Unformatted text preview: ree Distributions
The degree distribution, P (d ), of a network is a description of relative
frequencies of nodes that have diﬀerent degrees d .
For a given graph: P (d ) is a histogram, i.e., P (d ) is the fraction of
nodes with degree d .
For a random graph model: P (d ) is a probability distribution.
Two types of degree distributions:
P (d ) ≤ c e −αd , for some α > 0 and c > 0: The tail of the distribution
falls oﬀ faster than an exponential, i.e., large degrees are unlikely.
P (d ) = c d −γ , for some γ > 0 and c > 0: Powerlaw distribution:
The tail of the distribution is fat, i.e., there tend to be many more
nodes with very large degrees.
Appear in a wide variety of settings including networks describing
incomes, city populations, WWW, and the Internet
Also known as a scalefree distribution: a distribution that is unchanged
(within a multiplicative factor) under a rescaling of the variable
Appear linear on a log − log plot What is the degree distribution of the Erd¨sRenyi model?
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 Fall '09
 Acemoglu
 Graph Theory, Graph Theory and Social Networks

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