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

Dragomir R. Radev

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.

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

- 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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