We consider two competing first passage percolation processes started from uniformly chosen subsets of a random regular graph on @math vertices. The processes are allowed to spread with different rates, start from vertex subsets of different sizes or at different times. We obtain tight results regarding the sizes of the vertex sets occupied by each process, showing that in the generic situation one process will occupy @math vertices, for some @math . The value of @math is calculated in terms of the relative rates of the processes, as well as the sizes of the initial vertex sets and the possible time advantage of one process. The motivation for this work comes from the study of viral marketing on social networks. The described processes can be viewed as two competing products spreading through a social network (random regular graph). Considering the processes which grow at different rates (corresponding to different attraction levels of the two products) or starting at different times (the first to market advantage) allows to model aspects of real competition. The results obtained can be interpreted as one of the two products taking the lion share of the market. We compare these results to the same process run on @math dimensional grids where we show that in the generic situation the two products will have a linear fraction of the market each.
As mentioned earlier, diffusion and growth processes have been studied intensely in the past few years in relation to many areas such as sociology, economics and engineering. Among the models studied are (see, for example @cite_44 , @cite_49 , @cite_30 ), which was first introduced by Clifford and Sudbury in @cite_5 and has been much studied since in, for example, @cite_23 , @cite_47 , @cite_15 , (see, for example, @cite_22 ), (see @cite_48 , @cite_11 ), and (see @cite_37 ).
@cite_30 @cite_37 @cite_22 @cite_15 @cite_48 @cite_44 @cite_23 @cite_49 @cite_5 @cite_47 @cite_11
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Models of networked diffusion that are motivated by analogy with the spread of infectious disease have been applied to a wide range of social and economic adoption processes, including those related to new products, ideas, norms and behaviors. However, it is unknown how accurately these models account for the empirical structure of diffusion over networks. Here we describe the diffusion patterns arising from seven online domains, ranging from communications platforms to networked games to microblogging services, each involving distinct types of content and modes of sharing. We find strikingly similar patterns across all domains. In particular, the vast majority of cascades are small, and are described by a handful of simple tree structures that terminate within one degree of an initial adopting "seed." In addition we find that structures other than these account for only a tiny fraction of total adoptions; that is, adoptions resulting from chains of referrals are extremely rare. Finally, even for the largest cascades that we observe, we find that the bulk of adoptions often takes place within one degree of a few dominant individuals. Together, these observations suggest new directions for modeling of online adoption processes. Topological properties of "scale-free" networks are investigated by determining their spectral dimensions d(S), which reflect a diffusion process in the corresponding graphs. Data bases for citation networks and metabolic networks together with simulation results from the growing network model [A.-L. Barabasi and R. Albert, Science 286, 509 (1999)] are probed. For completeness and comparisons lattice, random and small-world models are also investigated. We find that d(S) is around 3 for citation and metabolic networks, which is significantly different from the growing network model, for which d(S) is approximately 7.5. This signals a substantial difference in network topology despite the observed similarities in vertex-order distributions. In addition, the diffusion analysis indicates that the citation networks are treelike in structure, whereas the metabolic networks contain many loops. (Less) Platforms such as Twitter have provided researchers with ample opportunities to analytically study social phenomena. There are however, significant computational challenges due to the enormous rate of production of new information: researchers are therefore, often forced to analyze a judiciously selected “sample” of the data. Like other social media phenomena, information diffusion is a social process–it is affected by user context, and topic, in addition to the graph topology. This paper studies the impact of different attribute and topology based sampling strategies on the discovery of an important social media phenomena–information diffusion. We examine several widely-adopted sampling methods that select nodes based on attribute (random, location, and activity) and topology (forest fire) as well as study the impact of attribute based seed selection on topology based sampling. Then we develop a series of metrics for evaluating the quality of the sample, based on user activity (e.g. volume, number of seeds), topological (e.g. reach, spread) and temporal characteristics (e.g. rate). We additionally correlate the diffusion volume metric with two external variables–search and news trends. Our experiments reveal that for small sample sizes (30 ), a sample that incorporates both topology and user context (e.g. location, activity) can improve on naive methods by a significant margin of 15-20 . In [N. Alon, M. Feldman, A.D. Procaccia, M. Tennenholtz, A note on competitive diffusion through social networks, Inform. Process. Lett. 110 (2010) 221-225], the authors introduced a game-theoretic model of diffusion process through a network. They showed a relation between the diameter of a given network and existence of pure Nash equilibria in the game. Theorem 1 of their paper says that a pure