Spontaneous emergence of social influence in online systems

Abstract

Social influence drives both offline and online human behavior. It pervades cultural markets, and manifests itself in the adoption of scientific and technical innovations as well as the spread of social practices. Prior empirical work on the diffusion of innovations in spatial regions or social networks has largely focused on the spread of one particular technology among a subset of all potential adopters. 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 behavior of all individuals who can influence each other in this context. By extending standard fluctuation scaling methods, we analyze the collective behavior induced by 100 million application installations, and show that two distinct regimes of behavior emerge in the system. Once applications cross a particular threshold of popularity, social influence processes induce highly correlated adoption behavior 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.

Fig. 1.

Facebook users and applications. (A) The users (round nodes) form a social network (solid lines) which influences their behavior in adopting applications (hexagons). The dashed lines connecting users to applications indicate which applications each user has installed. (B) Number of users n_i(t) as a function of time t for four applications of which “Texas HoldEm Poker” is the most popular one at the end when t = T. (C) Number of users n_i(T) sorted in descending order for the 2,123 applications that have n_i(T) > 0 (Zipf plot). (D) Probability density distribution P(n(T)) versus n(T) is fat-tailed. The dashed line ∼n(T)⁻² corresponds to the limit where the mean of the distribution diverges.

Fig. 2.

Fluctuation scaling. (A) The concept of FS can be illustrated by considering tossing coins in two ways (32). (i) We toss a group of k coins independently with sides corresponding to 0 and 1 and let f_k equal their sum. (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 and SD σ_k of f_k as shown in the schematic. In both cases whereas in i but σ_k ∼ k in ii. 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 local and global signals (SI Text). (C) Of the 2,705 Facebook applications in the empirical dataset, 2,562 with μ_i > 0 and σ_i > 0 are plotted here (SI Text). Two qualitatively different regimes emerge, and 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 dataset consists of 2,705 time series, of which 2,163 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.

Fig. 3.

Effect of application lifetime on scaling. Visual inspection shows that any interval of log-μ values contains a roughly equal number of red, green, and blue markers, indicating that the time of introduction and, hence, application lifetime are not related to its scaling properties. The histograms at the bottom of the panel show exactly how many applications from each of the three periods (red, green, blue) fall in the [−2, −1), [−1,0), [0, 1), [1, 2), and [2, 3) intervals, demonstrating clearly that there is no age trend in the scaling plot.

Fig. 4.

Schematic of the construction of the synthetic time series . (A) The empirical data consist of t = 1,…,7 observations for three applications. The data points have been connected with dashed black lines to guide the eye. For the most popular application at time t − 1, the change in the number of users between t − 1 and t is indicated by the height of the vertical red bar at time t, which corresponds to in the text. Similarly, and are indicated by the green and blue bars, respectively. An easy way to understand the process is first to compute the difference in the number of users for all applications given by f_i(t) = n_i(t) − n_i(t − 1) and then color the difference based on r_i(t − 1), the rank of the application at time t − 1. (B) The synthetic time series are seeded by the initial values taken from the empirical data such that , and of the empirical data and they are constructed by adding together the difference bars of the same color. Overlapping bars have been shifted slightly horizontally for clarity of presentation.