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# 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.

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

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

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.

• 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:

[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.

(4) Software

### List of packages

• 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