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Abstract

Extreme events are emergent phenomena in multi-particle transport processes on complex networks. In practice, such events could range from power blackouts to call drops in cellular networks to traffic congestion on roads. All the earlier studies of extreme events on complex networks had focused only on the nodal events. If random walks are used to model the transport process on a network, it is known that degree of the nodes determines the extreme event properties. In contrast, in this work, it is shown that extreme events on the edges display a distinct set of properties from that of the nodes. It is analytically shown that the probability for the occurrence of extreme events on an edge is independent of the degree of the nodes linked by the edge and is dependent only on the total number of edges on the network and the number of walkers on it. Further, it is also demonstrated that non-trivial correlations can exist between the extreme events on the nodes and the edges. These results are in agreement with the numerical simulations on synthetic and real-life networks.

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... In the last decade we have assisted with a wave of new works studying extreme (rare) events associated to dynamical processes evolving on complex networks [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The necessity to understand unlikely events comes from the fact that, although rare, their appearance determines the future of the system under study, which may be potentially catastrophic, e.g., earthquakes [1]. ...
... We show how to derive ARW via a large-deviation representation of MERW and study its dynamics on synthetic and real-world networks. In the last decade we have assisted with a wave of new works studying extreme (rare) events associated to dynamical processes evolving on complex networks [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The necessity to understand unlikely events comes from the fact that, although rare, their appearance determines the future of the system under study, which may be potentially catastrophic, e.g., earthquakes [1]. ...
... The necessity to understand unlikely events comes from the fact that, although rare, their appearance determines the future of the system under study, which may be potentially catastrophic, e.g., earthquakes [1]. In this context, researchers have focused on random walks and their load and flow properties [2,3,15,17] for models of traffic in transportation [11,16] and communication networks [4], or on epidemic models and extinction events [6], or again on general order-disorder [8] and percolation transitions [9,10] to corroborate the robustness of networks. In these settings, rare events are often driven by internal noise, and their understanding could provide us with control mechanisms to keep away from harmful scenarios [5,6,8,10]. ...
Article
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Maximum entropy random walks (MERWs) are maximally dispersing and play a key role in optimizing information spreading in various contexts. However, building MERWs comes at the cost of knowing beforehand the global structure of the network, a requirement that makes them totally inadequate in real-case scenarios. Here, we propose an adaptive random walk (ARW), which instead maximizes dispersion by updating its transition rule on the local information collected while exploring the network. We show how to derive ARW via a large-deviation representation of MERW and study its dynamics on synthetic and real-world networks.
... In the last decade we have assisted to a wave of new works studying extreme (rare) events associated to dynamical processes evolving on complex networks [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The necessity to understand unlikely events comes from the fact that, although rare, their appearance determines the future of the system under study, which may be potentially catastrophic, e.g., earthquakes [1]. ...
... The necessity to understand unlikely events comes from the fact that, although rare, their appearance determines the future of the system under study, which may be potentially catastrophic, e.g., earthquakes [1]. In this context, researchers have focused on random walks and their load and flow properties on nodes and edges [2,3,15,17] as models of traffic in transportation [11,16] and communication networks [4], or on epidemic models and extinction events [6], or again on general order-disorder [8] and percolation transitions [9,10] to corroborate the robustness of networks. In these settings, rare events are often driven by internal noise, e.g., inherent randomness in a random walk, rather than environmental changes, and their understanding could provide us with control mechanisms to keep away from harmful scenarios [5,6,8,10]. ...
... which is known to be cardinal to construct the MERW in Eq. (14). In fact, in a large deviation context, one seeks to solve Eq. (15) to determine the SCGF Ψ(1) via Eq. (8). ...
Preprint
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Random walks are the most versatile tool to explore a complex network. These include maximum entropy random walks (MERWs), which are maximally dispersing and therefore play a key role as they optimize information spreading. However, building a MERW comes at the cost of knowing beforehand the global structure of the network to be explored. Here, we propose an adaptive random walk (ARW), which instead maximizes dispersion by updating its transition rule on the local information collected while exploring the network. We show how to derive the ARW via a large-deviation representation of the MERW as a finite-time rare event of an unbiased random walk and study its dynamics on synthetic and real-world networks.
... The emergence of rare events may be bolstered or hindered by the hosting complex environment, often conveniently modeled as a complex network [2][3][4]. Large fluctuations in complex networks have been studied across a variety of processes, including percolation [5][6][7][8], spreading [9,10], and transport [11][12][13][14]. A stream of research has focused on random walks as a versatile model of diffusion in discrete spaces [15][16][17][18][19] and on their rare event properties [20][21][22]. ...
