Preferential Attachment

Power Law Distributions

Preferential attachment is one (but not the only) mechanism that has been proposed to explain the dynamics and structure of several biological and social phenomena whose distributions look different from normal or bell curve distributions and instead are highly skewed and said to follow a power law. While the distribution of some phenomena, like height, follow a normal bell curve distribution where even the extremes of human height remain fairly tightly distributed around the average, other phenomena, such as several kinds of social networks, have been found to have properties that are highly skewed and are described as a power law or long-tail distribution. These distributions are called long tail because of the shape of the graph that is generated when two characteristics of an event or phenomenon, like size and frequency, are plotted on a graph. The distribution of the population of U.S. cities, for instance, has a long-tail distribution. While there are a fairly large number of small and medium-sized cities, just a few cities with population sizes several orders of magnitude larger than the rest account for most of the population of the United States. The distribution of wealth also follows a power law distribution. It is sometimes called the 80–20 rule (or a Pareto distribution for the Italian economist who first formulated it), wherein 20 percent of the population hold 80 percent of the wealth in a society.

[Page 719]Power law distributions have been found in social network analysis where a network is analyzed by mapping the distribution of links or connections among all of the nodes (individuals, Web pages, or social media pages) in the network. In several social networks, the distribution of links among nodes is not normal, as a few nodes are extremely well connected while the majority of nodes have very few connections. Social networks such as film actor networks, some communication networks, scientific collaboration networks, and some online social networks can be characterized by power law distributions. In fact, the sheer number of natural and social phenomena that have been found to follow a power law distribution is one of the reasons why there has been so much interest in power laws and the mechanisms, such as preferential attachment, that have been proposed as generating power law distributions. Some scientists have even suggested that if one defines “normal” by the sheer quantity of occurrence of the phenomenon, the power law distribution is so widespread that it makes more sense to consider the power law distribution as the normal distribution.

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