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

1 / 19

# COMS 6998-06 Network Theory Week 2: January 31, 2008 - PowerPoint PPT Presentation

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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'COMS 6998-06 Network Theory Week 2: January 31, 2008' - mayten

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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

Thursdays, 6-8 PM

233 Mudd

Spring 2008

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
• 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 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.
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.
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
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.
Giant component size
• Similarly one can prove that

[Newman 2003]

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?
Diameter (cont’d)

For D equal to the diameter of the graph:

At all distances:

In other words (after taking a logarithm):

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
• 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).
Some real networks
• From Newman 2002:
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.
List of packages