Nash equilibrium exists if the diameter is at most two. However, we have an example which does not admit a pure Nash equilibrium even if the diameter is two. Hence we correct the statement of Theorem 1 of their paper. Predicting the diffusion of information on social networks is a key problem for applications like Opinion Leader Detection, Buzz Detection or Viral Marketing. Many recent diffusion models are direct extensions of the Cascade and Threshold models, initially proposed for epidemiology and social studies. In such models, the diffusion process is based on the dynamics of interactions between neighbor nodes in the network (the social pressure), and largely ignores important dimensions as the content of the piece of information diffused. We propose here a new family of probabilistic models that aims at predicting how a content diffuses in a network by making use of additional dimensions: the content of the piece of information diffused, user's profile and willingness to diffuse. These models are illustrated and compared with other approaches on two blog datasets. The experimental results obtained on these datasets show that taking into account the content of the piece of information diffused is important to accurately model the diffusion process. Diffusion in disordered systems does not follow the classical laws which describe transport in ordered crystalline media, and this leads to many anomalous physical properties. Since the application of percolation theory, the main advances in the understanding of these processes have come from fractal theory. Scaling theories and numerical simulations are important tools to describe diffusion processes (random walks: the 'ant in the labyrinth') on percolation systems and fractals. Different types of disordered systems exhibiting anomalous diffusion are presented (the incipient infinite percolation cluster, diffusion-limited aggregation clusters, lattice animals, and random combs), and scaling theories as well as numerical simulations of greater sophistication are described. Also, diffusion in the presence of singular distributions of transition rates is discussed and related to anomalous diffusion on disordered structures. Based on the empirical analysis of the dependency network in 18 Java projects, we develop a novel model of network growth which considersboth: an attachment mechanism and the addition of new nodes with a heterogeneous distribution of their initial degree, k0. Empirically we find that the cumulative degree distributions of initial degrees and of the final network, follow power-law behaviors: P(k0) ∝ k01−α, and P(k) ∝ k1−γ, respectively. For the total number of links as a function of the network size, we find empirically K(N) ∝ Nβ, where β is (at the beginning of thenetwork evolution) between 1.25 and 2, while converging to ∼ 1 for large N. This indicates a transition from a growth regime with increasing network density towards a sustainable regime, which revents a collapse because of ever increasing dependencies. Our theoretical framework is able to predict relations between the exponents α, β, γ, which also link issues of software engineering and developer activity. These relations are verified by means of computer simulations and empirical investigations. They indicate that the growth of real Open Source Software networks occurs on the edge between two regimes, which are either dominated by the initial degree distribution of added nodes, or by the preferential attachment mechanism. Hence, the heterogeneous degree distribution of newly added nodes, found empirically, is essential to describe the laws of sustainable growth in networks. We pose a fundamental question in understanding how to identify and design successful communities: What factors predict whether a community will grow and survive in the long term? Social scientists have addressed this question extensively by analyzing offline groups which endeavor to attract new members, such as social movements, finding that new individuals are influenced strongly by their ties to members of the group. As a result, prior work on the growth of communities has treated growth primarily as a diffusion processes, leading to findings about group evolution which can be difficult to explain. The proliferation of online social networks and communities, however, has created new opportunities to study, at a large scale and with very fine resolution, the mechanisms which lead to the formation, growth, and demise of online groups. In this paper, we analyze data from several thousand online social networks built on the Ning platform with the goal of understanding the factors contributing to the growth and longevity of groups within these networks. Specifically, we investigate the role that two types of growth (growth through diffusion and growth by other means) play during a group's formative stages from the perspectives of both the individual member and the group. Applying these insights to a population of groups of different ages and sizes, we build a model to classify groups which will grow rapidly over the short-term and long-term. Our model achieves over 79 accuracy in predicting group growth over the following two months and over 78 accuracy in predictions over the following two years. We utilize a similar approach to predict which groups will die within a year. The results of our combined analysis provide insight into how both early non-diffusion growth and a complex set of network constraints appear to contribute to the initial and continued growth and success of groups within social networks. Finally we discuss implications of this work for the design, maintenance, and analysis of online communities. The percolation phase transition and the mechanism of the emergence of the giant component through the critical scaling window for random graph models, has been a topic of great interest in many dierent communities ranging from statistical