... where we remind the reader that fixing s corresponds to fixing a fluctuation t (on average) according to (8). ...
Article
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Although higher-order interactions are known to affect the typical state of dynamical processes giving rise to new collective behavior, how they drive the emergence of rare events and fluctuations is still an open problem. We investigate how fluctuations of a dynamical quantity of a random walk exploring a higher-order network arise over time. In the quenched case, where the hypergraph structure is fixed, through large deviation theory we show that the appearance of rare events is hampered in nodes with many higher-order interactions, and promoted elsewhere. Dynamical fluctuations are further boosted in an annealed scenario, where both the diffusion process and higher-order interactions evolve in time. Here, extreme fluctuations generated by optimal higher-order configurations can be predicted in the limit of a saddle-point approximation. Our study lays the groundwork for a wide and general theory of fluctuations and rare events in higher-order networks. Published by the American Physical Society 2024
... By applying harmonic pump modulation to a fiber laser, the emergence of rogue waves has been identified [2,[19][20][21]. The EEs in stochastic transport on networks has been demonstrated using multiple random walks on complex networks [22,23]. Now, the interesting question is whether extreme events can be induced by nonchaotic signals. ...
... Furthermore, we found that the negative values of LE near zero exhibit strange nonchaotic behavior; extreme events are seen in this region. The literature has shown that the EEs occur under chaotic dynamics [16] through distinct routes, and stochastic processes such as stochastic transport on networks have been demonstrated using multiple random walks on complex networks [22,23]. Among the various routes, the occurrence of EEs in nonchaotic dynamics is different. ...
Article
Extreme events are unusual and rare large-amplitude fluctuations can occur unexpectedly in nonlinear dynamical systems. Events above the extreme event threshold of the probability distribution of a nonlinear process characterize extreme events. Different mechanisms for the generation of extreme events and their prediction measures have been reported in the literature. Based on the properties of extreme events, such as those that are rare in the frequency of occurrence and extreme in amplitude, various studies have shown that extreme events are both linear and nonlinear in nature. Interestingly, in this Letter, we report on a special class of extreme events which are nonchaotic and nonperiodic. These nonchaotic extreme events appear in between the quasiperiodic and chaotic dynamics of the system. We report the existence of such extreme events with various statistical measures and characterization techniques.
... Occurrence of the dragon-king-like extreme events due to occasional irregular unison firing in two coupled Hindmarsh-Rose bursting neurons is inspected in reference [10]. The role of edges in the origination of extreme events is scrutinized in references [11,12]. However, the study of extreme events has been less explored in coupled maps. ...
... We have numerically determined the range of ε for the occurrence of extreme events in section 4. GEV distribution further corroborates the statistical signature of these events. We have evaluated the probability of the appearance of extreme events for some specific values of ε using equation (12) and the procedure given in section 3. For instance, this probability is 0.0027 for the system (1) with N = 200 coupled identical logistic maps f(x) = 4x(1 − x) and ε = 0.4995. ...
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Understanding and predicting uncertain things are the central themes of scientific evolution. Human beings revolve around these fears of uncertainties concerning various aspects like a global pandemic, health, finances, to name but a few. Dealing with this unavoidable part of life is far tougher due to the chaotic nature of these unpredictable activities. In the present article, we consider a global network of identical chaotic maps, which splits into two different clusters, despite the interaction between all nodes are uniform. The stability analysis of the spatially homogeneous chaotic solutions provides a critical coupling strength, before which we anticipate such partial synchronization. The distance between these two chaotic synchronized populations often deviates more than eight times of standard deviation from its long-term average. The probability density function of these highly deviated values fits well with the Generalized Extreme Value distribution. Meanwhile, the distribution of recurrence time intervals between extreme events resembles the Weibull distribution. The existing literature helps us to characterize such events as extreme events using the significant height. These extremely high fluctuations are less frequent in terms of their occurrence. We determine numerically a range of coupling strength for these extremely large but recurrent events. On-off intermittency is the responsible mechanism underlying the formation of such extreme events. Besides understanding the generation of such extreme events and their statistical signature, we furnish forecasting these events using the powerful deep learning algorithms of an artificial recurrent neural network. This Long Short-Term Memory (LSTM) can offer handy one-step forecasting of these chaotic intermittent bursts. We also ensure the robustness of this forecasting model with two hundred hidden cells in each LSTM layer.
