We develop a theoretical framework for the study of epidemic-like social contagion in large scale social systems. We consider the most general setting in which different communication platforms or categories form multiplex networks. Specifically, we propose a contact-based information spreading model, and show that the critical point of the multiplex system associated to the active phase is determined by the layer whose contact probability matrix has the largest eigenvalue. The framework is applied to a number of different situations, including a real multiplex system. Finally, we also show that when the system through which information is disseminating is inherently multiplex, working with the graph that results from the aggregation of the different layers is inaccurate. PACS numbers: 89.75.Hc,89.75.Kd Social contagion processes such as the adoption of a belief, the propagation of opinions and behaviors, and the massive social movements that have recently unfolded worldwide [1][2][3][4][5][6][7] are determined by many factors, among which the structure of the underlying topology and the dynamics of information spreading [8]. The advent of new communication platforms such as online social networks (OSN), has made the study of social contagion more challenging. Today, individuals are increasingly exposed to many diverse sources of information, all of which they value differently [9], giving raise to new communication patterns that directly impact both the dynamics of information spreading and the structure of the social networks [10][11][12][13]. Admittedly, the commonplace multi-channel information spreading that characterizes the way we exchange information nowadays has not been studied so far. One way to address the latter is to consider that the process of contagion occurs in a system made up of different layers, i.e., in a multiplex network [14][15][16][17][18][19][20][21][22][23]. Although many studies have dealt with social contagion and information spreading on social networks, they all consider the case in which transmission occurs along the contacts of a simplex, i.e., single-layer, system. Here we aim at filling this existing gap.The dynamics of this kind of processes can be modeled using different classes of approaches. Threshold models [24][25][26][27][28][29] assume that individuals enroll in the process being modeled if a given intrinsic propensity level, the threshold, is surpassed. Although this class of models is useful to address the emergence of collective behavior, they are generally designed to simulate a single contagion process and therefore individuals, once they are active, remain so forever. This is not convenient in many situations that are characterized by self-sustained activity patterns [6,7]. For instance, think of an online social network in which tags are used to identify the topic of the information being transmitted (like hashtags in Twitter): individuals can use the same tag many times, but they can also decide not to use it after a number of times, thus being again suscept...

We report a high-precision finite-size scaling study of the critical behavior of the three-dimensional Ising Edwards-Anderson model (the Ising spin glass). We have thermalized lattices up to L = 40 using the Janus dedicated computer. Our analysis takes into account leading-order corrections to scaling. We obtain T c = 1.1019(29) for the critical temperature, ν = 2.562(42) for the thermal exponent, η = −0.3900(36) for the anomalous dimension, and ω = 1.12(10) for the exponent of the leading corrections to scaling. Standard (hyper)scaling relations yield α = −5.69(13), β = 0.782(10), and γ = 6.13(11). We also compute several universal quantities at T c .

Online social media have greatly affected the way in which we communicate with each other. However, little is known about what are the fundamental mechanisms driving dynamical information flow in online social systems. Here, we introduce a generative model for online sharing behavior that is analytically tractable and which can reproduce several characteristics of empirical micro-blogging data on hashtag usage, such as (time-dependent) heavy-tailed distributions of meme popularity. The presented framework constitutes a null model for social spreading phenomena which, in contrast to purely empirical studies or simulation-based models, clearly distinguishes the roles of two distinct factors affecting meme popularity: the memory time of users and the connectivity structure of the social network.

We compare the critical behavior of the short-range Ising spin glass with a spin glass with long-range interactions which fall off as a power σ of the distance. We show that there is a value of σ of the long-range model for which the critical behavior is very similar to that of the short-range model in four dimensions. We also study a value of σ for which we find the critical behavior to be compatible with that of the three-dimensional model, although we have much less precision than in the four-dimensional case.

Spin glasses are a longstanding model for the sluggish dynamics that appear at the glass transition. However, spin glasses differ from structural glasses in a crucial feature: they enjoy a time reversal symmetry. This symmetry can be broken by applying an external magnetic field, but embarrassingly little is known about the critical behavior of a spin glass in a field. In this context, the space dimension is crucial. Simulations are easier to interpret in a large number of dimensions, but one must work below the upper critical dimension (i.e., in d < 6) in order for results to have relevance for experiments. Here we show conclusive evidence for the presence of a phase transition in a four-dimensional spin glass in a field. Two ingredients were crucial for this achievement: massive numerical simulations were carried out on the Janus special-purpose computer, and a new and powerful finite-size scaling method. T he glass transition differs from standard phase transitions in that the equilibration time of glass formers (supercooled liquids, polymers, proteins, superconductors, etc.) diverges without dramatic changes in their structural properties (1-3). The reconciliation of the dynamic slowdown with the apparent immutability of glass formers is a major challenge for condensed matter physics.Spin glasses [which are disordered magnetic alloys (4)] enjoy a privileged status in this context, as they provide the simplest model system both for theoretical and experimental studies of a glassy dynamics. On the experimental side, time-dependent magnetic fields provide a wonderful tool to probe the dynamic response, which can be accurately measured with a SQUID (for instance, see ref. 5). On the theoretical side, magnetic systems are notably easier to model and to simulate numerically. In fact, special-purpose computers have been built for the simulation of spin glasses (6-9).Yet, spin glasses differ from most glassy systems in a crucial feature: like all magnetic systems, they enjoy time-reversal symmetry in the absence of an applied magnetic field. In fact, we now know that their glassy dynamics are due to a bona fide phase transition in which the time-reversal symmetry is spontaneously broken (10-12). Yet, in the presence of an applied magnetic field, the experimental spin-glass dynamics is just as glassy, although the field explicitly breaks the symmetry.However, whether spin glasses in a magnetic field undergo a phase transition has been a long-debated and still open question (see refs. 13, 14 for recent, opposed views). In the mean-field approximation, which is valid for large spatial dimension down to the upper critical dimension d u ¼ 6 (15), the de AlmeidaThouless line separates the high-temperature paramagnetic phase from the glassy phase (16)*. Yet, recent numerical simulations in spatial dimensions below d u did not find the transition in a field (18,19). Experimental studies have been conducted as well, with conflicting conclusions (20-23). In spite of these difficulties, it has been argued that the would-be spi...