physics, combinatorics, computer science, social networks and probability theory. The last few years have witnessed an explosion of models which couple random aggregation rules, that specify how one adds edges to existing congurations, and choice, wherein one selects from a ‘limited’ set of edges at random to use in the conguration. These models exhibit fascinating new phenomenon, ranging from delaying or speeding up the emergence of the giant component, to percolation", where the diameter of the scaling window is several orders of magnitude smaller than that for standard random graph models. While an intense study of such models has ensued, understanding the actual emergence of the giant component and merging dynamics in the critical scaling window has remained impenetrable to a rigorous analysis. In this work we take an important step in the analysis of such models by studying one of the standard examples of such processes, namely the Bohman-Frieze model, and provide rst results on the asymptotic dynamics, through the critical scaling window, that lead to the emergence of the giant component for such models. We identify the scaling window and show that through this window, the component sizes properly rescaled converge to the standard multiplicative coalescent. Proofs hinge on a careful analysis of certain innite-type branching processes with types taking values in the space of RCLL paths, and stochastic analytic techniques to estimate susceptibility functions of the components all the way through the scaling window where these functions explode. Previous approaches for analyzing random graphs at criticality have relied largely on classical breadth-rst search techniques that exploit asymptotic connections with Brownian excursions. For dynamic random graph models evolving via general Markovian rules, such approaches fail and we develop a quite dierent set of tools that can potentially be used for the study of critical dynamics for all bounded size rules. Diffusion of information, spread of rumors and infectious diseases are all instances of stochastic processes that occur over the edges of an underlying network. Many times networks over which contagions spread are unobserved, and such networks are often dynamic and change over time. In this paper, we investigate the problem of inferring dynamic networks based on information diffusion data. We assume there is an unobserved dynamic network that changes over time, while we observe the results of a dynamic process spreading over the edges of the network. The task then is to infer the edges and the dynamics of the underlying network. We develop an on-line algorithm that relies on stochastic convex optimization to efficiently solve the dynamic network inference problem. We apply our algorithm to information diffusion among 3.3 million mainstream media and blog sites and experiment with more than 179 million different pieces of information spreading over the network in a one year period. We study the evolution of information pathways in the online media space and find interesting insights. Information pathways for general recurrent topics are more stable across time than for on-going news events. Clusters of news media sites and blogs often emerge and vanish in matter of days for on-going news events. Major social movements and events involving civil population, such as the Libyan' s civil war or Syria' s uprise, lead to an increased amount of information pathways among blogs as well as in the overall increase in the network centrality of blogs and social media sites. Even though people in our contemporary technological society are depending on communication, our understanding of the underlying laws of human communicational behavior continues to be poorly understood. Here we investigate the communication patterns in 2 social Internet communities in search of statistical laws in human interaction activity. This research reveals that human communication networks dynamically follow scaling laws that may also explain the observed trends in economic growth. Specifically, we identify a generalized version of Gibrat's law of social activity expressed as a scaling law between the fluctuations in the number of messages sent by members and their level of activity. Gibrat's law has been essential in understanding economic growth patterns, yet without an underlying general principle for its origin. We attribute this scaling law to long-term correlation patterns in human activity, which surprisingly span from days to the entire period of the available data of more than 1 year. Further, we provide a mathematical framework that relates the generalized version of Gibrat's law to the long-term correlated dynamics, which suggests that the same underlying mechanism could be the source of Gibrat's law in economics, ranging from large firms, research and development expenditures, gross domestic product of countries, to city population growth. These findings are also of importance for designing communication networks and for the understanding of the dynamics of social systems in which communication plays a role, such as economic markets and political systems. The spread of new ideas, behaviors or technologies has been extensively studied using epidemic models. Here we consider a model of diffusion where the individualsʼ behavior is the result of a strategic choice. We study a simple coordination game with binary choice and give a condition for a new action to become widespread in a random network. We also analyze the possible equilibria of this game and identify conditions for the coexistence of both strategies in large connected sets. Finally we look at how can firms use social networks to promote their goals with limited information. Our results differ strongly from the one derived with epidemic models and show that connectivity plays an ambiguous role: while it allows the diffusion to spread, when the network is highly connected, the