Why So Many People Have Friends With More Friends Than They Have:
Positive Skewness, High Variation, and Positive Assortativity
There are three very different reasons why people tend to find that their friends have more friends on average than they have, depending upon three very different characteristics of the distributions of friends and their friends. Each of those characteristics of the distributions of friends and their friends contributes to the proportion of people who find that their friends have more friends on average than they do. It is common for each of these characteristics to be such that one can generally expect that an overwhelming proportion of people find a difference in that direction. The particular proportion will depend upon the extent of each of these conditions as well as somewhat on very specific conditions of the situation. This paper is about these three conditions that increase the extent to which people find that their friends have more friends than they do:
1)The skewness of the distribution of friends such that there is a long tail to the right implies that the mean of the distribution is above the median, the 50th percentile. If everyone’s friends have the mean number of friends for friends, then the proportion of people whose friends have a mean greater than they have is exactly the percentile corresponding to the mean. That is always greater than 50%, but can range from slightly greater to nearly 100% if one person is friends with everyone else.
2) The variation in the distribution relative to the mean increases the mean number of friends of friends above the mean number of friends for people, which (as stated above) is always greater than the median. The percentile for people of the mean for friends determines that proportion of people whose friends have more friends than they do.
3) However, if there is any association between the number of friends of a person and his/her friends mean number of friends, then that has further implications for the proportion of people whose friends, on average, have more friends than they do Specifically, if people with more friends tend to have friends with more friends, then the mean of the mean differences between people and their friends will be less than the mean difference between friends and people. However, the variation around that mean difference will also be less. The proportion of people whose friends have mean numbers greater than they is largely determined by the mean difference divided by the standard deviation of the differences. If there is a strong association of numbers of friends of people and their friends, nearly everyone may have friends with a mean slightly higher than the number they have
We present analytic results, simulations, and empirical results to show how each of these factors affects the proportion of people whose friends have friends with more friends on average than they have.