... Indeed, this first-passage time plays a significant role in various contexts, e.g. cell biology [38][39][40], finance [41][42][43][44][45], climate studies [46][47][48], transport phenomena [49][50][51][52][53], transition path times on a reaction coordinate [54,55], and bio-chemical reactions [56][57][58][59][60][61][62]. It also lies at the heart of extreme-value [63][64][65][66] and record statistics [67][68][69][70][71], which aim to describe the properties of rare events occuring in dynamical systems [72]. ...
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Gated first-passage processes, where completion depends on both hitting a target and satisfying additional constraints, are prevalent across various fields. Despite their significance, analytical solutions to basic problems remain unknown, e.g. the detection time of a diffusing particle by a gated interval, disk, or sphere. In this paper, we elucidate the challenges posed by continuous gated first-passage processes and present a renewal framework to overcome them. This framework offers a unified approach for a wide range of problems, including those with single-point, half-line, and interval targets. The latter have so far evaded exact solutions. Our analysis reveals that solutions to gated problems can be obtained directly from the ungated dynamics. This, in turn, reveals universal properties and asymptotic behaviors, shedding light on cryptic intermediate-time regimes and refining the notion of high-crypticity for continuous-space gated processes. Moreover, we extend our formalism to higher dimensions, showcasing its versatility and applicability. Overall, this work provides valuable insights into the dynamics of continuous gated first-passage processes and offers analytical tools for studying them across diverse domains.
... Extreme events, characterized by rare but intensely fluctuating properties, are ubiquitous in both engineering system and natural phenomena [1]. For instance, turbulent gusts over an aircraft can result in bumpy flights [2], severe weather can disrupt communication systems [3], rare but large cascades in electrical power grids may lead to failures [4], extreme ocean temperature oscillations could impact agriculture and ecosystems [5], rare but significant fluctuations in brain network could cause seizures [6], and sudden increases in traffic flow can trigger network paralysis [7]. In these scenarios, the real-time prediction of extreme events is crucial for enabling proactive measures to avert potential issues [8,9]. ...
Article
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Predicting extreme events in chaotic systems, characterized by rare but intensely fluctuating properties, is of great importance due to their impact on the performance and reliability of a wide range of systems. Some examples include weather forecasting, traffic management, power grid operations, and financial market analysis, to name a few. Methods of increasing sophistication have been developed to forecast events in these systems. However, the boundaries that define the maximum accuracy of forecasting tools are still largely unexplored from a theoretical standpoint. Here, we address the question: What is the minimum possible error in the prediction of extreme events in complex, chaotic systems? We derive the minimum probability of error in extreme event forecasting along with its information-theoretic lower and upper bounds. These bounds are universal for a given problem, in that they hold regardless of the modeling approach for extreme event prediction: from traditional linear regressions to sophisticated neural network models. The limits in predictability are obtained from the cost-sensitive Fano’s and Hellman’s inequalities using the Rényi entropy. The results are also connected to Takens’ embedding theorem using the information can’t hurt inequality. Finally, the probability of error for a forecasting model is decomposed into three sources: uncertainty in the initial conditions, hidden variables, and suboptimal modeling assumptions. The latter allows us to assess whether prediction models are operating near their maximum theoretical performance or if further improvements are possible. The bounds are applied to the prediction of extreme events in the Rössler system and the Kolmogorov flow.
... Extreme events, characterized by rare but intensely fluctuating properties, are ubiquitous in both engineering system and natural phenomena [1]. For instance, turbulent gusts over an aircraft can result in bumpy flights [2], severe weather can disrupt communication systems [3], rare but large cascades in electrical power grids may lead to failures [4], extreme ocean temperature oscillations could impact agriculture and ecosystems [5], rare but significant fluctuations in brain network could cause seizures [6], and sudden increases in traffic flow can trigger network paralysis [7]. In these scenarios, the real-time prediction of extreme events is crucial for enabling proactive measures to avert potential issues [8,9]. ...
Preprint
Full-text available
Predicting extreme events in chaotic systems, characterized by rare but intensely fluctuating properties, is of great importance due to their impact on the performance and reliability of a wide range of systems. Some examples include weather forecasting, traffic management, power grid operations, and financial market analysis, to name a few. Methods of increasing sophistication have been developed to forecast events in these systems. However, the boundaries that define the maximum accuracy of forecasting tools are still largely unexplored from a theoretical standpoint. Here, we address the question: What is the minimum possible error in the prediction of extreme events in complex, chaotic systems? We derive lower bounds for the minimum probability of error in extreme event forecasting using the information-theoretic Fano's inequality. The limits obtained are universal, in that they hold regardless of the modeling approach: from traditional linear regressions to sophisticated neural network models. The approach also allows us to assess whether reduced-order models are operating near their theoretical maximum performance or if further improvements are theoretically possible.
