Power Law

Abstract

A quantity x is said to obey a power law is it is drawn from a probability distribution given by $p(x) \propto x^{-\alpha }$ where $\alpha$ is a constant parameter known as exponent. In this chapter we will look at ways to determine whether or not a certain set of values follow a power law. We will learn graph models that can exhibit power-law, mainly focusing on the preferential attachment model that has power-law degree distribution. We will then look at the rich-get-richer phenomenon and how this is prevalent in citation networks and population growth of cities. Finally, we will cover densification power laws and shrinking diameters which are properties observed from temporal social networks, which have given rise to the forest fire model.

Problems

Given that the probability density function (PDF) of a power-law distribution is given by Eq. 11.13.

$$\begin{aligned} P(X=x) = \frac{\alpha -1}{x_{min}}\left( \frac{x}{x_{min}}\right) ^{-\alpha } \end{aligned}$$

(11.13)

where $x_{min}$ is the minimum value that X can take.

60

Derive an expression of $P(X\ge x)$, the Complementary Cumulative Distribution Function (CCDF), in terms of $\alpha$.

61

Show how to generate a random sample from the power-law distribution using the CCDF derived and a uniform random sample $u \sim U(0,1)$.

62

Using this sampling technique, create a dataset of 10000 samples following the power-law distribution with exponent $\alpha =2$ and $x_{min}=1$. Plot the empirical distribution on a log-log scale by first rounding each sample to the nearest integer and then plotting the empirical PDF over these rounded values. Also plot the true probability density function for the power law (this will help verify that the data was generated correctly).