Abstract Individual’s decisions, from what product to buy to whether to engage in risky behavior, often depend on the choices, behaviors, or states of other people. People, however, rarely have global knowledge of the states of others, but must estimate them from the local observations of their social contacts. Network structure can significantly distort individual’s local observations. Under some conditions, a state that is globally rare in a network may be dramatically over-represented in the local neighborhoods of many individuals. This effect, which we call the “majority illusion,” leads individuals to systematically overestimate the prevalence of that state, which may accelerate the spread of social contagions. We develop a statistical model that quantifies this effect and validate it with measurements in synthetic and real-world networks. We show that the illusion is exacerbated in networks with a heterogeneous degree distribution and disassortative structure.

Citation: Lerman K, Yan X, Wu X-Z (2016) The "Majority Illusion" in Social Networks. PLoS ONE 11(2): e0147617. https://doi.org/10.1371/journal.pone.0147617 Editor: Frederic Amblard, Université Toulouse 1 Capitole, FRANCE Received: September 17, 2015; Accepted: January 6, 2016; Published: February 17, 2016 Copyright: © 2016 Lerman et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability: All data are publicly available on the following web sites: (a) https://snap.stanford.edu/data/, (b) http://www.isi.edu/~lerman/downloads/digg2009.html, and (c) http://www-personal.umich.edu/~mejn/netdata/. http://www.reactome.org/pages/download-data, http://www.cs.cmu.edu/~enron/. The data located at http://www.isi.edu/~lerman/downloads/digg2009.html has also been uploaded to Figshare and access details can be found in the Supporting Information files. Funding: This work was supported in part by Air Force Office of Scientific Research (contract FA9550-10-1-0569), by the National Science Foundation (grant CIF-1217605) and by Defense Advanced Research Projects Agency (contract W911NF-12-1-0034). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist.

Introduction An individual’s attitudes and behaviors are shaped by his or her perceptions of the choices, attitudes, and behaviors of others [1–6]. This phenomenon is manifested daily in the decisions people make to adopt a new technology [7, 8] or idea [5, 9], listen to music [3], engage in risky behavior [10], abuse alcohol [11, 12], or join a social movement [1, 2]. As a result, a variety of behaviors are said to be “contagious”, because they spread through the population as people perceive others adopting the behavior and then adopt it themselves. In some cases, “social contagion” will spread from a small number of initial adopters to a large portion of the population, resulting in a fad, hit song, successful political campaign, or a prevailing social norm. Researchers have linked the onset of such global outbreaks to the topology of the underlying network [6, 13], the presence of highly connected individuals [14, 15] and small clusters of inter-connected people [4, 5]. Network structure, however, can systematically bias social perceptions and the inferences people make about their peers. Socially connected individuals tend to be similar [16]. This exposes people to a biased sample of the population, giving rise to the “selective exposure” [17] effect that leads individuals to overestimate the prevalence of their features in a population [18]. Moreover, individuals may selectively divulge or conceal their attributes or behaviors from peers, especially if these deviate from prevailing norms. Such “selective disclosure” [17, 19] will further bias social perceptions, leading individuals to incorrectly infer the prevalence of the behavior in the population. Social perception biases can alter the dynamics of social contagions and stabilize unpopular attitudes and behaviors [20, 21]. Beyond the effects described above, network structure may further distort social perceptions by biasing individual’s observations. One of these network biases is the friendship paradox, which states that, on average, most people have fewer friends than their friends have [22]. Despite its almost nonsensical nature, the friendship paradox has been used to design efficient strategies for vaccination [23], social intervention [24], and early detection of contagious outbreaks [25, 26]. In a nutshell, rather than monitoring random people to catch a contagious outbreak in its early stages, the friendship paradox suggests monitoring their random network neighbors, because they are more likely to be better connected and not only to get sick earlier, but also to infect more people once sick. Recently, friendship paradox was generalized for attributes other than degree, i.e., number of network neighbors. For example, your co-authors are cited more often than you [27], and the people you follow on Twitter post more frequently than you do [28]. In fact, any attribute that is correlated with degree will produce a paradox [27, 29]. We describe a novel variation of the friendship paradox that is essential for understanding social contagion. The paradox applies to networks in which individuals have attributes, in the simplest case a binary attribute, such as “has red hair” vs “does not have red hair,” “purchased an iPhone” vs “did not purchase an iPhone,” “Democrat” vs “Republican.” We refer to individuals with this attribute as “active”, and the rest as “inactive.” We show that under some conditions many individuals will observe a majority of their neighbors in the active state, even when it is globally rare. For example, though few people have red hair, many may observe that a majority of their friends are red-headed. For this reason, we call this effect the “majority illusion.” As a simple illustration of the “majority illusion” paradox, consider the two networks in Fig 1. The networks are identical, except for which of the few nodes are colored. Imagine that colored nodes are active and the rest of the nodes are inactive. Despite this apparently small difference, the two networks are profoundly different: in the first network, every inactive node will examine its neighbors to observe that “at least half of my neighbors are active,” while in the second network no node will make this observation. Thus, even though only three of the 14 nodes are active, it appears to all the inactive nodes in the first network that most of their neighbors are active. PPT PowerPoint slide

