We consider the classic Moran process modeling the spread of genetic mutations, as extended to structured populations by (Nature, 2005). In this process, individuals are the vertices of a connected graph G. Initially, there is a single mutant vertex, chosen uniformly at random. In each step, a random vertex is selected for reproduction with a probability proportional to its fitness: mutants have fitness r > 1, while non-mutants have fitness 1. The vertex chosen to reproduce places a copy of itself to a uniformly random neighbor in G, replacing the individual that was there. The process ends when the mutation either reaches fixation (i.e., all vertices are mutants), or gets extinct. The principal quantity of interest is the probability with which each of the two outcomes occurs. A problem that has received significant attention recently concerns the existence of families of graphs, called strong amplifiers of selection, for which the fixation probability tends to 1 as the order n of the graph increases, and the existence of strong suppressors of selection, for which this probability tends to 0. For the case of directed graphs, it is known that both strong amplifiers and suppressors exist. For the case of undirected graphs, however, the problem has remained open, and the general belief has been that neither strong amplifiers nor suppressors exist. In this paper we disprove this belief, by providing the first examples of such graphs. The strong amplifier we present has fixation probability 1 − ˜ O(n^ −1 3), and the strong suppressor has fixation probability ˜O(n^−1 4). Both graph constructions are surprisingly simple. We also prove a general upper bound of 1 − ˜Ω(n^−1 3) on the fixation probability of any undirected graph. Hence, our strong amplifier is existentially optimal.
Finally, similar stochastic processes have been use to model dynamics other than evolutionary ones, such as the spread of influence in social networks @cite_23 , the spread of epidemics @cite_29 , the emergence of monopolies @cite_33 , or interacting particle systems @cite_26 .
@cite_29 @cite_26 @cite_33 @cite_23
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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. The systematic study of large-scale networks has unveiled the ubiquitous presence of connectivity patterns characterized by large-scale heterogeneities and unbounded statistical fluctuations. These features affect dramatically the behavior of the diffusion processes occurring on networks, determining the ensuing statistical properties of their evolution pattern and dynamics. In this article, we present a stochastic computational framework for the forecast of global epidemics that considers the complete worldwide air travel infrastructure complemented with census population data. We address two basic issues in global epidemic modeling: (i) we study the role of the large scale properties of the airline transportation network in determining the global diffusion pattern of emerging diseases; and (ii) we evaluate the reliability of forecasts and outbreak scenarios with respect to the intrinsic stochasticity of disease transmission and traffic flows. To address these issues we define a set of quantitative measures able to characterize the level of heterogeneity and predictability of the epidemic pattern. These measures may be used for the analysis of containment policies and epidemic risk assessment. This paper reports statistical analyses performed on simulated data from a stochastic multi-agent model of speculative behaviour in a financial market. The price dynamics resulting from this artificial market process exhibits the same type of scaling laws as do empirical data from stock markets and foreign exchange markets: (i) one observes scaling in the probability distribution of relative price changes with a Pareto exponent around 2.6, (ii) volatility shows significant long-range correlation with a self-similarity parameter H around 0.85. This happens although we assume that news about the intrinsic or fundamental value of the asset follows a white noise process and, hence, incorporation of news about fundamental factors is insufficient to explain either of the characteristics (i) or (ii). As a consequence, in our model, the main stylised facts of financial data originate from the working of the market itself and one need not resort to scaling in unobservable extraneous signals as an explanation of the source of scaling in financial prices. The emergence of power laws can be explained by the existence of a critical state which is repeatedly approached in the course of the system's development. Evolutionary dynamics has been traditionally studied in the context of homogeneous populations, mainly described by the Moran process [P. Moran, Random processes in genetics, Proceedings of the Cambridge Philosophical Society 54 (1) (1958) 60-71]. Recently, this approach has been generalized in [E. Lieberman, C. Hauert, M.A. Nowak, Evolutionary dynamics on graphs, Nature 433 (2005) 312-316] by arranging individuals on the nodes of a network (in general, directed). In this setting, the existence of directed arcs enables the simulation of extreme phenomena, where the fixation probability of a randomly placed mutant (i.e., the probability that the offspring of the mutant eventually spread over the whole population) is arbitrarily small or large. On the other hand, undirected networks (i.e., undirected graphs) seem to have a smoother behavior, and thus it is more challenging to find suppressors amplifiers of selection, that is, graphs with smaller greater fixation probability than the complete