... Specific events in natural life, especially extreme events, have attracted much attention from academia and the public. 1 Because the concerned specific events occurred non-continuously, the classical linear and nonlinear statistical methods may not be able to obtain the correlation of events. As an effective method to quantify the correlation of extreme events, event synchrony measurement has gradually become the focus of many different research studies and methods. ...
Article
The problem of synchronicity quantification, based on event occurrence time, has become the research focus in different fields. Methods of synchrony measurement provide an effective way to explore spatial propagation characteristics of extreme events. Using the synchrony measurement method of event coincidence analysis, we construct a directed weighted network and innovatively explore the direction of correlations between event sequences. Based on trigger event coincidence, the synchrony of traffic extreme events of base stations is mea-sured. Analyzing topology characteristics of the network, we study the spatial propagation characteristics of traffic extreme events in the communication system, including the propagation area, propagation influence, and spatial aggregation. This study provides a framework of network modeling to quantify the propagation characteristics of extreme events, which is helpful for further research on the prediction of extreme events. In particular, our framework is effective for events that occurred in time aggregation. In addition, from the perspective of a directed network, we analyze differences between the precursor event coincidence and the trigger event coincidence and the impact of event aggregation on the synchrony measurement methods. The precursor event coincidence and the trigger event coincidence are consistent when identifying event synchronization, while there are differences when measuring the event synchronization extent. Our study can provide a reference for the analysis of extreme climatic events such as rainstorms, droughts, and others in the climate field.
... An important question to ask in all of these scenarios is when does X(t) cross a threshold for the first time. This question carries invaluable insights in the context of cell biology [35][36][37], finance [38][39][40][41][42], climate studies [43][44][45], transport phenomena [46][47][48][49][50], transition path times on a reaction coordinate [51], and bio-chemical reactions [52][53][54][55][56][57][58], and lies at the heart of extreme-value [59][60][61][62] and record statistics [63][64][65][66][67], which aim to describe the properties of rare events in dynamical systems [68]. However, oftentimes X(t) cannot be observed at all times, and thus the observations are intermittent. ...
Preprint
A first-passage process, as the name implies, is a process that completes when it hits a certain target value for the first time. Oftentimes, however, the completion relies on additional constraints that have to be simultaneously satisfied. Such processes are then called gated first-passage processes. Despite their ubiquity, and the importance of continuous processes, a unified framework which allows their systematic study is missing. While the use of a renewal approach has provided deep insight into the dynamics of gated reactions in discrete space, a straight forward analogy in continuous space is not possible due to the ill-defined nature of the first-return process in continuous space. In this work, we carefully explain this pathology, its implications, and its remedy. The formalism developed herein allows us to obtain the full statistics of the gated first-passage time. Notably, our general formalism extends beyond single-point targets. We first present a formalism that accounts for interval targets. Then we show how one can obtain the single-point target formalism as a limiting case of the interval target. The other limit, of an infinite target, is the problem of threshold crossing, which is a central theme in extremal statistics. In cases where the underlying process is heavy-tailed, we provide the asymptotics of the gated process in terms of the ungated one. Furthermore, our analysis of gated first-passage processes beyond the single-point target case reveals the need for a refinement of the definition of crypticity in gated process. Lastly, we demonstrate how the 1-dimensional formalism can be utilized to obtain results in higher dimensions via the mapping between a 1-dimensional diffusion in a logarithmic potential and the Bessel process in any dimension. All of our findings are corroborated on a wide set of examples.
... But, practical experiences suggest that jamming like situation can arise not only in nodes, but also on connecting edges. This study has been recently perceived by Kumar et al. [378]. The extreme event probability in the case of edges is solely dependent on the total number of walkers and the total number of edges. ...