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larger image TIFF original image Download: Fig 1. An illustration of the “majority illusion” paradox. The two networks are identical, except for which three nodes are colored. These are the “active” nodes and the rest are “inactive.” In the network on the left, all “inactive” nodes observe that at least half of their neighbors are “active,” while in the network on the right, no “inactive” node makes this observation. https://doi.org/10.1371/journal.pone.0147617.g001 The “majority illusion” can dramatically impact collective phenomena in networks, including social contagions. One of the more popular models describing the spread of social contagions is the threshold model [2, 13, 30]. At each time step in this model, an inactive individual observes the current states of its k neighbors, and becomes active if more than ϕk of the neighbors are active; otherwise, it remains inactive. The fraction 0 ≤ ϕ ≤ 1 is the activation threshold. It represents the amount of social proof an individual requires before switching to the active state [2]. Threshold of ϕ = 0.5 means that to become active, an individual has to have a majority of neighbors in the active state. Though the two networks in Fig 1 have the same topology, when the threshold is ϕ = 0.5, all nodes will eventually become active in the network on the left, but not in the network on the right. This is because the “majority illusion” alters local neighborhoods of the nodes, distorting their observations of the prevalence of the active state. Thus, “majority illusion” provides an alternate mechanism for social perception biases. For example, if heavy drinkers also happen to be more popular (they are the red nodes in the figure above), then, while most people drink little at parties, many people will examine their friends’ alcohol use to observe a majority drinking heavily. This may explain why adolescents overestimate their peers’ alcohol consumption and drug use [11, 12, 31]. The magnitude of the “majority illusion” paradox, which we define as the fraction of nodes more than half of whose neighbors are active, depends on structural properties of the network and the distribution of active nodes. Network configurations that exacerbate the paradox include those in which low-degree nodes tend to connect to high-degree nodes (i.e., networks are disassortative by degree). Activating the high-degree nodes in such networks biases the local observations of many nodes, which in turn impacts collective phenomena emerging in networks, including social contagions and social perceptions. We develop a statistical model that quantifies the strength of this effect in any network and evaluate the model using synthetic networks. These networks allow us to systematically investigate how network structure and the distribution of active nodes affect observations of individual nodes. We also show that structure of many real-world networks creates conditions for the “majority illusion” paradox.

Materials and Methods We used the configuration model [32, 33], as implemented by the SNAP library (https://snap.stanford.edu/data/) to create a scale-free network with a specified degree sequence. We generated a degree sequence from a power law of the form p(k)∼k−α. Here, p k is the fraction of nodes that have k half-edges. The configuration model proceeded by linking a pair of randomly chosen half-edges to form an edge. The linking procedure was repeated until all half-edges have been used up or there were no more ways to form an edge. To create Erdős-Rényi-type networks, we started with N = 10,000 nodes and linked pairs at random with some fixed probability. These probabilities were chosen to produce average degree similar to the average degree of the scale-free networks. The statistics of real-world networks we studied, including the collaboration network of high energy physicist (HepTh), Human protein–protein interactions network from Reactome project (http://www.reactome.org/pages/download-data/), Digg follower graph (DOI:10.6084/m9.figshare.2062467), Enron email network (http://www.cs.cmu.edu/∼enron/), Twitter user voting graph [34], and a network of political blogs (http://www-personal.umich.edu/∼mejn/netdata/) are summarized in Table 1. PPT PowerPoint slide

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larger image TIFF original image Download: Table 1. Network properties. Size of networks studied in this paper, along with their average degree 〈k〉 and degree assortativity coefficient r kk . https://doi.org/10.1371/journal.pone.0147617.t001

