Social influence (1) drives both offline and online human behaviour. It pervades cultural markets, and manifests itself in the adoption of scientific and technical innovations as well as the spread of social practices (2). Prior empirical work on the diffusion of innovations in spatial regions or social networks has largely focused on the spread of one particular technology (3, 4) among a subset of all potential adopters (5). It has also been difficult to determine whether the observed collective behaviour is driven by natural influence processes, or whether it follows external signals such as media or marketing campaigns (6). Here, we choose an online context that allows us to study social influence processes by tracking the popularity of a complete set of applications installed by the user population of a social networking site, thus capturing the behaviour of all individuals who can influence each other in this context. By extending standard fluctuation scaling methods (7, 8) , we analyse the collective behaviour induced by 100 million application installations, and show that two distinct regimes of behaviour emerge in the system. Once applications cross a particular threshold of popularity, social influence processes induce highly correlated adoption behaviour among the users, which propels some of the applications to extraordinary levels of popularity. Below this threshold, the collective effect of social influence appears to vanish almost entirely in a manner that has not been observed in the offline world. Our results demonstrate that even when external signals are absent, social influence can spontaneously assume an on-off nature in a digital environment. It remains to be seen whether a similar outcome could be observed in the offline world if equivalent experimental conditions could be replicated. Social influence captures the ways in which people affect each others' beliefs, feelings, and behaviors. It has traditionally been in the domain of social psychology with principal focus on micro-level processes among individuals (1), but it also plays a prominent role across the social sciences, for example, in the study of contagion in sociology (2), herding behavior in economics (9), speculative bubbles in financial markets (10), voting behavior (11), and interpersonal health (12). Social influence plays an especially important role in cultural markets (13), for products such as books and music, and generally pervades any arena of life where the attitudes and tastes of individuals are influenced by others. It is often useful to distinguish between local and global sources of influence, which typically are identified with an individual's interpersonal environment and the mass media, respectively (14). The overall social influence arises from a mixture of local and global influences, which themselves emerge from different signals. The fact that these two processes operate at very different scales poses considerable challenges for the empirical study of social influence. For the purposes of our study, we define (i) local signal as information on the behavior of individuals who are friends or acquaintances of ego, the person whose behavior is being analyzed, 2 and (ii) global signal as information on the aggregate behavior of the population. Note that these definitions rely on the potentially observable behaviors of others as opposed to the nonobservable ones, such as their intentions or feelings. This framework incorporating local and global signals is very generic and possible behaviors range from the consumption of cultural products to making lifestyle choices. The structures of social influence are most naturally addressed from the perspective of social network analysis (15). The notion of local influence presupposes that individuals are embedded in a social network that channels and directs how behaviors spread. Examples of such behaviors include innovation adoption among physicians (3), as well as other empirical and theoretical studies of diffusion (4, 17, 18, 16) . The notion of global influence, on the other hand, presupposes that individuals have information on the aggregate popularity of products and behaviors. While a given social network can be used as a proxy for communicating the behavioral signals, one should ideally have access to a network that accurately represents the potential communication channels for a given local signal, and these channels may vary between different behaviors. In addition, individuals are often selective as to what information they choose to disclose to their friends, resulting in the local signal being necessarily incomplete, biased, or misrepresented (19). Similarly, while accurate population level statistics exist for popular items, it is much harder to find statistics for more marginal products and behaviors. A novel opportunity to study human behavior in a setting that overcomes these methodological limitations is provided by certain online environments. These systems have the advantage of allowing access to complete sub-populations of agents. When combined with appropriate tools of analysis, they enable the direct study of collective macro-level social behavior in very large social systems without sampling. We study a complete online social system with welldefined local and global signals by harnessing data from Facebook (21), a hugely popular social networking site (SNS), which at the time of data collection had approximately 50 million ac-3 tive users worldwide. In addition to the current popular interest in social networks, scholars have recognized the potential of these and other social websites for research (22, 23, 25, 24, 26), reflecting the current move to utilising rich large-scale datasets on human behavior and communication (27,28). Facebook users, in line with other SNSs, can construct a public or semi-public profile within a bounded system, articulate a list of other users, "Facebook friends", with whom they share a connection, and view and traverse their list of connections and those made by others within the system (29). Facebook users can also install (and uninstall) applications ( Consequently, users with many Facebook friends are then, at least in principle, in a position to influence a larger number of other users. In addition, everyone has access at all times to an all-inclusive listing of applications ranked by their global popularity, which acts as an effective "best seller" list. Although applications are free of charge, popular applications have the advantage of being readily discoverable (low search cost), and are more likely to be of higher quality both with respect to reliability (exhaustively tested) and functionality (superior features). The applications provide recreational value and can be seen as cultural goods, and the different ways the users process the local and global signals in choosing applications reflect their personal preferences, i.e. the underlying heterogeneity of the population. In addition to the distinction between local and global signals, it is important to classify systems into two separate categories based on whether their dynamics are endogenous without external drivers, or exogenous and driven externally. Epidemic spreading in a closed system is an example of an endogenous process with local transmission, since the pathogens need to be passed from one person to another in close physical proximity. Similarly, it is possible (D) Probability density distribution P (n(T )) vs. n(T ) is fat-tailed. The dashed line ∼ n(T ) −2 is intended to guide the eye and corresponds to the limit where the mean of the distribution diverges. 