Article
Extreme events gain the attention of researchers due to their utmost importance in various contexts ranging from climate to brain. An observable that deviates significantly from its long-time average will have adverse consequences for the system. This brings such recurrent events to the limelight of attention in interdisciplinary research. There is a need for research efforts in many systems in the real world to find solutions that can predict and mitigate the unfavorable effects of these recurring events. A comprehensive review of recent progress is provided to capture recent improvements in analyzing such very high-amplitude events from the point of view of dynamical systems and random walkers. We emphasize, in detail, the mechanisms responsible for the emergence of such events in complex systems. Several mechanisms that contribute to the occurrence of extreme events have been elaborated that investigate the sources of instabilities leading to them. In addition, we discuss the prediction of extreme events from two different contexts, using dynamical instabilities and data-based machine learning algorithms. Tracking of instabilities in the phase space is not always feasible and a precise knowledge of the dynamics of extreme events does not necessarily help in forecasting extreme events. Moreover, in most of the studies on high-dimensional systems, only a few degrees of freedom participate in extreme events’ formation. Thus, a notable inclusion of prediction through machine learning is of enormous significance, particularly for those cases where the governing equations of the model are explicitly unavailable. Besides, random walks on complex networks can represent several transport processes, and exceedances of the flux of walkers above a prescribed threshold may describe extreme events. We unveil theoretical studies on random walkers with their enormous potential for applications in reducing extreme events. We cover the possible controlling strategies, which may be helpful to mitigate extreme events in physical situations like traffic jams, heavy load of web requests, competition for shared resources, floods in the network of rivers, and many more. This review presents an overview of the current trend of research on extreme events in dynamical systems and networks, including random walkers, and discusses future possibilities. We conclude this review with an extended outlook and compelling perspective, along with the non-trivial challenges for further investigation.
... The central quantity of interest is the first detection time distribution (FDTD) of the threshold crossing event by the sensor. This can be thought of as first detection of an extreme event [26][27][28] or a general threshold activated process [29][30][31] under intermittent sensing. A schematic is displayed in Fig. 1(a), where a stochastic process X (t ) is evolving in time, whereas a sensor, which monitors whether the stochastic process has crossed a predefined threshold X * or not, switches between inactive (red background) and active (green) states. ...
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The time taken by a random variable to cross a threshold for the first time, known as the first passage time, is of interest in many areas of sciences and engineering. Conventionally, there is an implicit assumption that the notional “sensor” monitoring the threshold crossing event is always active. In many realistic scenarios, the sensor monitoring the stochastic process works intermittently. Then, the relevant quantity of interest is the first detection time, which denotes the time when the sensor detects the random variable to be above the threshold for the first time. In this Letter, a birth-death process monitored by a random intermittent sensor is studied for which the first detection time distribution is obtained. In general, it is shown that the first detection time is related to and is obtainable from the first passage time distribution. Our analytical results display an excellent agreement with simulations. Furthermore, this framework is demonstrated in several applications—the susceptible infected susceptible compartmental and logistic models and birth-death processes with resetting. Finally, we solve the practically relevant problem of inferring the first passage time distribution from the first detection time.
... But, practical experiences suggest that jamming like situation can arise not only in nodes, but also on connecting edges. This study is recently perceived by Kumar et al. [370]. The extreme event probability in the case of edges is solely dependent on the total number of walkers and the total number of edges. ...
Preprint
Full-text available
Extreme events gain tremendous attention due to their utmost importance in a variety of diverse contexts ranging from climate science to neuroscience. Excursions of a relevant observable from its long-term average to extraordinary values have the capability of bringing adverse consequences. We provide here a comprehensive review to incorporate the recent efforts in understanding such extremely large-amplitude events from the perspective of dynamical systems and random walkers. We emphasize, in detail, the mechanisms responsible for the emergence of such events in the complex systems. We discuss the prediction of extreme events from two different contexts using (i) dynamical instabilities and (ii) machine learning algorithms. Tracking of instabilities in the phase space is not always feasible and precise knowledge of the dynamics of extreme events does not necessarily help in forecasting extreme events. Moreover, in most studies on high-dimensional systems, only a few degrees of freedom participate in extreme events' formation. Thus, the notable inclusion of prediction through machine learning is of enormous significance, particularly for those cases where the governing equations of the model are explicitly unavailable. Besides, random walk on the complex networks is capable of representing several transport processes, and exceedances of the flux of walkers above a prescribed threshold may describe extreme events. We unveil the theoretical studies on random walkers with their enormous potential for applications in reducing extreme events and also discuss possible controlling strategies. This review presents an overview of the current trend of research on extreme events in dynamical systems and networks, including random walkers, and discusses future possibilities. We conclude this review with the extended outlook and compelling perspective for further investigation.
... The central quantity of interest is the first detection time distribution (FDTD) of the threshold crossing event by the sensor. This can be thought of as first detection of an extreme event [26][27][28] or a general threshold activated process [29][30][31] under intermittent sensing. A schematic of the processes are displayed in Fig. 1(a), where a stochastic process X(t) is evolving in time, while a sensor, which monitors whether the stochastic process has crossed a pre-defined threshold X * or not, switches between inactive (red background) and active (green) states. ...