diffusion is also limited by high-degree nodes which are very stable. The website wheresgeorge.com invites its users to enter the serial numbers of their US dollar bills and track them across America and beyond. Why? “For fun and because it had not been done yet”, they say. But the dataset accumulated since December 1998 has provided the ideal raw material to test the mathematical laws underlying human travel, and that has important implications for the epidemiology of infectious diseases. Analysis of the trajectories of over half a million dollar bills shows that human dispersal is described by a ‘two-parameter continuous-time random walk’ model: our travel habits conform to a type of random proliferation known as ‘superdiffusion’. And with that much established, it should soon be possible to develop a new class of models to account for the spread of human disease. The dynamic spatial redistribution of individuals is a key driving force of various spatiotemporal phenomena on geographical scales. It can synchronize populations of interacting species, stabilize them, and diversify gene pools1,2,3. Human travel, for example, is responsible for the geographical spread of human infectious disease4,5,6,7,8,9. In the light of increasing international trade, intensified human mobility and the imminent threat of an influenza A epidemic10, the knowledge of dynamical and statistical properties of human travel is of fundamental importance. Despite its crucial role, a quantitative assessment of these properties on geographical scales remains elusive, and the assumption that humans disperse diffusively still prevails in models. Here we report on a solid and quantitative assessment of human travelling statistics by analysing the circulation of bank notes in the United States. Using a comprehensive data set of over a million individual displacements, we find that dispersal is anomalous in two ways. First, the distribution of travelling distances decays as a power law, indicating that trajectories of bank notes are reminiscent of scale-free random walks known as Levy flights. Second, the probability of remaining in a small, spatially confined region for a time T is dominated by algebraically long tails that attenuate the superdiffusive spread. We show that human travelling behaviour can be described mathematically on many spatiotemporal scales by a two-parameter continuous-time random walk model to a surprising accuracy, and conclude that human travel on geographical scales is an ambivalent and effectively superdiffusive process. Whether they are modeling bookmarking behavior in Flickr or cascades of failure in large networks, models of diffusion often start with the assumption that a few nodes start long chain reactions, resulting in large-scale cascades. While reasonable under some conditions, this assumption may not hold for social media networks, where user engagement is high and information may enter a system from multiple disconnected sources. Using a dataset of 262,985 Facebook Pages and their associated fans, this paper provides an empirical investigation of diffusion through a large social media network. Although Facebook diffusion chains are often extremely long (chains of up to 82 levels have been observed), they are not usually the result of a single chain-reaction event. Rather, these diffusion chains are typically started by a substantial number of users. Large clusters emerge when hundreds or even thousands of short diffusion chains merge together. This paper presents an analysis of these diffusion chains using zero-inflated negative binomial regressions. We show that after controlling for distribution effects, there is no meaningful evidence that a start node’s maximum diffusion chain length can be predicted with the user's demographics or Facebook usage characteristics (including the user's number of Facebook friends). This may provide insight into future research on public opinion formation. The population dynamics underlying the diffusion of ideas hold many qualitative similarities to those involved in the spread of infections. In spite of much suggestive evidence this analogy is hardly ever quantified in useful ways. The standard benefit of modeling epidemics is the ability to estimate quantitatively population average parameters, such as interpersonal contact rates, incubation times, duration of infectious periods, etc. In most cases such quantities generalize naturally to the spread of ideas and provide a simple means of quantifying sociological and behavioral patterns. Here we apply several paradigmatic models of epidemics to empirical data on the advent and spread of Feynman diagrams through the theoretical physics communities of the USA, Japan, and the USSR in the period immediately after World War II. This test case has the advantage of having been studied historically in great detail, which allows validation of our results. We estimate the effectiveness of adoption of the idea in the three communities and find values for parameters reflecting both intentional social organization and long lifetimes for the idea. These features are probably general characteristics of the spread of ideas, but not of common epidemics. Many of the world's most popular websites catalyze their growth through invitations from existing members. New members can then in turn issue invitations, and so on, creating cascades of member signups that can spread on a global scale. Although these diffusive invitation processes are critical to the popularity and growth of many websites, they have rarely been studied, and their properties remain elusive. For instance, it is not known how viral these cascades structures are, how cascades grow over time, or how diffusive growth affects the resulting distribution of member characteristics present on the site. In this paper, we study the diffusion of LinkedIn, an online professional network comprising over 332 million members, a