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**CS8803-NS Network Science Fall 2013**

Instructor: Constantine Dovrolis

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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.

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**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

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**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?

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**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

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**Reference point-1: ER random graphs**

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

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

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**Emergence of giant component**

See lecture notes for derivation of the following

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

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

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**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

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**The configuration model**

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**The configuration model**

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**For instance, power-law degree with exponential cutoff**

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Average path length

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**Clustering coefficient in random networks with given degree distribution**

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**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

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**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

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**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

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**Preferential attachment**

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**Preferential attachment**

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

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**Avg path length in PA model**

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Clustering in PA model

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

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

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**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

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**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

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**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

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**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

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**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

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**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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