Preprint
Full-text available
The time taken by a random variable to cross a threshold for the first time, known as the first passage time, is of interest in many areas of sciences and engineering. Conventionally, there is an implicit assumption that the notional 'sensor' monitoring the threshold crossing event is always active. In many realistic scenarios, the sensor monitoring the stochastic process works intermittently. Then, the relevant quantity of interest is the first detection time, which denotes the time when the sensor detects the threshold crossing event for the first time. In this work, a birth-death process monitored by a random intermittent sensor is studied, for which the first detection time distribution is obtained. In general, it is shown that the first detection time is related to, and is obtainable from, the first passage time distribution. Our analytical results display an excellent agreement with simulations. Further, this framework is demonstrated in several applications -- the SIS compartmental and logistic models, and birth-death processes with resetting. Finally, we solve the practically relevant problem of inferring the first passage time distribution from the first detection time.
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Extreme events have low occurrence probabilities and display pronounced deviation from their average behavior, such as earthquakes or power blackouts. Such extreme events occurring on the nodes of a complex network have been extensively studied earlier through the modeling framework of unbiased random walks. They reveal that the occurrence probability for extreme events on nodes of a network has a strong dependence on the nodal properties. Apart from these, a recent work has shown the independence of extreme events on edges from those occurring on nodes. Hence, in this work, we propose a more general formalism to study the properties of extreme events arising from biased random walkers on the edges of a network. This formalism is applied to biases based on a variety network centrality measures including PageRank. It is shown that with biased random walkers as the dynamics on the network, extreme event probabilities depend on the local properties of the edges. The probabilities are highly variable for some edges of the network, while they are approximately a constant for some other edges on the same network. This feature is robust with respect to different biases applied to the random walk algorithm. Further, using the results from this formalism, it is shown that a network is far more robust to extreme events occurring on edges when compared to those occurring on the nodes.
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With this webpage the authors intend to inform the readers of errors or mistakes found in the book after publication. We also give extensions for some material in the book. We acknowledge the contribution of many readers.
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Statistical physics has proven to be a very fruitful framework to describe phenomena outside the realm of traditional physics. The last years have witnessed the attempt by physicists to study collective phenomena emerging from the interactions of individuals as elementary units in social structures. Here we review the state of the art by focusing on a wide list of topics ranging from opinion, cultural and language dynamics to crowd behavior, hierarchy formation, human dynamics, social spreading. We highlight the connections between these problems and other, more traditional, topics of statistical physics. We also emphasize the comparison of model results with empirical data from social systems. Comment: 58 pages, 22 figures. Final version published in Reviews of Modern Physics
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Random walks on discrete lattices are fundamental models that form the basis for our understanding of transport and diffusion processes. For a single random walker on complex networks, many properties such as the mean first passage time and cover time are known. However, many recent applications such as search engines and recommender systems involve multiple random walkers on complex networks. In this work, based on numerical simulations, we show that the fraction of nodes of scale-free network not visited by W random walkers in time t has a stretched exponential form independent of the details of the network and number of walkers. This leads to a power-law relation between nodes not visited by W walkers and by one walker within time t. The problem of finding the distinct nodes visited by W walkers, effectively, can be reduced to that of a single walker. The robustness of the results is demonstrated by verifying them on four different real-world networks that approximately display scale-free structure.
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We introduce and study a metapopulation model of random walkers interacting at the nodes of a complex network. The model integrates random relocation moves over the links of the network with local interactions depending on the node occupation probabilities. The model is highly versatile, as the motion of the walkers depends on the topological properties of the nodes, such as their degree, while any general nonlinear function of the occupation probability of a node can be considered as local reaction term. In addition to this, the relative strength of reaction and relocation can be tuned at will, depending on the specific application being examined. We derive an analytical expression for the occupation probability of the walkers at equilibrium in the most general case. We show that it depends on different order derivatives of the local reaction functions, on the degree of a node, and on the average degree of its neighbors at various distances. For such a reason, reactive random walkers are very sensitive to the structure of a network and are a powerful way to detect network properties such as symmetries or degree-degree correlations. As possible applications, we first discuss how the occupation probability of reactive random walkers can be used to define novel measures of functional centrality for the nodes of a network. We then illustrate how network components with the same symmetries can be revealed by tracking the evolution of reactive walkers. Finally, we show that the dynamics of our model is influenced by the presence of degree-degree correlations, so that assortative and disassortative networks can be classified by quantitative indicators based on reactive walkers.
