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

The structure and dynamics of multilayer networks

Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems

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

https://digital.library.unt.edu/ark:/67531/metadc891681/m2/1/high_res_d/886897.pdf

Survey of sampling-based methods for uncertainty and sensitivity analysis

We explore the network topology of the interbank payments transferred between commercial banks over the Fedwire ® Funds Service. We find that the network has both a low average path length and low connectivity. The network includes a tightly connected core of banks to which most other banks connect. The degree distribution is scale free over a substantial range. We find that the properties of the network changed considerably in the immediate aftermath of the events of September 11, 2001.

https://www.econstor.eu/bitstream/10419/60649/1/511440952.pdf

The topology of interbank payment flows

Targeted social distancing to mitigate pandemic influenza can be designed through simulation of influenza's spread within local community social contact networks. We demonstrate this design for a stylized community representative of a small town in the United States. The critical importance of children and teenagers in transmission of influenza is first identified and targeted. For influenza as infectious as 1957-58 Asian flu (≈50% infected), closing schools and keeping children and teenagers at home reduced the attack rate by >90%. For more infectious strains, or transmission that is less focused on the young, adults and the work environment must also be targeted. Tailored to specific communities across the world, such design would yield local defenses against a highly virulent strain in the absence of vaccine and antiviral drugs.

https://stacks.cdc.gov/view/cdc/16257/cdc_16257_DS1.pdf

Targeted social distancing design for pandemic influenza.

Modern society's physical health depends vitally upon a number of real, interdependent, critical infrastructure networks that deliver power, petroleum, natural gas,water, and communications. Its economic health depends on a number of other infrastructure networks, some virtual and some real, that link residences, industries, commercial sectors, and transportation sectors. The continued prosperity and national security of the US depends on our ability to understand the vulnerabilities of and analyze the performance of both the individual infrastructures and the entire interconnected system of infrastructures. Only then can we respond to potential disruptions in a timely and effective manner. Collaborative efforts among Sandia, other government agencies, private industry, and academia have resulted in realistic models for many of the individual component infrastructures. In this paper, we propose an innovative modeling and analysis framework to study the entire system of physical and economic infrastructures. That framework uses the existing individual models together with system dynamics, functional models, and nonlinear optimization algorithms. We describe this framework and demonstrate its potential use to analyze, and propose a response for, a hypothetical disruption.

Toward modeling and simulation of critical national infrastructure interdependencies.

Modern society's physical health depends vitally upon a number of real, interdependent, critical infrastructure networks that deliver power, petroleum, natural gas, water, and communications. Its economic health depends on a number of other infrastructure networks, some virtual and some real, that link residences, industries, commercial sectors, and transportation sectors. The continued prosperity and national security of the US depends on our ability to understand the vulnerabilities of and analyze the performance of both the individual infrastructures and the entire interconnected system of infrastructures. Only then can we respond to potential disruptions in a timely and effective manner. Collaborative efforts among Sandia, other government agencies, private industry, and academia have resulted in realistic models for many of the individual component infrastructures. In this paper, we propose an innovative modeling and analysis framework to study the entire system of physical and economic infrastructur...

Toward modeling and simulation of critical national infrastructure interdependencies

Infrastructures are a complex set of interconnected, interdependent, adaptive systems on which the nation, manufacturing systems and individuals depend. Understanding the potential consequences of infrastructure interdependencies, as the infrastructures evolve and the regulations governing their operation change, is at the heart of our infrastructure interdependencies research program. This program includes development of analysis methods and simulation tools for evaluating the potential effects of disruptions and for prioritising risks. Fundamental infrastructures simulated using these tools include; transportation, telecommunications, electric power, banking and finance, water, agriculture, emergency services, fossil fuels, and government. The complexity of the infrastructures and their interactions prevent us from knowing a priori how these interactions will influence individuals, states or the nation; the consequences of policy decisions; vulnerabilities due to interdependencies, natural disasters, malevolent threats and aging; or vulnerabilities that need to be eliminated in order to assure individual, state or national economic security. The goal of the interdependency analyses is to identify significant risks to critical systems, arising from interconnection, and effective mechanisms for mitigating those risks. This article presents the infrastructure interdependency assessment process, modelling tools developed to support that process and examples of assessment results.

Walter E. Beyeler

Papers

The topology of interbank payment flows

Targeted social distancing design for pandemic influenza.

Toward modeling and simulation of critical national infrastructure interdependencies.

Toward modeling and simulation of critical national infrastructure interdependencies

Assessing infrastructure interdependencies: the challenge of risk analysis for complex adaptive systems