Modeling Uncertain and Dynamic Interdependencies of Infrastructure Systems Using Stochastic Block Models

We use the Stein‒Chen method to obtain compound Poisson approximations for the distribution of the number of subgraphs in a generalised stochastic block model which are isomorphic to some fixed graph. This model generalises the classical stochastic block model to allow for the possibility of multiple edges between vertices. We treat the case that the fixed graph is a simple graph and that it has multiple edges. The former results apply when the fixed graph is a member of the class of strictly balanced graphs and the latter results apply to a suitable generalisation of this class to graphs with multiple edges. We also consider a further generalisation of the model to pseudo-graphs, which may include self-loops as well as multiple edges, and establish a parameter regime in the multiple edge stochastic block model in which Poisson approximations are valid. The results are applied to obtain Poisson and compound Poisson approximations (in different regimes) for subgraph counts in the Poisson stochastic block model and degree corrected stochastic block model of Karrer and Newman (2011).

Critical infrastructures (CI) are designated sectors that if incapacitated or destroyed by natural disasters would have a serious impact on national security and economic and social welfare. Due to the interdependency of critical infrastructures failure of one infrastructure during a natural disaster such as earthquake or flood may cause failure of another and so on through a cascade or escalating effect. Quantification of these types of interdependencies between critical infrastructures is essential for effective response and management of resources for rescue, recovery, and restoration during times of crises. This paper proposes a new mathematical framework based on an asymmetric relation matrix constructed in a bottom-up approach for modeling and analyzing interdependencies of critical infrastructures. Asymmetric dependency matrices can be constructed using the asymmetric incidence coefficient based on node-level relationships defined between nodes for measuring the strength of interdependency between node and node, node and network, and networks and networks. These asymmetric matrices are further analyzed for ranking infrastructures in terms of their relative importance and for identifying nodes and infrastructure networks that play a critical role in chain effects among infrastructures involved in geo-disaster events such as flooding. Examples of interdependency analysis for the identification of vulnerabilities among fifteen national defense-related infrastructure sectors by the Australian government and a simulated example using the newly developed GIS-based network simulator GeoPN are used to validate and demonstrate the implementation and effectiveness of interdependency analysis methods in analyzing infrastructure interdependency during a flooding event.

There are a wide range of different frameworks and methodologies for analysing Critical Infrastructure (CI) resilience, covering organisational, technological and social resilience. However, there is a lack of a clear methodology combining these three resilience domains into one framework. The final goal of the ongoing EU-project IMPROVER, ‘Improved risk evaluation and implementation of resilience concepts to Critical Infrastructure,’ is to develop one single improved and easy-to-use critical infrastructure resilience analysis tool which will be applicable within all resilience domains and to all types of critical infrastructure. This article presents part of this work, in which IMPROVER comprehensively evaluated, by demonstration and comparison, a selection of existing resilience methodologies in order to integrate their best features into the new methodology. The selected methodologies were The Benchmark Resilience Tool (BRT) (Lee et al., 2013), Guidelines for Critical Infrastructures Resilience Evaluation (CIRE) (Bertocchi et al., 2016) and the Critical Infrastructure Resilience Index (CIRI). The latter was developed within the consortium (Pursiainen et al., 2017). The results show that it is hard to evaluate and compare the different methodologies considering that the methodologies are not aiming to achieve the same thing. However, this evaluation shows that all the methodologies have pros and cons, and that the IMPROVER project should aim at combining, in so far as is possible and commensurable, the identified pros while avoiding the identified cons into a Critical Infrastructure resilience assessment framework compatible with the current guidelines for risk assessment in the Member States.

The stochastic block model is widely used for detecting community structures in network data. However, the research interest in much of the literature focuses on the study of one sample of stochastic block models. Detecting the difference between the two community structures is a less studied issue for stochastic block models. In this article, we propose a novel test statistic based on the largest singular value of a residual matrix obtained by subtracting the geometric mean of two estimated block mean effects from the sum of two observed adjacency matrices. We prove that the null distribution of the proposed test statistic converges in distribution to the Tracy-Widom distribution with index 1, and we show the difference between the two samples for stochastic block models can be tested via the proposed method. We show that the proposed test is asymptotically powerful against alternative models. Further, we extend the proposed method to the degree-corrected stochastic block model. Both simulation studies and real-world data examples indicate that the proposed method works well.

