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

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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  1. COMS 6998-06 Network TheoryWeek 2: January 31, 2008 Dragomir R. Radev Thursdays, 6-8 PM 233 Mudd Spring 2008

  2. (3) Random graphs

  3. 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.

  4. 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

  5. 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.

  6. 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.

  7. 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

  8. 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.

  9. Giant component size • Similarly one can prove that [Newman 2003]

  10. 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?

  11. Diameter (cont’d) For D equal to the diameter of the graph: At all distances: In other words (after taking a logarithm):

  12. 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

  13. 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).

  14. Some real networks • From Newman 2002:

  15. [Newman 2002]

  16. 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.

  17. (4) Software

  18. 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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