CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu http://www.cc.gatech.edu/~dovrolis/Courses/NetSci/

Disclaimers The following slides include only the figures or videos that we use in class; they do not include detailed explanations, derivations or descriptions covered 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.

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

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?

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

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

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

Emergence of giant component See lecture notes for derivation of the following

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

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

The configuration model

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

For instance, power-law degree with exponential cutoff

Average path length

Clustering coefficient in random networks with given degree distribution

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

Here is a more important question: 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

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

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

Preferential attachment

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

Avg path length in PA model

Clustering in PA model

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

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

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

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

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

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

Discussion about network models Random? Stochastic evolution? Optimization-based? How to choose? When does it matter? How do we compare two models that seem equally realistic? “All models are wrong but some are useful” But when is a model useful?