large fraction of whom joined the site as part of a signup cascade. First we analyze the structural patterns of these signup cascades, and find them to be qualitatively different from previously studied information diffusion cascades. We also examine how signup cascades grow over time, and observe that diffusion via invitations on LinkedIn occurs over much longer timescales than are typically associated with other types of online diffusion. Finally, we connect the cascade structures with rich individual-level attribute data to investigate the interplay between the two. Using novel techniques to study the role of homophily in diffusion, we find striking differences between the local, edge-wise homophily and the global, cascade-level homophily we observe in our data, suggesting that signup cascades form surprisingly coherent groups of members. The spread of information cascades over social networks forms the diffusion networks. The latent structure of diffusion networks makes the problem of extracting diffusion links difficult. As observing the sources of information is not usually possible, the only available prior knowledge is the infection times of individuals. We confront these challenges by proposing a new method called to extract the diffusion networks by using the time-series data. We model the diffusion process on information networks as a Markov random walk process and develop an algorithm to discover the most probable diffusion links. We validate our model on both synthetic and real data and show the low dependency of our method to the number of transmitting cascades over the underlying networks. Moreover, The proposed model can speed up the extraction process up to 300 times with respect to the existing state of the art method. Unlike traditional epidemic virus spreading, behavior diffusion in social networks is conducted by rational users, who would make strategic choices instead of being randomly infected with some probability. Specifically, individuals always try to maximize their utilities through rationally selecting specific behaviors (adopting a new product, or spreading a rumor, etc.). However, utility obtained by an individual, naturally contains uncertainty, and it may stem from two sources: users’ imperfect and incomplete knowledge about others, and the inherently stochastic property in human behavior. Thus, it is imperative to model and analyze the diffusion pattern under the resulting uncertainty in social networks, which, however, has not yet been deeply examined by the existing works. This paper deeply explores the pattern of gossip diffusion in social networks when uncertainty exists in users’ decision making. In detail, the innovative results provided in this paper are: first, inspired by random utility theory, we formulate the diffusion model based on mixed logit model that allows for user’s uncertainty in determining whether to adopt a specific strategy; second, the formal analysis framework characterizing the diffusion process is derived through the approximation method of mean field theory; finally, we explore the extensive applicability of our proposed analysis framework through modeling rumor diffusion in social networks as a coordination game. Our findings are, for various structural characteristics, small uncertainty can significantly speed up the diffusion of gossip; furthermore, social networks with scale-free property can facilitate the gossip diffusion in the easiest way, but, the range of uncertainty factor that can maximize gossip diffusion is the smallest. The obtained results perfectly comply with the philosophical saying about rumor diffusion in real social life: easy come, easy go. Diffusion is known to be an important process governing the behaviours observed in network environments like social networks, contact networks, etc. For modeling the diffusion process, the Independent Cascade Model (IC Model) is commonly adopted and algorithms have been proposed for recovering the hidden diffusion network based on observed cascades. However, the IC Model assumes the effects of multiple neighbors on a node to be independent and does not consider the structural diversity of nodes' neighbourhood. In this paper, we propose an extension of the IC Model with the community structure of node neighbours incorporated. We derive an expectation maximization (EM) algorithm to infer the model parameters. To evaluate the effectiveness and efficiency of the proposed method, we compared it with the IC model and its variants that do not consider the structural properties. Our empirical results based on the MemeTracker dataset, shows that after incorporating the structural diversity, there is a significant improvement in the modelling accuracy, with reasonable increase in run-time. We prove that the extremal process of branching Brownian motion, in the limit of large times, converges weakly to a cluster point process. The limiting process is a (randomly shifted) Poisson cluster process, where the positions of the clusters is a Poisson process with intensity measure with exponential density. The law of the individual clusters is characterized as branching Brownian motions conditioned to perform “unusually large displacements”, and its existence is proved. The proof combines three main ingredients. First, the results of Bramson on the convergence of solutions of the Kolmogorov–Petrovsky–Piscounov equation with general initial conditions to standing waves. Second, the integral representations of such waves as first obtained by Lalley and Sellke in the case of Heaviside initial conditions. Third, a proper identification of the tail of the extremal process with an auxiliary process (based on the work of Chauvin and Rouault), which fully captures the large time asymptotics of the extremal process. The analysis through the auxiliary process is a rigorous formulation of the cavity method developed in the study of mean field spin glasses.