CS8803-NSNetwork Science

Fall 2013

Instructor: Constantine Dovrolisconstantine@gatech.edu

http://www.cc.gatech.edu/~dovrolis/Courses/NetSci/

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.

Disclaimers

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

• 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.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?