9/14/22, 6:49 AM L4: Power-law Degree Distribution : Network Science – CS-7280-O01
L4: Power-law Degree Distribu on
A ¡°power-law network¡± has a degree distribution that is defined by the following equation:
In other words, the probability that the degree of a node is equal to a positive integer is proportional to where >0.
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The proportionality coefficient c is calculated so that the sum of all degree probabilities is equal to one for a given and a given minimum degree (the minimum degree may not always be 1).
The calculation of c can be simplified if we approximate the discrete degree distribution with a continuous distribution:
which gives:
So, the complete equation for a power-law degree distribution is:
The Complementary Cumulative Distribution Function (C-CDF) is:
Note that the exponent of the CCDF function is , instead of . So, if the probability that a node has degree decays with a power-law exponent of 3, the probability that we see nodes with degree greater than decays with an exponent of 2.
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9/14/22, 6:49 AM L4: Power-law Degree Distribution : Network Science – CS-7280-O01
For directed networks, we can have that the in-degree or the out-degree or both follow a power-law distribution (with potentially different exponents).
Food for Thought
Prompt 1: Repeat the derivations given here in more detail.
Prompt 2: Can you think of a network with n nodes in which all nodes have about the same in-degree but the out-degrees are highly skewed?
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