Friendship paradox
[2][3][4][5] The same observation can be applied more generally to social networks defined by other relations than friendship: for instance, most people's sexual partners have had (on the average) a greater number of sexual partners than they have.
[6][7] The friendship paradox is an example of how network structure can significantly distort an individual's local observations.
[8][9] In spite of its apparently paradoxical nature, the phenomenon is real, and can be explained as a consequence of the general mathematical properties of social networks.
[10] Formally, Feld assumes that a social network is represented by an undirected graph G = (V, E), where the set V of vertices corresponds to the people in the social network, and the set E of edges corresponds to the friendship relation between pairs of people.
The friendship between x and y is therefore modeled by the edge {x, y}, and the number of friends an individual has corresponds to a vertex's degree.
This amounts to choosing, uniformly at random, an edge of the graph (representing a pair of friends) and an endpoint of that edge (one of the friends), and again calculating the degree of the selected endpoint.
The halving factor simply comes from the fact that each edge has two vertices.
This allows us to compute the desired expected value as For a graph that has vertices of varying degrees (as is typical for social networks),
is strictly positive, which implies that the average degree of a friend is strictly greater than the average degree of a random node.
However, this conclusion is not a mathematical certainty; there exist undirected graphs (such as the graph formed by removing a single edge from a large complete graph) that are unlikely to arise as social networks but in which most vertices have higher degree than the average of their neighbors' degrees.
Note that this can count 2-hop neighbors multiple times, but so does Feld's analysis.
Feld considered the following "micro average" quantity.
However, there is also the (equally legitimate) "macro average" quantity, given by The computation of MacroAvg can be expressed as the following pseudocode.
[11] In a 2023 paper, a parallel paradox, but for negative, antagonistic, or animosity ties, termed the "enmity paradox," was defined and demonstrated by Ghasemian and Christakis.
This paper also documented diverse phenomena is "mixed worlds" of both hostile and friendly ties.
The analysis of the friendship paradox implies that the friends of randomly selected individuals are likely to have higher than average centrality.
This observation has been used as a way to forecast and slow the course of epidemics, by using this random selection process to choose individuals to immunize or monitor for infection while avoiding the need for a complex computation of the centrality of all nodes in the network.
[13][14][15] In a similar manner, in polling and election forecasting, friendship paradox has been exploited in order to reach and query well-connected individuals who may have knowledge about how numerous other individuals are going to vote.
[16] However, when utilized in such contexts, the friendship paradox inevitably introduces bias by over-representing individuals with many friends, potentially skewing resulting estimates.
[17][18] A study in 2010 by Christakis and Fowler showed that flu outbreaks can be detected almost two weeks before traditional surveillance measures would do so by using the friendship paradox in monitoring the infection in a social network.
[19] They found that using the friendship paradox to analyze the health of central friends is "an ideal way to predict outbreaks, but detailed information doesn't exist for most groups, and to produce it would be time-consuming and costly.
[21][13][22] This observation has been explained with the argument that individuals with more social connections may be the driving forces behind the spread of these ideas and beliefs, and as such can be used as early-warning signals.
[18] Friendship paradox based sampling (i.e., sampling random friends) has been theoretically and empirically shown to outperform classical uniform sampling for the purpose of estimating the power-law degree distributions of scale-free networks.
[23][24] The reason is that sampling the network uniformly will not collect enough samples from the characteristic heavy tail part of the power-law degree distribution to properly estimate it.
However, sampling random friends incorporates more nodes from the tail of the degree distribution (i.e., more high degree nodes) into the sample.
Hence, friendship paradox based sampling captures the characteristic heavy tail of a power-law degree distribution more accurately and reduces the bias and variance of the estimation.
[28] The same effect has also been demonstrated for Subjective Well-Being by Bollen et al. (2017),[29] who used a large-scale Twitter network and longitudinal data on subjective well-being for each individual in the network to demonstrate that both a Friendship and a "happiness" paradox can occur in online social networks.
The friendship paradox has also been used as a means to identify structurally influential nodes within social networks, so as to magnify social contagion of diverse practices relevant to human welfare and public health.
This has been shown to be possible in several large-scale randomized controlled field trials conducted by Christakis et al., with respect to the adoption of multivitamins[30] or maternal and child health practices[31][32] in Honduras, or of iron-fortified salt in India.
[33] This technique is valuable because, by exploiting the friendship paradox, one can identify such influential nodes without the expense and delay of actually mapping the whole network.
