Social behaviours are often contagious. To adopt a new technology or idea, listen to music, engage in risky behaviour, or join a social movement, people usually observe others before making a decision themselves. Why do some behaviours spread like wildfire from a small number of initial adopters to a large portion of the population, while others remain mostly unseen? Content itself cannot often explain the difference.

One interesting finding is that the configuration of initial adopters on a social network can systematically skew the observations people make of their friends’ behaviour. This can create the illusion that something is much more popular than it actually is, thus creating conditions for its spread.

Network scientists have known about the paradoxical nature of social networks for some time. One well-known counter-intuitive property of networks is the friendship paradox, which states that, on average, most people have fewer friends than their own friends. In fact, any attribute that is correlated with degree will produce a paradox. For example, your co-authors are cited more often than you, and the people you follow on Twitter post more frequently than you do.

in “The Majority Illusion in Social Networks”⁽¹⁾ Kristina Lerman, Xiaoran Yan, and Xin-Zeng Wu describe a novel variation of the friendship paradox that is essential for understanding contagious behaviours. It applies to networks in which nodes have simple binary attributes, such as “has red hair” vs “does not have red hair” or “purchased an iPhone” vs “did not purchase an iPhone”. Under some conditions, a large fraction of nodes will observe most of their neighbours in one of those states, even when such state is globally rare. For this reason, they call the paradox the “majority illusion.”

A simple illustration of the “majority illusion” paradox, is shown in Figure above. The networks are identical, except for the few nodes which are red. Despite this apparently small difference, the two networks are very different: in the first network, every white node will examine its neighbours to observe that “at least half of my neighbours are red,” while in the second network, no node will make this observation. Thus, even though only three of the 14 nodes are red, it appears to all white nodes in the first network that most of their neighbours are red.

A simple math model. No cheating. Appearances can be deceiving!

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(1) Kristina Lerman, Xiaoran Yan, & Xin-Zeng Wu (2015). The Majority Illusion in Social Networks arxiv arXiv: 1506.03022v1