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These notes give a summary of the techniques used in large deviation theory to study the fluctuations of time-additive quantities, called dynamical observables, defined in the context of Langevin-type equations, which model equilibrium and nonequilibrium processes driven by external forces and noise sources. These fluctuations are described by large deviation functions, obtained by solving a dominant eigenvalue problem similar to the problem of finding the ground state energy of quantum systems. This analogy is used to explain the differences that exist between the fluctuations of equilibrium and nonequilibrium processes. An example involving the Ornstein-Uhlenbeck process is worked out in detail to illustrate these methods. Exercises, at the end of the notes, also complement the theory.
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We investigate multiple random walks traversing independently and concurrently on complex networks and introduce the concept of mean first parallel passage time (MFPPT) to quantify their search efficiency. The mean first parallel passage time represents the expected time required to find a given target by one or some of the multiple walkers. We develop a general theory that allows us to calculate the MFPPT analytically. Interestingly, we find that the global MFPPT follows a harmonic law with respect to the global mean first passage times of the associated walkers. Remarkably, when the properties of multiple walkers are identical, the global MFPPT decays in a power law manner with an exponent of unity, irrespective of network structure. These findings are confirmed by numerical and theoretical results on various synthetic and real networks. The harmonic law reveals a universal principle governing multiple random walks on networks that uncovers the contribution and role of the combined walkers in a target search. Our paradigm is also applicable to a broad range of random search processes.
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Preface to the special issue on Extreme Events and its Applications.
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Do the movements of animals, including humans, follow patterns that can be described quantitatively by simple laws of motion? If so, then why? These questions have attracted the attention of scientists in many disciplines, and stimulated debates ranging from ecological matters to queries such as 'how can there be free will if one follows a law of motion?' This is the first book on this rapidly evolving subject, introducing random searches and foraging in a way that can be understood by readers without a previous background on the subject. It reviews theory as well as experiment, addresses open problems and perspectives, and discusses applications ranging from the colonization of Madagascar by Austronesians to the diffusion of genetically modified crops. The book will interest physicists working in the field of anomalous diffusion and movement ecology as well as ecologists already familiar with the concepts and methods of statistical physics. © G. M. Viswanathan, M.G.E. da Luz, E.P. Raposo and H.E. Stanley 2011.
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The Nobel Foundation asks that the Nobel lecture cover the work for which the Prize is awarded. The announcement of this year's Prize cites empirical work in asset pricing. I interpret this to include work on efficient capital markets and work on developing and testing asset pricing models—the two pillars, or perhaps more descriptive, the Siamese twins of asset pricing. I start with efficient markets and then move on to asset pricing models.
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Significant, and usually unwelcome, surprises, such as floods, financial crisis, epileptic seizures, or material rupture, are the topics of Extreme Events in Nature and Society. The book, authored by foremost experts in these fields, reveals unifying and distinguishing features of extreme events, including problems of understanding and modelling their origin, spatial and temporal extension, and potential impact. The chapters converge towards the difficult problem of anticipation: forecasting the event and proposing measures to moderate or prevent it.
Chapter
We examine the relationship between PageRank and several invariants occurring in the study of random walks and electrical networks. We consider a generalized version of hitting time and effective resistance with an additional parameter which controls the ‘speed’ of diffusion. We will establish their connection with PageRank. Through these connections, a combinatorial interpretation of Page-Rank is given in terms of rooted spanning forests by using a generalized version of the matrix-tree theorem. Using PageRank, we will illustrate that the generalized hitting time leads to finding sparse cuts and efficient approximation algorithms for PageRank can be used for approximating hitting time and effective resistance.
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We review the general problem of random searches in the context of biological encounters. We analyze deterministic and stochastic aspects of searching in general and address the destructive and nondestructive cases specifically. We discuss the concepts of Lévy walks as adaptive strategies and explore possible examples. We also review Lévy searches in other media and spaces, including lattices and networks as opposed to continuous environments. We analyze empirical evidence supporting the Lévy flight foraging hypothesis, as well as the more general idea of superdiffusive foraging. We compare these hypothesis with alternative theories of random searches. Finally, we comment on several issues relevant to the practical application of models of Lévy and superdiffusive strategies to the general question of biological foraging.
