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 ...

Mean-field analysis is an important tool for understanding dynamics on complex networks. However, surprisingly little attention has been paid to the question of whether mean-field predictions are accurate, and this is particularly true for real-world networks with clustering and modular structure. In this paper, we compare mean-field predictions to numerical simulation results for dynamical processes running on 21 real-world networks and demonstrate that the accuracy of the theory depends not only on the mean degree of the networks but also on the mean first-neighbor degree. We show that mean-field theory can give (unexpectedly) accurate results for certain dynamics on disassortative real-world networks even when the mean degree is as low as z = 4.

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The spread of ideas across a social network can be studied using complex contagion models, in which agents are activated by contact with multiple activated neighbors. The investigation of complex contagions can provide crucial insights into social influence and behavior-adoption cascades on networks. In this paper, we introduce a model of a multi-stage complex contagion on networks. Agents at different stages-which could, for example, represent differing levels of support for a social movement or differing levels of commitment to a certain product or ideaexert different amounts of influence on their neighbors. We demonstrate that the presence of even one additional stage introduces novel dynamical behavior, including interplay between multiple cascades, which cannot occur in single-stage contagion models. We find that cascades-and hence collective action-can be driven not only by high-stage influencers but also by low-stage influencers. Studying models of cascades allows one to gain insights into a variety of processes ranging from the spread of fads and ideas in social networks to the appearance of cascading failures in infrastructure networks. To date, researchers have mostly considered single-stage cascade models wherein the propagation of a cascade is characterized by a single subpopulation of active agents, 1-5 though some multi-stage models have been examined recently. 6,7 In the usual approach, it is assumed that all active agents exhibit the same amount of influence on their peers. In reality, however, supporters of a cause can vary significantly in their desire and ability to recruit new members. In this paper, we introduce a model of multi-stage cascading dynamics in which agents can exert different amounts of influence on their peers depending on the stage of their adoption (i.e., on the level of their commitment to a certain idea or product). We investigate the dynamics of our multi-stage cascade model on various networks and observe an interplay between cascadese.g., one cascade driving the other one or vice versathat cannot be observed in single-stage cascade models. We also provide an analytical method for solving the model that gives a good prediction for the cascade sizes on configuration-model networks.

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A widespread approach to investigating the dynamical behaviour of complex social systems is via agent-based models (ABMs). In this paper, we describe how such models can be dynamically calibrated using the ensemble Kalman filter (EnKF), a standard method of data assimilation. Our goal is twofold. First, we want to present the EnKF in a simple setting for the benefit of ABM practitioners who are unfamiliar with it. Second, we want to illustrate to data assimilation experts the value of using such methods in the context of ABMs of complex social systems and the new challenges these types of model present. We work towards these goals within the context of a simple question of practical value: how many people are there in Leeds (or any other major city) right now? We build a hierarchy of exemplar models that we use to demonstrate how to apply the EnKF and calibrate these using open data of footfall counts in Leeds.

Agent-based modelling is a valuable approach for modelling systems whose behaviour is driven by the interactions between distinct entities, such as crowds of people. However, it faces a fundamental di iculty: there are no established mechanisms for dynamically incorporating real-time data into models. This limits simulations that are inherently dynamic, such as those of pedestrian movements, to scenario testing on historic patterns rather than real-time simulation of the present. This paper demonstrates how a particle filter could be used to incorporate data into an agent-based model of pedestrian movements at run time. The experiments show that although it is possible to use a particle filter to perform online (real time) model optimisation, the number of individual particles required (and hence the computational complexity) increases exponentially with the number of agents. Furthermore, the paper assumes a one-to-one mapping between observations and individual agents, which would not be the case in reality. Therefore this paper lays some of the fundamental groundwork and highlights the key challenges that need to be addressed for the real-time simulation of crowd movements to become a reality. Such success could have implications for the management of complex environments both nationally and internationally such as transportation hubs, hospitals, shopping centres, etc.

We propose a unified framework to represent a wide range of continuous-time discrete-state Markov processes on networks, and show how many network dynamics models in the literature can be represented in this unified framework. We show how a particular sub-set of these models, referred to here as single-vertex-transition (SVT) processes, lead to the analysis of quasi-birth-and-death (QBD) processes in the theory of continuous-time Markov chains. We illustrate how to analyse a number of summary statistics for these processes, such as absorption probabilities and first-passage times. We extend the graph-automorphism lumping approach [Kiss, Miller, Simon, Mathematics of Epidemics on Networks, 2017; Simon, Taylor, Kiss, J. Math. Bio. 62(4), 2011], by providing a matrix-oriented representation of this technique, and show how it can be applied to a very wide range of dynamical processes on networks. This approach can be used not only to solve the master equation of the system, but also to analyse the summary statistics of interest. We also show the interplay between the graph-automorphism lumping approach and the QBD structures when dealing with SVT processes. Finally, we illustrate our theoretical results with examples from the areas of opinion dynamics and mathematical epidemiology.