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

This paper presents a polynomial dimensional decomposition (PDD) method for global sensitivity analysis of stochastic systems subject to independent random input following arbitrary probability distributions. The method involves Fourier-polynomial expansions of lower-variate component functions of a stochastic response by measure-consistent orthonormal polynomial bases, analytical formulae for calculating the global sensitivity indices in terms of the expansion coefficients, and dimension-reduction integration for estimating the expansion coefficients. Due to identical dimensional structures of PDD and analysis-of-variance decomposition, the proposed method facilitates simple and direct calculation of the global sensitivity indices. Numerical results of the global sensitivity indices computed for smooth systems reveal significantly higher convergence rates of the PDD approximation than those from existing methods, including polynomial chaos expansion, random balance design, state-dependent parameter, improved Sobol’s method, and sampling-based methods. However, for non-smooth functions, the convergence properties of the PDD solution deteriorate to a great extent, warranting further improvements. The computational complexity of the PDD method is polynomial, as opposed to exponential, thereby alleviating the curse of dimensionality to some extent. Mathematical modeling of complex systems often requires sensitivity analysis to determine how an output variable of interest is influenced by individual or subsets of input variables. A traditional local sensitivity analysis entails gradients or derivatives, often invoked in design optimization, describing changes in the model response due to the local variation of input. Depending on the model output, obtaining gradients or derivatives, if they exist, can be simple or difficult. In contrast, a global sensitivity analysis (GSA), increasingly becoming mainstream, characterizes how the global variation of input, due to its uncertainty, impacts the overall uncertain behavior of the model. In other words, GSA constitutes the study of how the output uncertainty from a mathematical model is divvied up, qualitatively or quantitatively, to distinct sources of input variation in the model [1].

Reliability Engineering and System Safety

The statistical tools of Complex Network Analysis are of useful to understand salient properties of complex systems, may these be natural or pertaining human engineered infrastructures. One of these that is receiving growing attention for its societal relevance is that of electricity distribution. In this paper, we present a survey of the most relevant scientific studies investigating the properties of different Power Grids infrastructures using Complex Network Analysis techniques and methodologies. We categorize and explore the most relevant literature works considering general topological properties, physical properties, and differences between the various graph-related indicators and reliability aspects. We also trace the evolution in such field of the approach of study during the years to see the improvement achieved in the analysis.

/pdf/the-power-grid-as-a-complex-network-a-survey-1ypu7fix7j.pdf

The Power Grid as a Complex Network: a Survey

Purpose – Supply chain risk management (SCRM) has recently gained increasing attention in the supply chain context, both from the practitioners' perspective and as a research area. Given the relevance of the topic, the aim of the present paper is to present a focused literature review, investigating the process of knowledge creation, transfer and development from a dynamic perspective within the context of SCRM.Design/methodology/approach – A review of the literature on SCRM was undertaken. The new proposed methodology combines the systematic literature review approach to identify the most relevant articles to be included in the study with the citation network analysis in order to unfold the dynamics of the field under study. The authors define this new methodology as systematic literature network analysis.Findings – The paper demonstrates that there are a number of key themes in the field of SCRM. The contributions that influenced the field were identified and, by analysing the evolution over time of key...

Supply chain risk management: a new methodology for a systematic literature review

The statistical tools of Complex Network Analysis are of great use to understand salient properties of complex systems, may these be natural or pertaining human engineered infrastructures. One of these that is receiving growing attention for its societal relevance is that of electricity distribution. In this paper, we present a survey of the most important scientific studies investigating the properties of different Power Grids infrastructures using Complex Network Analysis techniques and methodologies. We categorize and explore the most relevant literature works considering general topological properties, differences between the various graph-related indicators and reliability aspects.

In this article, we model electric power delivery networks as graphs, and conduct studies of two power transmission grids, i.e., the Nordic and the western states (U.S.) transmission grid. We calculate values of topological (structural) characteristics of the networks and compare their error and attack tolerance (structural vulnerability), i.e., their performance when vertices are removed, with two frequently used theoretical reference networks (the Erdos-Renyi random graph and the Barabasi-Albert scale-free network). Further, we perform a structural vulnerability analysis of a fictitious electric power network with simple structure. In this analysis, different strategies to decrease the vulnerability of the system are evaluated. Finally, we present a discussion on the practical applicability of graph modeling.