graph (i.e., the homogeneous population). In this paper we focus on undirected graphs. We present the first class of undirected graphs which act as suppressors of selection, by achieving a fixation probability that is at most one half of that of the complete graph, as the number of vertices increases. Moreover, we provide some generic upper and lower bounds for the fixation probability of general undirected graphs. As our main contribution, we introduce the natural alternative of the model proposed in [E. Lieberman, C. Hauert, M.A. Nowak, Evolutionary dynamics on graphs, Nature 433 (2005) 312-316]. In our new evolutionary model, all individuals interact simultaneously and the result is a compromise between aggressive and non-aggressive individuals. We prove that our new model of mutual influences admits a potential function, which guarantees the convergence of the system for any graph topology and any initial fitness vector of the individuals. Furthermore, we prove fast convergence to the stable state for the case of the complete graph, as well as we provide almost tight bounds on the limit fitness of the individuals. Apart from being important on its own, this new evolutionary model appears to be useful also in the abstract modeling of control mechanisms over invading populations in networks. We demonstrate this by introducing and analyzing two alternative control approaches, for which we bound the time needed to stabilize to the ''healthy'' state of the system. We introduce a general stochastic model for the spread of rumours, and derive mean-field equations that describe the dynamics of the model on complex social networks (in particular, those mediated by the Internet). We use analytical and numerical solutions of these equations to examine the threshold behaviour and dynamics of the model on several models of such networks: random graphs, uncorrelated scale-free networks and scale-free networks with assortative degree correlations. We show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system. In the case of scale-free networks, on the other hand, this threshold becomes vanishingly small in the limit of infinite system size. We find that the initial rate at which a rumour spreads is much higher in scale-free networks than in random graphs, and that the rate at which the spreading proceeds on scale-free networks is further increased when assortative degree correlations are introduced. The impact of degree correlations on the final fraction of nodes that ever hears a rumour, however, depends on the interplay between network topology and the rumour spreading rate. Our results show that scale-free social networks are prone to the spreading of rumours, just as they are to the spreading of infections. They are relevant to the spreading dynamics of chain emails, viral advertising and large-scale information dissemination algorithms on the Internet. We introduce stochastic time-dependency in evolving graphs: starting from an arbitrary initial edge probability distribution, at every time step, every edge changes its state (existing or not) according to a two-state Markovian process with probabilities p (edge birth-rate) and q (edge death-rate). If an edge exists at time t then, at time t+1, it dies with probability q. If instead the edge does not exist at time t, then it will come into existence at time t+1 with probability p. Such evolving graph model is a wide generalization of time-independent dynamic random graphs [6] and will be called edge-Markovian dynamic graphs. We investigate the speed of information dissemination in such dynamic graphs. We provide nearly tight bounds (which in fact turn out to be tight for a wide range of probabilities p and q) on the completion time of the flooding mechanism aiming to broadcast a piece of information from a source node to all nodes. In particular, we provide: i) A tight characterization of the class of edge-Markovian dynamic graphs where flooding time is constant and, thus, it does not asymptotically depend on the initial probability distribution. ii) A tight characterization of the class of edge-Markovian dynamic graphs where flooding time does not asymptotically depend on the edge death-rate q. The spread of epidemics and malware is commonly modeled by diffusion processes on networks. Protective interventions such as vaccinations or installing anti-virus software are used to contain their spread. Typically, each node in the network has to decide its own strategy of securing itself, and its benefit depends on which other nodes are secure, making this a natural game-theoretic setting. There has been a lot of work on network security game models, but most of the focus has been either on simplified epidemic models or homogeneous network structure. We develop a new formulation for an epidemic containment game, which relies on the characterization of the SIS model in terms of the spectral radius of the network. We show in this model that pure Nash equilibria (NE) always exist, and can be found by a best response strategy. We analyze the complexity of finding NE, and derive rigorous bounds on their costs and the Price of Anarchy or PoA (the ratio of the cost of the worst NE to the optimum social cost) in general graphs as well as in random graph models. In particular, for arbitrary power-law graphs with exponent β > 2, we show that the PoA is bounded by O(T2(β -1))), where T = γ α is the ratio of the recovery rate to the transmission rate in the SIS model. We prove that this bound is tight up to a constant factor for the Chung-Lu random power-law graph model. We study the characteristics of Nash equilibria