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COMS 6998-06 Network Theory Week 2: January 31, 2008. Dragomir R. Radev Thursdays, 6-8 PM 233 Mudd Spring 2008. (3) Random graphs. Statistical analysis of networks. We want to be able to describe the behavior of networks under certain assumptions.

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COMS 6998-06 Network Theory Week 2: January 31, 2008

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Coms 6998 06 network theory week 2 january 31 2008 l.jpg

COMS 6998-06 Network TheoryWeek 2: January 31, 2008

Dragomir R. Radev

Thursdays, 6-8 PM

233 Mudd

Spring 2008


Slide2 l.jpg

(3) Random graphs


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Statistical analysis of networks

  • We want to be able to describe the behavior of networks under certain assumptions.

  • The behavior is described by the diameter, clustering coefficient, degree distribution, size of the largest connected component, the presence and count of complete subgraphs, etc.

  • For statistical analysis, we need to introduce the concept of a random graph.


Erdos renyi model l.jpg

Erdos-Renyi model

  • A very simple model with several variants.

  • We fix n and connect each candidate edge with probability p. This defines an ensemble Gn,p

  • The two examples below are specific instances of G10,0.2. In other models, m is fixed. There are also versions in which some graphs are more likely than others, etc.

Try Pajek


Erdos renyi model5 l.jpg

Erdos-Renyi model

  • We are interested in the computation of specific properties of E-R random graphs.

  • The number ofcandidate edges is:

  • The actual number of edges mis on average:

  • We will look at the actual distribution in a bit.


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Properties

  • The expected value of a Poisson-distributed random variable is equal to λ and so is its variance.

  • The mode of a Poisson-distributed random variable with non-integer λ is equal to floor(λ), which is the largest integer less than or equal to λ. When λ is a positive integer, the modes are λ and λ − 1.


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Degree distribution

  • The probability p(k) that a node has a degree k is Binomial:

  • In practice, this is the Poisson distribution for large n (n >> kz)where l is the mean degree

  • Average degree = l= 2m/n = p(n-1) ≈ pn


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Giant component size

  • Let v be the number of nodes that are not in the giant component. Then u=v/n is the fraction of the graph outside of the giant component.

  • If a node is outside of the giant component, its k neighbors are too. The probability of this happening is uk.

  • Let S=1-u. We now haveFor l<1, the only non-negative solution is S=0For l>1 (after the phase transition), the only non-negative solution is the size of the giant component

  • At the phase transition, the component sizes are distributed according to a power law with exponent 5/2.


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Giant component size

  • Similarly one can prove that

[Newman 2003]


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Diameter

  • A given vertex i has Ni1 first neighbors. The expected value of this number is l.

  • But we also know that l = pn.

  • Now move to Ni2. This is the number of second neighbors of i. Let’s make the assumption that these are the neighbors of the first neighbors. So,

  • What does this remind you of?

  • When must the procedure end?


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Diameter (cont’d)

For D equal to the diameter of the graph:

At all distances:

In other words (after taking a logarithm):


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Are E-R graphs realistic?

  • They have small world properties (diameter is logarithmic in the size of the graph)

  • But low clustering coefficient. Example for autonomous internet systems, compare 0.30 with 0.0004 [Pastor-Satorras and Vespignani]

  • And unrealistic degree distributions

  • Not to mention skinny tails


Clustering coefficient l.jpg

Clustering coefficient

  • Given a vertex i and its two real neighbors j and k, what is the probability that the graph contains an edge between j and k.

  • Ci = #triangles at i / #triples at I

  • C = average over all Ci

  • Typical value in real graphs can be as high as 50% [Newman 2002].

  • In random graphs, C = p (ignoring the fact that j and k share a neighbor (i).


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Some real networks

  • From Newman 2002:


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[Newman 2002]


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Graphs with predetermined degree sequences

  • Bender and Canfield introduced this concept.

  • For a given degree sequence, gie the same statistical weight to all graphs in the ensemble.

  • Generate a random sequence in proportion to the predefined sequence

  • Note that the sum of degrees must be even.


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(4) Software


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List of packages

  • Pajek: http://vlado.fmf.uni-lj.si/pub/networks/pajek/

  • Jung: http://jung.sourceforge.net/

  • Guess: http://graphexploration.cond.org/

  • Networkx: https://networkx.lanl.gov/wiki

  • Pynetconv: http://pynetconv.sourceforge.net/

  • Clairlib: http://www.clairlib.org

  • UCINET


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