Some features of the spread of epidemics and information on a random graph

Durrett, Rick

doi:10.1073/pnas.0914402107

Cited by 133 publications

(129 citation statements)

References 69 publications

(48 reference statements)

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“…In this expressios β = λ k is the per capita spreading rate that takes into account the rate of contacts k . While the expression might be different in the case of static networks [31][32][33] the topological properties of the underlying network have critical effects on the threshold. In time-varying networks the analytical study of contagion processes is hindered by the difficulties in dealing with the concurrent time scales of the contagion and network evolution processes.…”

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Controlling Contagion Processes in Activity Driven Networks

et al. 2014

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The vast majority of strategies aimed at controlling contagion processes on networks considers the connectivity pattern of the system as either quenched or annealed. However, in the real world many networks are highly dynamical and evolve in time concurrently to the contagion process. Here, we derive an analytical framework for the study of control strategies specifically devised for time-varying networks. We consider the removal/immunization of individual nodes according the their activity in the network and develop a block variable mean-field approach that allows the derivation of the equations describing the evolution of the contagion process concurrently to the network dynamic. We derive the critical immunization threshold and assess the effectiveness of the control strategies. Finally, we validate the theoretical picture by simulating numerically the information spreading process and control strategies in both synthetic networks and a large-scale, real-world mobile telephone call dataset.The spreading of infectious diseases, malwares, scientific ideas, memes are just few example of real world phenomena that can be modeled as contagion processes on networks [1][2][3][4]. It has long been acknowledged that network structures and connectivity patterns are a relevant factor in determining the properties of spreading processes. A number of strategies aimed at controlling them have been proposed with the aim of improving on the random removal strategies that originally defined the concept of herd immunity. Those strategies target nodes according the number of connections (degree), the k-core or the betweenness of nodes, just to mention a few examples [5,6]. The efficiency of each strategy is then measured by its effect on the contagion process when a fraction p of nodes is removed from the system. More precisely, the smaller is the fraction p of nodes removed in order to halt the contagion process and the more effective is the strategy. Although most real world networks show a high level of dynamic activity, the large majority of theoretical results concerning the control of contagion processes have been obtained by using a complete timescale separation between the contagion process, τ P , and the change in network's structure, τ G . In these approaches the dynamical process takes place in either static (τ P τ G ) or annealed (τ P τ G ) networks. However, when the two time scales are comparable these convenient approximations might introduce uncontrolled biases in the correct characterization of the dynamical properties of the contagion process . Here we investigate the effect of time-varying connectivity pattern of networks on contagion control strategies by considering the specific class of activity driven network models [23]. In particular, we consider the susceptibleinfected-susceptible (SIS) contagion model [30] and derive analytically its critical immunization threshold in three different control strategies. We also validate the findings obtained in synthetic networks by studying the effect of each strategy...

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Controlling Contagion Processes in Activity Driven Networks

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“…For related evolutions on random graphs, see ref. 25, for example. Some of the results mentioned below have extensions to cases in which Z d is replaced by a general countable set.…”

Section: Models For Interacting Systemsmentioning

confidence: 99%

Stochastic models for large interacting systems and related correlation inequalities

Liggett

2010

Proc. Natl. Acad. Sci. U.S.A.

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A very large and active part of probability theory is concerned with the formulation and analysis of models for the evolution of large systems arising in the sciences, including physics and biology. These models have in their description randomness in the evolution rules, and interactions among various parts of the system. This article describes some of the main models in this area, as well as some of the major results about their behavior that have been obtained during the past 40 years. An important technique in this area, as well as in related parts of physics, is the use of correlation inequalities. These express positive or negative dependence between random quantities related to the model. In some types of models, the underlying dependence is positive, whereas in others it is negative. We give particular attention to these issues, and to applications of these inequalities. Among the applications are central limit theorems that give convergence to a Gaussian distribution.contact process | exclusion process | Glauber dynamics | voter models Models for Interacting SystemsDuring the past half century, mathematical models for the evolution of large interacting systems arising in a number of scientific areas have been proposed and analyzed. Here are some of these areas, together with a sampling of the many papers and books in which such models have been discussed: magnetic systems (1), high energy scattering (2), dynamics of mutation in a structured population (3), tumor growth (4, 5), competition between different strains of viruses (6), mutations of pathogens (7), biopolymers (8), epidemics (9, 10), ecology (11, 12), hydrology (13), cooperative behavior (14-16), spatial distribution of unemployment (17), and the analysis of traffic flow (18)(19)(20).The main objective in the study of these models is to describe their long-time behavior. Usually, the models contain one or more parameters. An important issue is to determine how the long-time behavior depends on these parameters. Often there is a sharp transition in the nature of the behavior at some particular parameter value. This situation is described by saying that a phase transition occurs there.Some of the analysis of these systems has been mathematical, whereas other approaches have been based on simulations and heuristics. Dobrushin (21) and Spitzer (22) are usually credited with initiating the mathematical developments about 40 years ago. The modern theory of models of this type is treated in my two monographs (23,24). Typically, the model is a random process η t with state space f0;1g Z d of binary configurations on the d-dimensional integer lattice Z d . The interpretation of the values 0 and 1 at a site x ∈ Z d depends on the model, and on the area that motivated it. The process satisfies the Markov property, which means that once one knows the state of the system at a given time t, the evolution of the system after that time does not depend on its behavior before time t. It follows that the evolution rules can be described by specifying how the proc...

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“…The threshold of an SIS model in this case is tied to the spectral properties of the adjacency matrix A i j coding for the static network structure. Indeed, for an arbitrary network the epidemic threshold is inversely proportional to the largest eigenvalue λ 1 [47,48,49] of the matrix. In case of uncorrelated networks, λ 1 ∼ √ k max , thus the threshold is associated to the largest degree in the system.…”

Section: Introductionmentioning

confidence: 99%

Control Strategies of Contagion Processes in Time-Varying Networks

Karsai

Perra

2017

Temporal Network Epidemiology

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The vast majority of strategies aimed at controlling contagion processes on networks consider a timescale separation between the evolution of the system and the unfolding of the process. However, in the real world, many networks are highly dynamical and evolve, in time, concurrently to the contagion phenomena. Here, we review the most commonly used immunization strategies on networks. In the first part of the chapter, we focus on controlling strategies in the limit of timescale separation. In the second part instead, we introduce results and methods that relax this approximation. In doing so, we summarize the main findings considering both numerical and analytically approaches in real as well as synthetic time-varying networks.

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Some features of the spread of epidemics and information on a random graph

Cited by 133 publications

References 69 publications

Controlling Contagion Processes in Activity Driven Networks

Controlling Contagion Processes in Activity Driven Networks

Stochastic models for large interacting systems and related correlation inequalities

Control Strategies of Contagion Processes in Time-Varying Networks

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