empirically in different real communication and infrastructure networks, and find that our analytical results can help explain some of the empirical observations. This text offers a systematic, rigorous, and unified presentation of evolutionary game theory, covering the core developments of the theory from its inception in biology in the 1970s through recent advances. Evolutionary game theory, which studies the behavior of large populations of strategically interacting agents, is used by economists to make predictions in settings where traditional assumptions about agents' rationality and knowledge may not be justified. Recently, computer scientists, transportation scientists, engineers, and control theorists have also turned to evolutionary game theory, seeking tools for modeling dynamics in multiagent systems. Population Games and Evolutionary Dynamics provides a point of entry into the field for researchers and students in all of these disciplines. The text first considers population games, which provide a simple, powerful model for studying strategic interactions among large numbers of anonymous agents. It then studies the dynamics of behavior in these games. By introducing a general model of myopic strategy revision by individual agents, the text provides foundations for two distinct approaches to aggregate behavior dynamics: the deterministic approach, based on differential equations, and the stochastic approach, based on Markov processes. Key results on local stability, global convergence, stochastic stability, and nonconvergence are developed in detail. Ten substantial appendixes present the mathematical tools needed to work in evolutionary game theory, offering a practical introduction to the methods of dynamic modeling. Accompanying the text are more than 200 color illustrations of the mathematics and theoretical results; many were created using the Dynamo software suite, which is freely available on the author's Web site. Readers are encouraged to use Dynamo to run quick numerical experiments and to create publishable figures for their own research. Abstract Stochastic models such as Continuous-Time Markov Chains (CTMC) and Stochastic Hybrid Automata (SHA) are powerful formalisms to model and to reason about the dynamics of biological systems, due to their ability to capture the stochasticity inherent in biological processes. A classical question in formal modelling with clear relevance to biological modelling is the model checking problem, i.e. calculate the probability that a behaviour, expressed for instance in terms of a certain temporal logic formula, may occur in a given stochastic process. However, one may not only be interested in the notion of satisfiability, but also in the capacity of a system to maintain a particular emergent behaviour unaffected by the perturbations, caused e.g. from extrinsic noise, or by possible small changes in the model parameters. To address this issue, researchers from the verification community have recently proposed several notions of robustness for temporal logic providing suitable definitions of distance between a trajectory of a (deterministic) dynamical system and the boundaries of the set of trajectories satisfying the property of interest. The contributions of this paper are twofold. First, we extend the notion of robustness to stochastic systems, showing that this naturally leads to a distribution of robustness degrees. By discussing three examples, we show how to approximate the distribution of the robustness degree and the average robustness . Secondly, we show how to exploit this notion to address the system design problem , where the goal is to optimise some control parameters of a stochastic model in order to maximise robustness of the desired specifications. Evolutionary dynamics have been traditionally studied in the context of homogeneous or spatially extended populations1,2,3,4. Here we generalize population structure by arranging individuals on a graph. Each vertex represents an individual. The weighted edges denote reproductive rates which govern how often individuals place offspring into adjacent vertices. The homogeneous population, described by the Moran process3, is the special case of a fully connected graph with evenly weighted edges. Spatial structures are described by graphs where vertices are connected with their nearest neighbours. We also explore evolution on random and scale-free networks5,6,7. We determine the fixation probability of mutants, and characterize those graphs for which fixation behaviour is identical to that of a homogeneous population7. Furthermore, some graphs act as suppressors and others as amplifiers of selection. It is even possible to find graphs that guarantee the fixation of any advantageous mutant. We also study frequency-dependent selection and show that the outcome of evolutionary games can depend entirely on the structure of the underlying graph. Evolutionary graph theory has many fascinating applications ranging from ecology to multi-cellular organization and economics. Many people share their activities with others through online communities. These shared activities have an impact on other users' activities. For example, users are likely to become interested in items that are adopted (e.g. liked, bought and shared) by their friends. In this paper, we propose a probabilistic model for discovering latent influence from sequences of item adoption events. An inhomogeneous Poisson process is used for modeling a sequence, in which adoption by a user triggers the subsequent adoption of the same item by other users. For modeling adoption of multiple items, we