Heavy-tailed distributions of meme popularity occur naturally in a model of meme diffusion on social networks. Competition between multiple memes for the limited resource of user attention is identified as the mechanism that poises the system at criticality. The popularity growth of each meme is described by a critical branching process, and asymptotic analysis predicts power-law distributions of popularity with very heavy tails (exponent α < 2, unlike preferential-attachment models), similar to those seen in empirical data. DOI: 10.1103/PhysRevLett.112.048701 PACS numbers: 89.65.-s, 05.65.+b, 89.75.Fb, 89.75.Hc When people select from multiple items of roughly equal value, some items quickly become extremely popular, while other items are chosen by relatively few people [1]. The probability P n ðtÞ that a random item has been selected n times by time t is often observed to have a heavytailed distribution (n is called the popularity of the item at time t). In examples where the items are baby names [2], apps on Facebook [3], retweeted URLs or hashtags on Twitter [4-6], or video views on YouTube [7], the popularity distribution is found to scale approximately as a power law P n ∼ n −α over several decades. The exponent α in all these examples is less than two, and typically has a value close to 1.5. This range of α values is notably distinct from those obtainable from cumulative-advantage or preferential-attachment models of the Yule-Simon type-as used to describe power-law degree distributions of networks, for example [8-11]-which give α ≥ 2. Interestingly, the value α ¼ 1.5 is also found for the power-law distribution of avalanche sizes in self-organized criticality (SOC) models [12,13], suggesting the possibility that the heavy-tailed distributions of popularity in the examples above are due to the systems being somehow poised at criticality.In this Letter, we present an analytically tractable model of selection behavior, based on simplifying the model of Weng et al. [14] for the spreading of memes on a social network. We show that, in certain limits, the system is automatically poised at criticality-in the sense that meme popularities are described by a critical branching process [15]-and that the criticality can be ascribed to the competition between memes for the limited resource of user attention. We dub this mechanism "competitioninduced criticality" (CIC) and investigate the impact of the social network topology (degree distribution) and the age of the memes upon the distribution of meme popularities. We show that CIC gives rise to heavy-tailed distributions very similar to the distributions of avalanche sizes in SOC models [16,17], even though our competition mechanism is quite different from the sandpile paradigm of SOC. This Letter may, therefore, be of interest in other areas where SOC-like critical phenomena have been observed in experiments or simulations, such as economic models of competing firms [18,19], the evolution and extinction of competing species [20][21][22], and neural activity in ...

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.

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Background Reduced flexibility has been documented in athletes with lower limb injury, however stretching has limited evidence of effectiveness in preventing injury or reducing the risk of recurrence. In contrast, it has been proposed that eccentric training can not only improve strength and reduce the risk of injury, but also facilitate increased muscle flexibility via sarcomerogenesis. Objective This systematic review was undertaken to examine the evidence that eccentric training has demonstrated effectiveness as a means of improving lower limb flexibility. Design Systematic Review. 6 electronic databases were systematically searched by two independent reviewers to identify randomised clinical trials comparing the effectiveness of eccentric training to either a different intervention, or a no-intervention control group. Studies evaluating flexibility using both joint range of motion (ROM) and muscle fascicle length (FL) were included. 6 studies met the inclusion/exclusion criteria, and were appraised using the PEDro scale. Differences in the muscles studied, and the outcome measures used, did not allow for pooled data analysis. Results There was consistent, strong evidence from all six trials in three different muscle groups that eccentric training can improve lower limb flexibility, as assessed using either joint ROM or muscle FL. Conclusions The results support the hypothesis that eccentric training is an effective method of increasing lower limb flexibility. Therefore eccentric training is associated with improved flexibility, and not only with gains in strength, performance and injury reduction. Further research is required to compare the increased flexibility obtained after eccentric training to that obtained with static stretching and other exercise interventions.