Using Graph Models to Analyze the Vulnerability of Electric Power Networks

The infrastructure of a society consists of facilities such as communications, power supplies, transportation, water supplies, and the stock of buildings. In a broad definition of infrastructure, it is also possible to include basic societal functions like education, national defense, and financial and judicial systems. Here, the notion critical infrastructure will refer to the collection of large technical systems, for example electric power grids, which form the basis for most activities in a modern society, and are of great importance for the economic prosperity. Today, critical infrastructure protection is also considered to be a matter of national security.1

A Framework for Vulnerability Assessment of Electric Power Systems

1. Introduction: The Global Risk Arena, Technological Systems, and this Book 2. Reflections on Risks and Technology 3. Risks in the Past and Present 4. The Dangerous Steam Engine 5. Risks and Safety in Building Structures 6. Risks in the Conversion of Energy 7. Chemical and Physical Health Risks - Chemicals, Radiation, and Noise 8. Safety Factors and Exposure Limits 9. How Dangerous Is It To Travel? 10. IT - Risks and Security 11. Vulnerability of Infrastructures 12. Risk Management 13. Risk Analysis 14. Fire and Explosion 15. The Human Factor? 16. The Perceptions of Risks of Technology 17. The Value of a Life 18. Tacit Knowledge and Risks 19. How To Worry in Moderation

Risks in technological systems

Disturbances in the supply of electric power can have serious implications for everyday life as well as for national (homeland) security. A power outage can be initiated by natural disasters, adverse weather, technical failures, human errors, sabotage, terrorism, and acts of war. The vulnerability of a system is described as a sensitivity to threats and hazards, and is measured by P (Q(t) > q), i.e. the probability of at least one disturbance with negative societal consequences Q larger than some critical value q, during a given period of time (0,t]. The aim of the thesis is to present methods for quantitative vulnerability analysis of electric power delivery networks to enable effective strategies for prevention, mitigation, response, and recovery to be developed. Paper I provides a framework for vulnerability assessment of infrastructure systems. The paper discusses concepts and perspectives for developing a methodology for vulnerability analysis, and gives examples related to power systems. Paper II analyzes the vulnerability of power delivery systems by means of statistical analysis of Swedish disturbance data. It is demonstrated that the size of large disturbances follows a power law, and that the occurrence of disturbances can be modeled as a Poisson process. Paper III models electric power delivery systems as graphs. Statistical measures for characterizing the structure of two empirical transmission systems are calculated, and a structural vulnerability analysis is performed, i.e. a study of the connectivity of the graph when vertices and edges are disabled. Paper IV discusses the origin of power laws in complex systems in terms of their structure and the dynamics of disturbance propagation. A branching process is used to model the structure of a power distribution system, and it is shown that the disturbance size in this analytical network model follows a power law. Paper V shows how the interaction between an antagonist and the defender of a power system can be modeled as a game. A numerical example is presented, and it is studied if there exists a dominant defense strategy, and if there is an optimal allocation of resources between protection of components, and recovery.

Structural vulnerability analysis of electric power networks

The global citizen of the twenty-first century faces a wide spectrum of risks, such as from transportation, ionizing radiation, the propagation of contagious diseases, terrorism, natural disasters, and global climate change Some of them, for instance the risk of a motorcycle accident, affect only individuals and have merely occasional direct impacts on society Nevertheless the accumulation of these individual risks is leading to major long-term changes in risk patterns

Åke J. Holmgren

Papers

Using Graph Models to Analyze the Vulnerability of Electric Power Networks

A Framework for Vulnerability Assessment of Electric Power Systems

Risks in technological systems

Structural vulnerability analysis of electric power networks

Introduction: the Global Risk Arena, Technological Systems and This Book