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Page 1: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

CS8803-NSNetwork Science

Fall 2013

Instructor: Constantine [email protected]

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

Page 2: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@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

Page 3: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 4: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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?

Page 5: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 6: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

Reference point-1: ER random graphs

• G(n,m) and G(n,p) models (see lecture notes for derivations)

Page 7: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 8: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

Emergence of giant component

• See lecture notes for derivation of the following

Page 9: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

Emergence of giant connected component in G(n,p) as p increases

• https://www.youtube.com/watch?v=mpe44sTSoF8

Page 10: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 11: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 12: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

The configuration model

Page 13: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

The configuration model

http://mathinsight.org/generating_networks_desired_degree_distribution

Page 14: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

For instance, power-law degree with exponential cutoff

Page 15: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 16: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

Average path length

Page 17: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

Clustering coefficient in random networks with given degree distribution

Page 18: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 19: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 20: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 21: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

• 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

Page 22: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 23: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 24: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 25: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 26: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

Preferential attachment

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

Page 27: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

Preferential attachment

Page 28: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 29: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

Avg path length in PA model

Page 30: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

Clustering in PA model

Page 31: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 32: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 33: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 34: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 35: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 36: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 37: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 38: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 39: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 40: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 41: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 42: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 43: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 44: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 45: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 46: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 47: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 48: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 49: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 50: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 51: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 52: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 53: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 54: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 55: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 56: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 57: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 58: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci
Page 59: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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

Page 60: CS8803-NS Network Science Fall 2013 Instructor: Constantine Dovrolis constantine@gatech.edu dovrolis/Courses/NetSci

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?


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