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DOI:https://doi.org/10.1103/RevModPhys.15.1
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Complex networks have been studied intensively for a decade, but research still focuses on the limited case of a single, non-interacting network. Modern systems are coupled together and therefore should be modelled as interdependent networks. A fundamental property of interdependent networks is that failure of nodes in one network may lead to failure of dependent nodes in other networks. This may happen recursively and can lead to a cascade of failures. In fact, a failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system of several interdependent networks. A dramatic real-world example of a cascade of failures ('concurrent malfunction') is the electrical blackout that affected much of Italy on 28 September 2003: the shutdown of power stations directly led to the failure of nodes in the Internet communication network, which in turn caused further breakdown of power stations. Here we develop a framework for understanding the robustness of interacting networks subject to such cascading failures. We present exact analytical solutions for the critical fraction of nodes that, on removal, will lead to a failure cascade and to a complete fragmentation of two interdependent networks. Surprisingly, a broader degree distribution increases the vulnerability of interdependent networks to random failure, which is opposite to how a single network behaves. Our findings highlight the need to consider interdependent network properties in designing robust networks.
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A study of failures in interconnected networks highlights the vulnerability of tightly coupled infrastructures and shows the need to consider mutually dependent network properties in designing resilient systems. On 28 September 2003, Italy suffered a near-nationwide power cut (Sicily was spared) that also brought down the Internet. Buldyrev et al. take this event, typical of a number that have occurred worldwide in recent years, and examine how such a cascade of failures involving independent networks can occur. They find that, surprisingly, a broader degree of distribution increases the vulnerability of interdependent networks to random failure — the opposite of what happens in a single network. This highlights the need to consider interdependent network properties when designing robust networks if a random failure is not to have catastrophic results.
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The distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines. It is well known that many physical processes in nature and society display long-range correlations. Hence, in the last few years, considerable research effort has been directed towards studying the distribution of return intervals for long-range correlated time series. Based on numerical simulations, it was shown that the return interval distributions are of stretched exponential type. In this paper, we obtain an analytical expression for the distribution of return intervals in long-range correlated time series which holds good when the average return intervals are large. We show that the distribution is actually a product of power law and a stretched exponential form. We also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events.
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We investigate random walks on complex networks and derive an exact expression for the mean first-passage time (MFPT) between two nodes. We introduce for each node the random walk centrality C, which is the ratio between its coordination number and a characteristic relaxation time, and show that it determines essentially the MFPT. The centrality of a node determines the relative speed by which a node can receive and spread information over the network in a random process. Numerical simulations of an ensemble of random walkers moving on paradigmatic network models confirm this analytical prediction.
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A novel method is proposed for performing multilabel, interactive image segmentation. Given a small number of pixels with user-defined (or predefined) labels, one can analytically and quickly determine the probability that a random walker starting at each unlabeled pixel will first reach one of the prelabeled pixels. By assigning each pixel to the label for which the greatest probability is calculated, a high-quality image segmentation may be obtained. Theoretical properties of this algorithm are developed along with the corresponding connections to discrete potential theory and electrical circuits. This algorithm is formulated in discrete space (i.e., on a graph) using combinatorial analogues of standard operators and principles from continuous potential theory, allowing it to be applied in arbitrary dimension on arbitrary graphs.
Article
The theory of large deviations is concerned with the exponential decay of probabilities of large fluctuations in random systems. These probabilities are important in many fields of study, including statistics, finance, and engineering, as they often yield valuable information about the large fluctuations of a random system around its most probable state or trajectory. In the context of equilibrium statistical mechanics, the theory of large deviations provides exponential-order estimates of probabilities that refine and generalize Einstein's theory of fluctuations. This review explores this and other connections between large deviation theory and statistical mechanics, in an effort to show that the mathematical language of statistical mechanics is the language of large deviation theory. The first part of the review presents the basics of large deviation theory, and works out many of its classical applications related to sums of random variables and Markov processes. The second part goes through many problems and results of statistical mechanics, and shows how these can be formulated and derived within the context of large deviation theory. The problems and results treated cover a wide range of physical systems, including equilibrium many-particle systems, noise-perturbed dynamics, nonequilibrium systems, as well as multifractals, disordered systems, and chaotic systems. This review also covers many fundamental aspects of statistical mechanics, such as the derivation of variational principles characterizing equilibrium and nonequilibrium states, the breaking of the Legendre transform for nonconcave entropies, and the characterization of nonequilibrium fluctuations through fluctuation relations. Comment: v1: 89 pages, 18 figures, pdflatex. v2: 95 pages, 20 figures, text, figures and appendices added, many references cut, close to published version