employ multiple inhomogeneous Poisson processes, which share parameters, such as influence for each user and relations between users. The proposed model can be used for finding influential users, discovering relations between users and predicting item popularity in the future. We present an efficient Bayesian inference procedure of the proposed model based on the stochastic EM algorithm. The effectiveness of the proposed model is demonstrated by using real data sets in a social bookmark sharing service. In stochastic dynamical systems, different concepts of stability can be obtained in different limits. A particularly interesting example is evolutionary game theory, which is traditionally based on infinite populations, where strict Nash equilibria correspond to stable fixed points that are always evolutionarily stable. However, in finite populations stochastic effects can drive the system away from strict Nash equilibria, which gives rise to a new concept for evolutionary stability. The conventional and the new stability concepts may apparently contradict each other leading to conflicting predictions in large yet finite populations. We show that the two concepts can be derived from the frequency dependent Moran process in different limits. Our results help to determine the appropriate stability concept in large finite populations. The general validity of our findings is demonstrated showing that the same results are valid employing vastly different co-evolutionary processes. In this paper we tackle the following question: is it possible to predict the characteristics of the evolution of an epidemic process in a social network on the basis of the degree distribution alone? We answer this question affirmatively for several diffusion processes-- Push-Pull, Broadcast and SIR-- by showing that it is possible to predict with good accuracy their average evolution. We do this by developing a space efficient predictor that makes it possible to handle very large networks with very limited computational resources. Our experiments show that the prediction is surprisingly good for many instances of real-world networks. The class of real-world networks for which this happens can be characterized in terms of their neighbourhood function, which turns out to be similar to that of random networks. Finally, we analyse real instances of rumour spreading in Twitter and observe that our model describes qualitatively well their evolution. Evolution and structure of very large networks has attracted considerable attention in recent years. In this paper we study a possibility to simulate stochastic processes which move edges in a network leading to a scale-free structure. Scale-free networks are characterized by a ''fat-tail'' degree distribution with considerably higher presence of so called hubs - nodes with very high degree. To understand and predict very large networks it is important to study the possibility of parallel simulation. We consider a class of stochastic processes which keeps the number of edges in the network constant called equilibrium networks. This class is characterized by a preferential selection where the edge destinations are chosen according to a preferential function f(k) which depends on the node degree k. For this class of stochastic processes we prove that it is difficult if not impossible to design an exact parallel algorithm if the function f(k) is monotonous with an injective derivative. However, in the important case where f(k) is linear we present a fully scalable algorithm with almost linear speedup. The experimental results confirm the linear scalability on a large processor cluster. Volatility clustering, evaluated through slowly decaying auto-correlations, Hurst effect or 1 f noise for absolute returns, is a characteristic property ofmost financial assets return time series (1999). Statistical analysis alone is not able to provide a definite answer for the presence or absence of long-range dependence phenomenon in stock returns or volatility, unless economic mechanisms are proposed to understand the origin of such phenomenon Cont (2005); (1999). Whether results of statistical analysis correspond to longrange dependence is a difficult question and subject to an ongoing statistical debate Cont (2005). Extensive empirical analysis of the financial market data, supporting the idea that the longrange volatility correlations arise from trading activity, provides valuable background for further development of the long-ranged memory stochastic models (2003); (2001). The power-law behavior of the auto-regressive conditional duration process Sato (2004) based on the random multiplicative process and it’s special case the self-modulation process Takayasu (2003), exhibiting 1 f fluctuations, supported the idea of stochastic modeling with a power-law probability density function (PDF) and long-range memory. Thus the agent based economic models Kirman & Teyssiere (2002); Lux & Marchesi (2000) as well as the stochastic models Borland (2004); (2008; 2010); Queiros (2007) exhibiting long-range dependence phenomenon in volatility or trading volume are of great interest and remain an active topic of research. Properties of stochastic multiplicative point processes have been investigated analytically and numerically and the formula for the power spectrum has been derived Gontis & Kaulakys (2004). In themore recent papers (2006); Kaulakys A Ruseckas & Kaulakys (2010) the general form of the multiplicative stochastic differential equation (SDE) was derived in agreement with the model earlier proposed in Gontis & Kaulakys (2004). Since Gontis & Kaulakys (2004) a