We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Complex networks arise in a wide range of biological and sociotechnical systems. Epidemic spreading is central to our understanding of dynamical processes in complex networks, and is of interest to physicists, mathematicians, epidemiologists, and computer and social scientists. This review presents the main results and paradigmatic models in infectious disease modeling and generalized social contagion processes.

Epidemic processes in complex networks

The structure and dynamics of multilayer networks

Implementation science has progressed towards increased use of theoretical approaches to provide better understanding and explanation of how and why implementation succeeds or fails. The aim of this article is to propose a taxonomy that distinguishes between different categories of theories, models and frameworks in implementation science, to facilitate appropriate selection and application of relevant approaches in implementation research and practice and to foster cross-disciplinary dialogue among implementation researchers. Theoretical approaches used in implementation science have three overarching aims: describing and/or guiding the process of translating research into practice (process models); understanding and/or explaining what influences implementation outcomes (determinant frameworks, classic theories, implementation theories); and evaluating implementation (evaluation frameworks). This article proposes five categories of theoretical approaches to achieve three overarching aims. These categories are not always recognized as separate types of approaches in the literature. While there is overlap between some of the theories, models and frameworks, awareness of the differences is important to facilitate the selection of relevant approaches. Most determinant frameworks provide limited “how-to” support for carrying out implementation endeavours since the determinants usually are too generic to provide sufficient detail for guiding an implementation process. And while the relevance of addressing barriers and enablers to translating research into practice is mentioned in many process models, these models do not identify or systematically structure specific determinants associated with implementation success. Furthermore, process models recognize a temporal sequence of implementation endeavours, whereas determinant frameworks do not explicitly take a process perspective of implementation.

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Making sense of implementation theories, models and frameworks

We present the design, implementation, and evaluation of B4, a private WAN connecting Google's data centers across the planet. B4 has a number of unique characteristics: i) massive bandwidth requirements deployed to a modest number of sites, ii) elastic traffic demand that seeks to maximize average bandwidth, and iii) full control over the edge servers and network, which enables rate limiting and demand measurement at the edge.These characteristics led to a Software Defined Networking architecture using OpenFlow to control relatively simple switches built from merchant silicon. B4's centralized traffic engineering service drives links to near 100% utilization, while splitting application flows among multiple paths to balance capacity against application priority/demands. We describe experience with three years of B4 production deployment, lessons learned, and areas for future work.

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B4: experience with a globally-deployed software defined wan

Mean-field deterministic epidemic models have been successful in uncovering several important dynamic properties of stochastic epidemic spreading processes over complex networks. In particular, individual-based epidemic models isolate the impact of the network topology on spreading dynamics. In this paper, the existing models are generalized to develop a class of models that includes the spreading process in multilayer complex networks. We provide a detailed description of the stochastic process at the agent level where the agents interact through different layers, each represented by a graph. The set of differential equations that describes the time evolution of the state occupancy probabilities has an exponentially growing state-space size in terms of the number of the agents. Based on a mean-field type approximation, we developed a set of nonlinear differential equations that has linearly growing state-space size. We find that the latter system, referred to as the generalized epidemic mean-field (GEMF) model, has a simple structure characterized by the elements of the adjacency matrices of the network layers and the Laplacian matrices of the transition rate graphs. Finally, we present several examples of epidemic models, including spreading of virus and information in computer networks and spreading of multiple pathogens in a host population .

Generalized epidemic mean-field model for spreading processes over multilayer complex networks

This study extends the Susceptible-Infected-Susceptible (SIS) epidemic model for single-virus propagation over an arbitrary graph to an Susceptible-Infected by virus 1-Susceptible-Infected by virus 2-Susceptible (${\mathrm{SI}}_{1}{\mathrm{SI}}_{2}\mathrm{S}$) epidemic model of two exclusive, competitive viruses over a two-layer network with generic structure, where network layers represent the distinct transmission routes of the viruses. We find analytical expressions determining extinction, coexistence, and absolute dominance of the viruses after we introduce the concepts of survival threshold and absolute-dominance threshold. The main outcome of our analysis is the discovery and proof of a region for long-term coexistence of competitive viruses in nontrivial multilayer networks. We show coexistence is impossible if network layers are identical yet possible if network layers are distinct. Not only do we rigorously prove a region of coexistence, but we can quantitate it via interrelation of central nodes across the network layers. Little to no overlapping of the layers' central nodes is the key determinant of coexistence. For example, we show both analytically and numerically that positive correlation of network layers makes it difficult for a virus to survive, while in a network with negatively correlated layers, survival is easier, but total removal of the other virus is more difficult.

Competitive epidemic spreading over arbitrary multilayer networks

Electric power-systems are one of the most important critical infrastructures. In recent years, they have been exposed to extreme stress due to the increasing demand, the introduction of distributed renewable energy sources, and the development of extensive interconnections. We investigate the phenomenon of abrupt breakdown of an electric power-system under two scenarios: load growth (mimicking the ever-increasing customer demand) and power fluctuations (mimicking the effects of renewable sources). Our results on real, realistic and synthetic networks indicate that increasing the system size causes breakdowns to become more abrupt; in fact, mapping the system to a solvable statistical-physics model indicates the occurrence of a first order transition in the large size limit. Such an enhancement for the systemic risk failures (black-outs) with increasing network size is an effect that should be considered in the current projects aiming to integrate national power-grids into "super-grids".

/pdf/abruptness-of-cascade-failures-in-power-grids-fa1f1t2vwk.pdf

Abruptness of Cascade Failures in Power Grids

The spontaneous behavioral responses of individuals to the progress of an epidemic are recognized to have a significant impact on how the infection spreads. One observation is that, even if the infection strength is larger than the classical epidemic threshold, the initially growing infection can diminish as the result of preventive behavioral patterns adopted by the individuals. In order to investigate such dynamics of the epidemic spreading, we use a simple behavioral model coupled with the individual-based SIS epidemic model where susceptible individuals adopt a preventive behavior when sensing infection. We show that, given any infection strength and contact topology, there exists a region in the behavior-related parameter space such that infection cannot survive in long run and is completely contained. Several simulation results, including a spreading scenario in a realistic contact network from a rural district in the State of Kansas, are presented to support our analytical arguments.

/pdf/on-the-existence-of-a-threshold-for-preventive-behavioral-3uha7450ca.pdf

Caterina Scoglio

Papers

Generalized epidemic mean-field model for spreading processes over multilayer complex networks

Competitive epidemic spreading over arbitrary multilayer networks

Abruptness of Cascade Failures in Power Grids

On the existence of a threshold for preventive behavioral responses to suppress epidemic spreading.

An individual-based approach to SIR epidemics in contact networks