5 to model the spread of innovations such as the uptake of new hybrid crops by farmers as an endogenous social contagion process, and to try to distinguish between different types of local processes that may underlie the observed rate at which the innovation is adopted (20). However, studies of social influence which focus on local and endogenous processes such as word-ofmouth transmission are almost always open to the challenge that they neglect equally important exogenous effects such as marketing or mass advertising, and typically trying to separate these two confounding factors is highly problematic. For instance, a recent re-analysis (6) of the classic diffusion studies on how prescriptions for an antibiotic drug spread among physicians in different communities (3, 4) suggests that marketing efforts, in this context corresponding to external drivers, can account for most of the observed behavior. In the current setup, both the local and global signals are generated endogenously within the system, i.e. there is no exogenous driver (31). We downloaded the data from Facebook for all existing 2720 applications between June 25, 2007 and August 14, 2007, shortly after applications were introduced. These data consist of time series n i (t) with i = 1, 2, . . . , M = 2720 and t = 1, 2, . . . , T = 1208 corresponding to the aggregate number of users who have application i installed at time t (Fig. 1B) . Data for 15 applications were partly corrupted and were consequently omitted from the analysis, leaving us with 2705 applications, or 99% of the data. Importantly, studying all the applications avoids a selection bias, which is generated by examining the trajectories of those applications that spread successfully as tends to be done in most studies on social influence (5). Successful products in cultural markets have been found to be orders of magnitude more popular than the average cultural product (13). This finding is also manifest in the case of Facebook applications. The number of users at the end of the time horizon, n i (T ), sorted in descending order is shown in Fig. 1C . For the ten most popular applications these numbers vary between n (1) (T ) ≈ 12 million and n (10) (T ) ≈ 4.6 million, whereas n (100) (T ) ≈ 180, 000 and n (1000) (T ) ≈ 1, 300. The probability density distribution for the number of application installations (Fig. 1D ) has a very fat tail and decays so slowly that even its mean value diverges in the limit of infinite system size. Each new installation, in addition to increasing the overall user base of the application and thus its global signal, also generates a local signal, through which the adopter may in turn influence the future behavior of his friends. Each installation thus acts as a microscopic social stimulus and creates a form of positive feedback in the system. Note that the observable behavior which generates patterns of social influence in this case is restricted to the adoption of an application, rather than its use. Given that the users are part of a very large social network, the consequences of adopting an application are not limited to a user's immediate neighborhood, but may percolate further in the network. This underlines the importance of having data that reflects the behavior of the entire network even if the underlying microscopic data are not available. While the impact of a single installation is admittedly minute, the superposition of the observed 104 million application installations leaves behind a detectable footprint. To study the effect of social influence, i.e. the extent to which the behavior of an individual (his installing an application) is related to the behavior of others (their installing the same application), we turn to the method of fluctuation scaling (FS). This allows us to extract a key signature of the system's behavior purely on the basis of the above aggregate data. FS has been applied successfully to a number of complex systems whose interacting elements participate in some dynamic process. Examples of application domains range from fluctuations in population sizes in ecology to fluctuations in stock trading activity in financial markets (32, 7, 8) . Here we outline how FS can be utilized in the current problem, and refer the reader to Supplementary Information (SI) for details. For a given application i, the act of individual j regarding installation of the application is encoded by the random variable S i,j (t), where S i,j (t) = 1 corresponds to him installing the application at time t, and S i,j (t) = 0 corresponds to him doing nothing. From the stochastic process point of view, one can think of each individual tossing coins at every time step, one per application, to decide whether he will install the given application. Social influence, operating through the local and global signals, is likely to render the coin tosses dependent for any given application ( Fig. 2A,B) . To measure the strength of social influence, we define net activity f i (t) of application i at time t as which corresponds to the net increase in the number of installations for application i between times t − 1 and t. It can be expressed in terms of the individual constituent variables as shown, where the first sum is taken over all N individuals, whereas the latter sum is taken over potential new installers, with the subset of indices j 1 , j 2 , . . . , j N −n i (t) ∈ {1, 2, . . . , N } such that S i,j k (t − 1) = 0. In terms of the above analogy, once a user has installed a given application, he stops tossing the particular coin corresponding to that application. According to FS, the temporal average and standard deviation of f i (t) are related through the relationship σ i ∼ µ α i . This motivates us to identify a region in which the relationship between