CS8803-NS Network Science Fall 2013

1. CS8803-NSNetwork ScienceFall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu http://www.cc.gatech.edu/~dovrolis/Courses/NetSci/

2. Disclaimers The following slides include only the figures or videos that we use in class; they do not include detailed explanations, derivations or descriptionscovered in class. Many of the following figures are copied from open sources at the Web. I do not claim any intellectual property for the following material.

3. Outline • Network models – Why and how? • Random network models • ER or Poisson random graphs (covered last week) • Random graphs with given degree distribution • Watts-Strogatz model for small-world networks • Network models based on stochastic evolution • Preferential attachment • Variants of preferential attachment • Preferential attachment for weighted networks • Duplication-based models • Network models based on optimization • Fabrikant-Koutsoupias-Papadimitriou model • Application paper: modeling the evolution of the proteome using a duplication-based model • Discussion about network modeling

4. Network models – Why and how? • What does it mean to create a “network model”? • What is the objective of this exercise? • How do we know that a model is “realistic”? • How do we know that a model is “useful”? • How do we compare two models that seem equally realistic? • Do we need models in our “brave new world” of big data?

5. Outline • Network models – Why and how? • Random network models • ER or Poisson random graphs (covered last week) • Random graphs with given degree distribution • Watts-Strogatz model for small-world networks • Network models based on stochastic evolution • Preferential attachment • Variants of preferential attachment • Preferential attachment for weighted networks • Duplication-based models • Network models based on optimization • Fabrikant-Koutsoupias-Papadimitriou model • Application paper: modeling the evolution of the proteome using a duplication-based model • Discussion about network modeling

6. Reference point-1: ER random graphs • G(n,m) and G(n,p) models (see lecture notes for derivations)

7. Emergence of giant connected component in G(n,p) as p increases http://networkx.lanl.gov/archive/networkx-1.1/examples/drawing/giant_component.html

8. Emergence of giant component • See lecture notes for derivation of the following

9. Emergence of giant connected component in G(n,p) as p increases • https://www.youtube.com/watch?v=mpe44sTSoF8

10. Outline • Network models – Why and how? • Random network models • ER or Poisson random graphs (covered last week) • Random graphs with given degree distribution • Watts-Strogatz model for small-world networks • Network models based on stochastic evolution • Preferential attachment • Variants of preferential attachment • Preferential attachment for weighted networks • Duplication-based models • Network models based on optimization • Fabrikant-Koutsoupias-Papadimitriou model • Application paper: modeling the evolution of the proteome using a duplication-based model • Discussion about network modeling

12. The configuration model

13. The configuration model http://mathinsight.org/generating_networks_desired_degree_distribution

14. For instance, power-law degree with exponential cutoff

16. Average path length

17. Clustering coefficient in random networks with given degree distribution

18. Outline • Network models – Why and how? • Random network models • ER or Poisson random graphs (covered last week) • Random graphs with given degree distribution • Watts-Strogatz model for small-world networks • Network models based on stochastic evolution • Preferential attachment • Variants of preferential attachment • Preferential attachment for weighted networks • Duplication-based models • Network models based on optimization • Fabrikant-Koutsoupias-Papadimitriou model • Application paper: modeling the evolution of the proteome using a duplication-based model • Discussion about network modeling

21. Deriving an expression for the APL in this model has been proven very hard • Here is a more important question: • What is the minimum value of p for which we expect to see a small-world (logarithmic) path length? • p >> 1/N

23. Outline • Network models – Why and how? • Random network models • ER or Poisson random graphs (covered last week) • Random graphs with given degree distribution • Watts-Strogatz model for small-world networks • Network models based on stochastic evolution • Preferential attachment • Variants of preferential attachment • Preferential attachment for weighted networks • Duplication-based models • Network models based on optimization • Fabrikant-Koutsoupias-Papadimitriou model • Application paper: modeling the evolution of the proteome using a duplication-based model • Discussion about network modeling

26. Preferential attachment http://www3.nd.edu/~networks/Linked/newfile11.htm

27. Preferential attachment

28. Continuous-time model of PA(see class notes for derivations)

29. Avg path length in PA model

30. Clustering in PA model

32. “Statistical mechanics of complex networks” by R.Albert and A-L.Barabasi

34. Outline • Network models – Why and how? • Random network models • ER or Poisson random graphs (covered last week) • Random graphs with given degree distribution • Watts-Strogatz model for small-world networks • Network models based on stochastic evolution • Preferential attachment • Variants of preferential attachment • Preferential attachment for weighted networks • Duplication-based models • Network models based on optimization • Fabrikant-Koutsoupias-Papadimitriou model • Application paper: modeling the evolution of the proteome using a duplication-based model • Discussion about network modeling

39. Outline • Network models – Why and how? • Random network models • ER or Poisson random graphs (covered last week) • Random graphs with given degree distribution • Watts-Strogatz model for small-world networks • Network models based on stochastic evolution • Preferential attachment • Variants of preferential attachment • Preferential attachment for weighted networks • Duplication-based models • Network models based on optimization • Fabrikant-Koutsoupias-Papadimitriou model • Application paper: modeling the evolution of the proteome using a duplication-based model • Discussion about network modeling

44. Outline • Network models – Why and how? • Random network models • ER or Poisson random graphs (covered last week) • Random graphs with given degree distribution • Watts-Strogatz model for small-world networks • Network models based on stochastic evolution • Preferential attachment • Variants of preferential attachment • Preferential attachment for weighted networks • Duplication-based models • Network models based on optimization • Fabrikant-Koutsoupias-Papadimitriou model • Application paper: modeling the evolution of the proteome using a duplication-based model • Discussion about network modeling