Models of complex networks

Citation model (our)

Let q be the linking probability.

1 i chooses an ambassador a; 2 i selects xp ∼ G(

p 1−p) neighbors a1, . . . , axp;

i selects xq ∼ G(

q 1−q) neighbors and links to them;

3 a1, . . . , axp are taken as the ambassadors of i. y z a x i w v y z a x i w v

Networks are scale-free, small-world, degree disassortative etc. Nodes do not (necessarily) link to their ambassadors!

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Models of complex networks

Alternative models

y z a x i w v y z a x i w v

Forest Fire (Leskovec et al., 2007)

y z a x i w v y z a x i w v

Butterfly (McGlohon et al., 2008)

y z a x i w v y z a x i w v

Copying (Krapivsky and Redner, 2005)

y z a x i w v y z a x i w v

Citation model (our)

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Models of complex networks

Analysis of the models

S, T are the ambassadors and linked nodes. Forest Fire model: S = T Butterfly model: S ⊇ T Copying model: S ⊆ T Citation model: S, T arbitrary

y z a x i w v y z a x i w v y z a x i w v y z a x i w v y z a x i w v y z a x i w v y z a x i w v y z a x i w v

Why degree disassortativity? Linking to the ambassadors increases assortativity. Absence of such a process prevents assortativity. (Newman and Park, 2003) Heterogeneous networks are disassortative. (Johnson et al., 2010)

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Experimental analysis

Comparison of the models (k & r)

Only Citation model gives degree disassortative networks (i.e., r < 0).

0.1 0.2 0.3 0.4 2 4 6 8 10 12 14

Forest Fire Butterfly Copying Citation Burning probability p Network degree k

0.1 0.2 0.3 0.4

• 0.4
• 0.2

0.2 0.4 0.6 0.8

Forest Fire Butterfly Copying Citation Burning probability p Degree mixing r

0.2 0.4 0.6 0.8 2 4 6 8 10 12 14

Forest Fire Butterfly Copying Citation Linking probability q Network degree k

0.2 0.4 0.6 0.8

• 0.4
• 0.2

0.2 0.4 0.6 0.8

Forest Fire Butterfly Copying Citation Linking probability q Degree mixing r

Shaded regions show most likely parameter values. (Laurienti et al., 2011)

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Experimental analysis

Comparison of the models (l, C & Q)

All models give (scale-free) small-world networks with high modularity.

0.1 0.2 0.3 0.4 2 4 6 8 10 12 14 16 18

Forest Fire Butterfly Copying Citation Burning probability p Mean distance l

0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8

Forest Fire Butterfly Copying Citation Burning probability p Network clustering C

0.1 0.2 0.3 0.4 0.2 0.4 0.6 0.8 1

Forest Fire Butterfly Copying Citation Burning probability p Network modularity Q

0.2 0.4 0.6 0.8 2 4 6 8 10 12 14 16 18

Forest Fire Butterfly Copying Citation Linking probability q Mean distance l

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

Forest Fire Butterfly Copying Citation Linking probability q Network clustering C

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1

Forest Fire Butterfly Copying Citation Linking probability q Network modularity Q

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Experimental analysis

Parameter estimation

s is the number of ambassadors, s = |S|. s ≤ 1 − p 1 − 2p and k ≤ 2qs 1 − q − (1 − q)s+1 For a given k and fixed q, the system can be solved for p.

0.1 0.2 0.3 0.4 1 2 3 4 5

100 500 1000 Burning probability p # ambassadors s

0.2 0.4 0.6 0.8 5 10 15 20 25 30

100 500 1000 Linking probability q Network degree k

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Experimental analysis

Cora citation network

p q n m k r Cora network 23166 89157 7.697 −0.055 Forest Fire 0.46

• 23166 88828 7.669

0.211 Citation 0.37 0.59 23166 89888 7.760 −0.047 Percentage of papers considered is 66% (# references just 3.85)!

1 10 100 1000 0.0001 0.001 0.01 0.1

Forest Fire Citation Node degree k Degree distribution P(k) Cora network

1 10 100 10 100 1000

Forest Fire Citation Node degree k Neighbor degree kN Cora network

For other network properties see paper and (ˇ Subelj and Bajec, 2012).

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Experimental analysis

arXiv citation network

p q n m k r arXiv network 27400 352021 25.695 −0.030 Citation 0.46 0.67 27400 350699 25.598 −0.068 Percentage of papers considered is 49% (# references is 12.85)!

1 10 100 1000 0.0001 0.001 0.01 0.1

Citation Node degree k Degree distribution P(k) arXiv network

1 10 100 1000 10 100 1000

Citation Node degree k Neighbor degree kN arXiv network

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Conclusions

Conclusions

Model for citation networks with most common properties. (Non-social) degree non-assortative networks → nodes must not link to their ambassadors! Networks also show dichotomous mixing. (Hao and Li, 2011) Future work: extension to directed networks, network traversal (isolated nodes), analyses on reliable data (e.g., WoS).

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Questions & comments

lovro.subelj@fri.uni-lj.si http://lovro.lpt.fri.uni-lj.si/

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• D. Hao and C. Li. The dichotomy in degree correlation of biological networks. PLoS ONE, 6

(12):e28322, 2011. doi: 10.1371/journal.pone.0028322.

• S. Johnson, J. J. Torres, J. Marro, and M. A. Mu˜
• noz. Entropic origin of disassortativity in

complex networks. Phys. Rev. Lett., 104(10):108702, 2010. doi: 10.1103/PhysRevLett.104.108702.

• P. L. Krapivsky and S. Redner. Network growth by copying. Phys. Rev. E, 71(3):036118, 2005.

doi: 10.1103/PhysRevE.71.036118.

• P. J. Laurienti, K. E. Joyce, Q. K. Telesford, J. H. Burdette, and S. Hayasaka. Universal fractal

scaling of self-organized networks. Physica A, 390(20):3608–3613, 2011. doi: 16/j.physa.2011.05.011.

• J. Leskovec, J. Kleinberg, and C. Faloutsos. Graph evolution: Densification and shrinking
• diameters. ACM Trans. Knowl. Discov. Data, 1(1):1–41, 2007.
• M. McGlohon, L. Akoglu, and C. Faloutsos. Weighted graphs and disconnected components:

Patterns and a generator. In Proceedings of the ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, page 524–532, New York, NY, USA, 2008.

• M. E. J. Newman and J. Park. Why social networks are different from other types of networks.
• Phys. Rev. E, 68(3):036122, 2003. doi: 10.1103/PhysRevE.68.036122.
• M. V. Simkin and V. P. Roychowdhury. Read before you cite! Compl. Syst., 14:269–274, 2003.
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Subelj and M. Bajec. Clustering assortativity, communities and functional modules in real-world networks. e-print arXiv:12082518v1, pages 1–21, 2012.

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Subelj and M. Bajec. Model of complex networks based on citation dynamics. In Proceedings

• f the WWW Workshop on Large Scale Network Analysis, page 4, Rio de Janeiro, Brazil,

2013.

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