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For a given graph p d is a histogram ie p d is the

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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: Power-law 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 scale-free 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¨s-Renyi model? o 18...
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