log µ k and log σ k is linear. The value of the fluctuation scaling exponent α is given by the slope of the line. Although α lies in the rather narrow range [1/2, 1], its value is crucial as an indicator of statistical coupling in the system ( Fig. 2A,B) . If the behavior of a user is independent of the behavior of others, one would expect α = 1/2, whereas if her behavior is fully correlated with others one would expect α = 1 for all applications (33). As shown in (ii) We toss a single coin with sides 0 and k, which corresponds to tossing k fully coupled coins. (B) We perform the experiment several times and calculate the average f k and standard deviation σ k of f k as shown in the schematic. In both cases Varying the value of k produces a series of points in the log µ k , log σ k plane. From the FS point of view, this simple example resembles Facebook users making decisions on application adoption; the "coins" are now biased, reflecting individual heterogeneity, and the tosses are not independent but coupled via the local and global signals (see S2 in SOM). (C) Of the 2705 Facebook applications in the empirical data set, 2562 with µ i > 0 and σ i > 0 are plotted here (see S3 in SOM). Two qualitatively different regimes emerge and they are separated by a cross-over point located at log µ x = 0.36. The first, individual regime is characterized by the exponent α I ≈ 0.55, and the second, collective regime by α C ≈ 0.85. (D) The synthetic data set consists of 2705 time series of which 2163 have µ i > 0 and σ i > 0. We now obtain a single regime characterized by the exponent α S ≈ 0.84. Note that in C and D the exponents lie between 1/2 and 1, corresponding to the extremes of completely uncorrelated and correlated decisions of users to adopt applications. 9 qualitatively different regime for the less popular applications. This individual regime with log(µ i ) < log(µ x ) has α I ≈ 0.55, which is very close to the limiting case of α = 1/2, meaning that applications installations are nearly uncorrelated and social influence is negligible. The transition between the two regimes takes place at approximately log(µ x ) = 0.36, which translates into an average daily activity of 24 × 10 0.36 ≈ 55 new installations a day. We emphasize that theoretical considerations guided our choice to fit a linear function to the data in Fig. 2C as opposed to, say, trying to find the best fit among a class of curvilinear functions. While it would be interesting to resolve also the precise location and nature of the transition (sharp or continuous), we are unable to make this distinction on the basis of the empirical data. However, the central finding on the existence of two different regimes remains unaffected. The interpretation of FS exponents in terms of correlations assumes that the underlying stochastic processes is stationary (8). However, the fact that n i (t) increases over time demonstrates that the system cannot be stationary. The question then becomes whether the system is sufficiently close to stationarity so that the fluctuation scaling exponents can be given the above interpretation. Let us impose the stringent condition that the system is sufficiently close to stationarity when at most 1% of users have the application installed. We show in SI that even under this strict condition, 98% of the time series are stationary. This also means that the scaling in Fig. 2C holds for over two orders magnitude above the cross-over point. We conclude that the system is sufficiently stationary so that the temporal fluctuations may indeed be given the above interpretation. As a simple explanatory hypothesis for the observed behavior, one might suggest that the different scaling properties result from applications having different lifetimes. To test this, we divide the applications into three distinct groups based on their time of introduction such that each group covers an equally long time interval. We repeat the scaling plot by choosing randomly 300 applications from each group with the red, green, and blue colors indicating 10 whether the application was introduced in the first, second, or third interval (Fig. 3) . Since any interval of x-values contains an approximately equal number of markers of different colors, the time of introduction and, hence, application lifetime, does not explain its scaling properties. Should the transition from one regime to the other be attributed to the popularity of applications reaching a certain threshold value, or should it be attributed to the system itself? If the former is true, then one might think that the transition corresponds to a phase transition or to the crossing of an epidemic threshold, essentially a density threshold effect, resulting in an epidemic of popularity. To isolate the effects of popularity, we construct rank-order preserving synthetic time series from the empirical ones. This deterministic process (apart from ties) cuts the empirical time series into pieces and then recombines the pieces using a rank based rule (see Methods). As shown in Fig. 2D , the transition disappears for the synthetic data. Statistical tests also support the existence of a single regime (see SI) and, in addition, the Pearson's linear correlation coefficient between log σ and log µ is 0.99. The consequences of this are threefold. First, the lack of two regimes for the synthetic data demonstrates that the transition from one regime to the other is not a result of the popularity of an application exceeding a certain threshold, so the phenomenon is not analogous to crossing an epidemic threshold. Second, it demonstrates that the collective (correlated) regime does not result from the system becoming saturated with users of a given application that would then induce correlations between the behaviors of the individuals. This is because all the synthetic time series obey the same scaling relation also for small values of log(µ) (corresponding to dilute limit), where the system is far from being saturated. Third, the synthetic regime has an exponent α S ≈ 0.84, which is very close to α C ≈ 0.85 that characterizes the collective regime for empirical data. This shows that we can recover the exponent of the collective regime by assuming that the future popularity of an application is driven by its current popularity, a finding that has also been used to predict popularity of online content (34).
These additional factors are often neglected in modeling, which is reasonable if there is enough evidence that some of them, for example exogenous influence, is negligible REF .
The Spontaneous Emergence of Social Influence in Online Systems
Proc.Natl.Acad.Sci. U.S.A. 43 (2010) 18375-18380
Computer Science Physics Medicine