Conclusion Local prevalence of some attribute among a node’s network neighbors can be very different from its global prevalence, creating an illusion that the attribute is far more common than it actually is. In a social network, this illusion may cause people to reach wrong conclusions about how common a behavior is, leading them to accept as a norm a behavior that is globally rare. In addition, it may also explain how global outbreaks can be triggered by very few initial adopters. This may also explain why the observations and inferences individuals make of their peers are often incorrect. Psychologists have, in fact, documented a number of systematic biases in social perceptions [43]. The “false consensus” effect arises when individuals overestimate the prevalence of their own features in the population [18], believing their type to be more common. Thus, Democrats believe that most people are also Democrats, while Republicans think that the majority are Republican. “Pluralistic ignorance” is another social perception bias. This effect arises in situations when individuals incorrectly believe that a majority has an attribute or accepts a norm that they themselves do not share. Pluralistic ignorance was invoked to explain why bystanders fail to act in emergencies [44], and why college students tend to overestimate alcohol use among their peers [11, 12, 31]. Psychologists proposed several explanations for these biases (see [17] for a concise review), many based on emotional or cognitive mechanisms. For example, when making social inferences, individuals may use themselves as examples for estimating the states of others (using the “availability” heuristic [45]). This leads them to mistakenly believe that majority shares their attitudes and behaviors. However, if instead of using themselves, individuals use their peers as examples to generalize about the population as a whole, network-based explanations for social perception bias are also possible. “Selective exposure” [17] is one such explanation. Social networks are homophilous [16], meaning that socially linked individuals tend to be similar. Homophily exposes people to a biased sample of the population, creating the false consensus effect [18]. A related mechanism is “selective disclosure” [17, 19], in which people selectively divulge or conceal their attributes or behaviors to peers, especially if these deviate from prevailing norms. This too can bias social perceptions, leading individuals to incorrectly infer the prevalence of the behavior in the population. The paradox described in this paper provides an alternate network-based mechanism for biases in social perceptions. We showed that under some conditions, individuals will grossly overestimate the prevalence of some attribute, making it appear more popular than it is. We quantified this paradox, which we call the “majority illusion”, and studied its dependence on network structure and attribute configuration. As in the friendship paradox [22, 27–29], “majority illusion” can ultimately be traced to the power of high degree nodes to skew the observations of many others. This is because such nodes are overrepresented in the local neighborhoods of other nodes. This, by itself is not surprising, given than high degree nodes are expected to have more influence and are often targeted by influence maximization algorithms [14]. However, the ability of high degree nodes to bias the observations of others depends on other aspects of network structure. Specifically, we showed that the paradox is much stronger in disassortative networks, where high degree nodes tend to link to low degree nodes. In other words, given the same degree distribution, the high degree nodes in a disassortative network will have greater power to skew the observations of others than those in an assortative network. This suggests that some network structures are more susceptible than others to influence manipulation and the spread of external shocks [13]. Furthermore, small changes in network topology, degree assortativity and degree–attribute correlation may further exacerbate the paradox even when there are no actual changes in the distribution of the attribute. This may explain the apparently sudden shifts in public attitudes witnessed during the Arab Spring and on the question of gay marriage. The “majority illusion” is an example of class size bias effect. When sampling data to estimate average class or event size, more popular classes and events will be over-represented in the sample, biasing estimates of their average size [46]. Thus, the average class size that students experience at college is larger than the college’s average class size. Similarly, people experience highways, restaurants, and events to be more crowded than they normally are. In networks, sampling bias affects estimates of network structure, including its degree distribution [41, 47]. Our work suggests that network bias also affects an individual’s local perceptions. Further work is required to understand how this bias impacts the dynamics of collective social phenomena.

Supporting Information S1 File. Friendship paradox. Derivation of the generalized friendship paradox for binary attribute networks. https://doi.org/10.1371/journal.pone.0147617.s001 (PDF) S1 Fig. Structural differences. Strength of the majority illusion in synthetic networks with identical degree sequence and assortativity, but with higher-order structural differences. To create these higher-order structural differences, we used the edge swapping procedure to change the network’s degree correlation matrix e(k, k′). https://doi.org/10.1371/journal.pone.0147617.s002 (EPS)

Acknowledgments Authors are grateful to Nathan Hodas and Farshad Kooti for their inputs into this work. This work was supported in part by AFOSR (contract FA9550-10-1-0569), by NSF (grant CIF-1217605) and by DARPA (contract W911NF-12-1-0034).

Author Contributions Conceived and designed the experiments: KL XY. Performed the experiments: XW. Analyzed the data: XW. Wrote the paper: KL XY XW.