Social networks have many counter-intuitive properties, including the "friendship paradox" that states, on average, your friends have more friends than you do. Recently, a variety of other paradoxes were demonstrated in online social networks. This paper explores the origins of these network paradoxes. Specifically, we ask whether they arise from mathematical properties of the networks or whether they have a behavioral origin. We show that sampling from heavy-tailed distributions always gives rise to a paradox in the mean, but not the median. We propose a strong form of network paradoxes, based on utilizing the median, and validate it empirically using data from two online social networks. Specifically, we show that for any user the majority of user's friends and followers have more friends, followers, etc. than the user, and that this cannot be explained by statistical properties of sampling. Next, we explore the behavioral origins of the paradoxes by using the shuffle test to remove correlations between node degrees and attributes. We find that paradoxes for the mean persist in the shuffled network, but not for the median. We demonstrate that strong paradoxes arise due to the assortativity of user attributes, including degree, and correlation between degree and attribute.
The friendship paradox and other paradoxes have been shown to occur in a variety of contexts, both online and offline. As many have pointed out, starting with Feld @cite_12 , these network paradoxes arise from users with high degree being overrepresented in the population of friends. Feld also claimed that the friendship paradox, that your friends have higher degree than you, holds with both the mean and the median. In this paper we build on existing works which have documented paradoxes using the mean, including @cite_8 @cite_4 @cite_21 @cite_24 @cite_3 @cite_11 .
@cite_4 @cite_8 @cite_21 @cite_3 @cite_24 @cite_12 @cite_11
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Feld's friendship paradox states that "your friends have more friends than you, on average." This paradox arises because extremely popular people, despite being rare, are overrepresented when averaging over friends. Using a sample of the Twitter firehose, we confirm that the friendship paradox holds for >98 of Twitter users. Because of the directed nature of the follower graph on Twitter, we are further able to confirm more detailed forms of the friendship paradox: everyone you follow or who follows you has more friends and followers than you. This is likely caused by a correlation we demonstrate between Twitter activity, number of friends, and number of followers. In addition, we discover two new paradoxes: the virality paradox that states "your friends receive more viral content than you, on average," and the activity paradox, which states "your friends are more active than you, on average." The latter paradox is important in regulating online communication. It may result in users having difficulty maintaining optimal incoming information rates, because following additional users causes the volume of incoming tweets to increase super-linearly. While users may compensate for increased information flow by increasing their own activity, users become information overloaded when they receive more information than they are able or willing to process. We compare the average size of cascades that are sent and received by overloaded and underloaded users. And we show that overloaded users post and receive larger cascades and they are poor detector of small cascades. We report on a survey of undergraduates at the University of Chicago in which respondents were asked to assess their popularity relative to others. Popularity estimates were related to actual popularity, but we also found strong evidence of self-enhancement in self-other comparisons of popularity. In particular, self-enhancement was stronger for self versus friend comparisons than for self versus typical other comparisons; this is contrary to the reality demonstrated in Feld's friendship paradox and suggests that people are more threatened by the success of friends than of strangers. At the same time, people with relatively popular friends tended to make more self-serving estimates of their own popularity than did people with less popular friends. These results clarify how objective patterns of interpersonal contact work together with cognitive and motivational tendencies to shape perceptions of one's location in the social world. As online social networks expand their role beyond maintaining existing relationships, they may look to more faceted ratings to support the formation of new connections between their users. Our study focuses on one community employing faceted ratings, CouchSurfing.org, and combines data analysis of ratings, a large-scale survey, and in-depth interviews. In order to understand the ratings, we revisit the notions of friendship and trust and uncover an asymmetry: close friendship includes trust, but high levels of trust can be achieved without close friendship. To users, providing faceted ratings presents challenges, including differentiating and quantifying inherently subjective feelings such as friendship and trust, concern over a friend’s reaction to a rating, and knowledge of how ratings can affect others’ reputations. One consequence of these issues is the near absence of negative feedback, even though a small portion of actual experiences and privately held ratings are negative. We show how users take this into account when formulating and interpreting ratings, and discuss designs that could encourage more balanced feedback. It is reasonable to suppose that individuals use the number of friends that their friends have as one basis for determining whether they, themselves, have an adequate number of friends. This article shows that, if individuals compare themselves with their friends, it is likely that most of them will feel relatively inadequate. Data on friendship drawn from James Coleman's (1961) classic study The Adolescent Society are used to illustrate the phenomenon that most people have fewer friends than their friends have. The logic underlying the phenomenon is mathematically explored, showing that the mean number of friends of friends is always greater than the mean number of friends of individuals. Further analysis shows that the proportion of individuals who have fewer friends than the mean number of friends their own friends have is affected by the exact arrangement of friendships in a social network. This disproportionate experiencing of friends with many friends is related to a set of