model of trading activity, based on a SDE driven Poisson-like process, was presented (2008) and in the most recent paper (2010) we proposed a double stochastic model, whose return time series yield two power-law statistics, i.e., the PDF and the power spectral density (PSD) of absolute return, mimicking the empirical data for the one-minute trading return in the NYSE. In this chapter we present theoretical arguments and empirical evidence for the non-linear double stochastic model of return in financial markets. With empirical data from NYSE and Vilnius Stock Exchange (VSE) demonstrating universal scaling of return statistical properties, 27 Cascades are ubiquitous in various network environments. How to predict these cascades is highly nontrivial in several vital applications, such as viral marketing, epidemic prevention and traffic management. Most previous works mainly focus on predicting the final cascade sizes. As cascades are typical dynamic processes, it is always interesting and important to predict the cascade size at any time, or predict the time when a cascade will reach a certain size (e.g. an threshold for outbreak). In this paper, we unify all these tasks into a fundamental problem: cascading process prediction. That is, given the early stage of a cascade, how to predict its cumulative cascade size of any later time? For such a challenging problem, how to understand the micro mechanism that drives and generates the macro phenomena (i.e. cascading process) is essential. Here we introduce behavioral dynamics as the micro mechanism to describe the dynamic process of a node's neighbors getting infected by a cascade after this node getting infected (i.e. one-hop subcascades). Through data-driven analysis, we find out the common principles and patterns lying in behavioral dynamics and propose a novel Networked Weibull Regression model for behavioral dynamics modeling. After that we propose a novel method for predicting cascading processes by effectively aggregating behavioral dynamics, and present a scalable solution to approximate the cascading process with a theoretical guarantee. We extensively evaluate the proposed method on a large scale social network dataset. The results demonstrate that the proposed method can significantly outperform other state-of-the-art baselines in multiple tasks including cascade size prediction, outbreak time prediction and cascading process prediction. 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. Stochastic processes of common use in mathematical finance are presented throughout this book, which consists of eleven chapters, interlacing on the one hand financial concepts and instruments, such as arbitrage opportunities, admissible strategies, contingent claims, option pricing, default risk, ruin, and on the other hand, Brownian motion, diffusion processes, Levy processes, together with the basic properties of these processes. The first half of the book is devoted to continuous path processes whereas the second half deals with discontinuous processes. Only basic knowledge of probability theory is assumed; the book is organized so that the mathematical facts pertaining to a given financial question are gathered close to the study of that question. Information diffusion in online social networks is affected by the underlying network topology, but it also has the power to change it. Online users are constantly creating new links when exposed to new information sources, and in turn these links are alternating the way information spreads. However, these two highly intertwined stochastic processes, information diffusion and network evolution, have been predominantly studied separately, ignoring their co-evolutionary dynamics. We propose a temporal point process model, COEVOLVE, for such joint dynamics, allowing the intensity of one process to be modulated by that of the other. This model allows us to efficiently simulate interleaved diffusion and network events, and generate traces obeying common diffusion and network patterns observed in real-world networks. Furthermore, we also develop a convex optimization framework to learn the parameters of the model from historical diffusion and network evolution traces. We experimented with both synthetic data and data gathered from Twitter, and show that our model provides a good fit to the data as well as more accurate predictions than alternatives. Can evolving networks be inferred and modeled without directly observing their nodes and edges? In many applications, the edges of a dynamic network might not be observed, but one can observe the dynamics of stochastic cascading processes (e.g., information diffusion, virus propagation) occurring over the unobserved network. While there have been efforts to infer networks based on such data, providing a generative probabilistic model that is able to identify the underlying time-varying network remains an open question. Here we consider the problem of inferring generative dynamic network models based on network cascade diffusion data. We propose a novel framework for providing a non-parametric dynamic network model---based on a mixture of coupled hierarchical Dirichlet processes---based on data capturing cascade node infection times. Our approach allows us to infer the evolving community structure in networks and to obtain an explicit predictive distribution over the edges of the underlying network---including those that were not involved in transmission of any cascade, or are likely to appear in the future. We show the effectiveness of our approach using extensive experiments on synthetic as well as real-world networks.