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.
Current methods for the detection of contagious outbreaks give contemporaneous information about the course of an epidemic at best. It is known that individuals near the center of a social network are likely to be infected sooner during the course of an outbreak, on average, than those at the periphery. Unfortunately, mapping a whole network to identify central individuals who might be monitored for infection is typically very difficult. We propose an alternative strategy that does not require ascertainment of global network structure, namely, simply monitoring the friends of randomly selected individuals. Such individuals are known to be more central. To evaluate whether such a friend group could indeed provide early detection, we studied a flu outbreak at Harvard College in late 2009. We followed 744 students who were either members of a group of randomly chosen individuals or a group of their friends. Based on clinical diagnoses, the progression of the epidemic in the friend group occurred 13.9 days (95 C.I. 9.9–16.6) in advance of the randomly chosen group (i.e., the population as a whole). The friend group also showed a significant lead time (p<0.05) on day 16 of the epidemic, a full 46 days before the peak in daily incidence in the population as a whole. This sensor method could provide significant additional time to react to epidemics in small or large populations under surveillance. The amount of lead time will depend on features of the outbreak and the network at hand. The method could in principle be generalized to other biological, psychological, informational, or behavioral contagions that spread in networks. Recent research has focused on the monitoring of global-scale online data for improved detection of epidemics, mood patterns, movements in the stock market, political revolutions, box-office revenues, consumer behaviour and many other important phenomena. However, privacy considerations and the sheer scale of data available online are quickly making global monitoring infeasible, and existing methods do not take full advantage of local network structure to identify key nodes for monitoring. Here, we develop a model of the contagious spread of information in a global-scale, publicly-articulated social network and show that a simple method can yield not just early detection, but advance warning of contagious outbreaks. In this method, we randomly choose a small fraction of nodes in the network and then we randomly choose a "friend" of each node to include in a group for local monitoring. Using six months of data from most of the full Twittersphere, we show that this friend group is more central in the network and it helps us to detect viral outbreaks of the use of novel hashtags about 7 days earlier than we could with an equal-sized randomly chosen group. Moreover, the method actually works better than expected due to network structure alone because highly central actors are both more active and exhibit increased diversity in the information they transmit to others. These results suggest that local monitoring is not just more efficient, it is more effective, and it is possible that other contagious processes in global-scale networks may be similarly monitored. 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.  We study the structure of the social graph of active Facebook users, the largest social network ever analyzed. We compute numerous features of the graph including the number of users and friendships, the degree distribution, path lengths, clustering, and mixing patterns. Our results center around three main observations. First, we characterize the global structure of the graph, determining that the social network is nearly fully connected, with 99.91 of individuals belonging to a single large connected component, and we confirm the "six degrees of separation" phenomenon on a global scale. Second, by studying the average local clustering coefficient and degeneracy of graph neighborhoods, we show that while the Facebook graph as a whole is clearly sparse, the graph neighborhoods of users contain surprisingly dense structure. Third, we characterize the assortativity patterns present in the graph by studying the basic demographic and network properties of users. We observe clear degree assortativity and characterize the extent to which "your friends have more friends than you". Furthermore, we observe a strong effect of age on friendship preferences as well as a globally modular community structure driven by nationality, but we do not find any strong gender homophily. We compare our results with those from smaller social networks and find mostly, but not entirely, agreement on common structural network characteristics. 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 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.