Log-Normal Distribution

Log-normal distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. It produces a right-skewed, heavy-tailed distribution that is often statistically indistinguishable from a power law over realistic data ranges — a fact that has generated significant controversy in network science, where many claimed power-law degree distributions may actually be log-normal. The distribution arises naturally when a variable is the product of many independent positive factors, making it common in finance, biology, and reliability engineering. Unlike a true power law, the log-normal distribution has a well-defined mean and variance, and its tail is not scale-invariant. The distinction matters: networks with log-normal degree distributions may not exhibit the hub-driven dynamics characteristic of scale-free networks.

The ease with which log-normal data masquerades as power-law data on log-log plots is one of the most expensive statistical illusions in modern science.