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The friendship paradox is the phenomenon first observed by the sociologist Scott L. Feld in 1991 that most people have fewer friends than their friends have, on average.〔.〕 It can be explained as a form of sampling bias in which people with greater numbers of friends have an increased likelihood of being observed among one's own friends. In contradiction to this, most people believe that they have more friends than their friends have.〔.〕 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.〔.〕〔.〕 ==Mathematical explanation== 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. The mathematics behind this are directly related to the arithmetic-geometric mean inequality and the Cauchy–Schwarz inequality. Formally, Feld assumes that a social network is represented by an undirected graph , where the set of vertices corresponds to the people in the social network, and the set of edges corresponds to the friendship relation between pairs of people. That is, he assumes that friendship is a symmetric relation: if is a friend of , then is a friend of . He models the average number of friends of a person in the social network as the average of the degrees of the vertices in the graph. That is, if vertex has edges touching it (representing a person who has friends), then the average number of friends of a random person in the graph is : The average number of friends that a typical friend has can be modeled by 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. That is, mathematically, it is : where is the variance of the degrees in the graph. For a graph that has vertices of varying degrees (as is typical for social networks), both and are positive, which implies that the average degree of a friend is strictly greater than the average degree of a random node. Another way of understanding how the first term came is as follows. For each friendship , a node mentions that is a friend and has friends. There are such friends who mention this. Hence the square of term. We add this for all such friendships in the network from both the 's and 's perspective, which gives the numerator. The denominator is the number of total such friendships, which counts to total edges in the network twice (one from the 's perspective and the other from the 's). After this analysis, Feld goes on to make some more qualitative assumptions about the statistical correlation between the number of friends that two friends have, based on theories of social networks such as assortative mixing, and he analyzes what these assumptions imply about the number of people whose friends have more friends than they do. Based on this analysis, he concludes that in real social networks, most people are likely to have fewer friends than the average of their friends' numbers of friends. 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Friendship paradox」の詳細全文を読む スポンサード リンク
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