The proliferation of models for networks raises challenging problems of model selection: the data are sparse and globally dependent, and models are typically high-dimensional and have large numbers of latent variables. Together, these issues mean that the usual model-selection criteria do not work properly for networks. We illustrate these challenges, and show one way to resolve them, by considering the key network-analysis problem of dividing a graph into communities or blocks of nodes with homogeneous patterns of links to the rest of the network. The standard tool for undertaking this is the stochastic block model, under which the probability of a link between two nodes is a function solely of the blocks to which they belong. This imposes a homogeneous degree distribution within each block; this can be unrealistic, so degree-corrected block models add a parameter for each node, modulating its overall degree. The choice between ordinary and degree-corrected block models matters because they make very different inferences about communities. We present the first principled and tractable approach to model selection between standard and degree-corrected block models, based on new large-graph asymptotics for the distribution of log-likelihood ratios under the stochastic block model, finding substantial departures from classical results for sparse graphs. We also develop linear-time approximations for log-likelihoods under both the stochastic block model and the degree-corrected model, using belief propagation. Applications to simulated and real networks show excellent agreement with our approximations. Our results thus both solve the practical problem of deciding on degree correction and point to a general approach to model selection in network analysis.

Finding communities in complex networks is a challenging task and one promising approach is the Stochastic Block Model (SBM). But the influences from various fields led to a diversity of variants and inference methods. Therefore, a comparison of the existing techniques and an independent analysis of their capabilities and weaknesses is needed. As a first step, we review the development of different SBM variants such as the degree-corrected SBM of Karrer and Newman or Peixoto’s hierarchical SBM. Beside stating all these variants in a uniform notation, we show the reasons for their development. Knowing the variants, we discuss a variety of approaches to infer the optimal partition like the Metropolis-Hastings algorithm. We perform our analysis based on our extension of the Girvan-Newman test and the Lancichinetti-Fortunato-Radicchi benchmark as well as a selection of some real world networks. Using these results, we give some guidance to the challenging task of selecting an inference method and SBM variant. In addition, we give a simple heuristic to determine the number of steps for the Metropolis-Hastings algorithms that lack a usual stop criterion. With our comparison, we hope to guide researches in the field of SBM and highlight the problem of existing techniques to focus future research. Finally, by making our code freely available, we want to promote a faster development, integration and exchange of new ideas.

A Decision Support System for the Resilience of Critical Transport Infrastructure to Extreme Weather Events

Many statistical inference problems correspond to recovering the values of a set of hidden variables from sparse observations on them. For instance, in a planted constraint satisfaction problem such as planted 3-SAT, the clauses are sparse observations from which the hidden assignment is to be recovered. In the problem of community detection in a stochastic block model, the community labels are hidden variables that are to be recovered from the edges of the graph. Inspired by ideas from statistical physics, the presence of a stable fixed point for belief propogation has been widely conjectured to characterize the computational tractability of these problems. For community detection in stochastic block models, many of these predictions have been rigorously confirmed. In this work, we consider a general model of statistical inference problems that includes both community detection in stochastic block models, and all planted constraint satisfaction problems as special cases. We carry out the cavity method calculations from statistical physics to compute the regime of parameters where detection and recovery should be algorithmically tractable. At precisely the predicted tractable regime, we give: (i) a general polynomial-time algorithm for the problem of detection: distinguishing an input with a planted signal from one without; (ii) a general polynomial-time algorithm for the problem of recovery: outputting a vector that correlates with the hidden assignment significantly better than a random guess would. Analogous to the spectral algorithm for community detection [1], [2], the detection and recovery algorithms are based on the spectra of a matrix that arises as the derivatives of the belief propagation update rule. To devise a spectral algorithm in our general model, we obtain bounds on the spectral norms of certain families of random matrices with correlated and matrix valued entries. We then demonstrate how eigenvectors of various powers of the matrix can be used to partially recover the hidden variables.

When we examine the impacts of space weather on society, do we really understand the risks? Can past experiences reliably predict what will happen in the future? As the complexity of technology increases, there is the potential for it to become more fragile, allowing for a single point of failure to bring down the entire system. Take the Global Positioning System (GPS) as an example. GPS positioning, navigation, and timing have become an integral part of daily life, supporting transportation and communications systems vital to the aviation, merchant marine, cargo, cellular phone, surveying, and oil exploration industries. Everyday activities such as banking, mobile phone operations, and even the control of power grids are facilitated by the accurate timing provided by GPS. Understanding the risks of space weather to GPS and the many economic sectors reliant upon it, as well as how to build resilience, was the focus of a policy workshop organized by the American Meteorological Society (AMS) and held on 13–14 October 2010 in Washington, D. C. The workshop brought together a select group of policy makers, space weather scientists, and GPS experts and users. Arif Alikhan, assistant secretary for policy development in the U.S. Department of Homeland Security, kicked off the meeting by stressing that authorities should see GPS and resilience through an all-hazards lens rather than plan for each emergency scenario on a case-by-case basis. Whether a GPS signal error or disruption is caused by a solar storm, an intentional jammer, or some other source, government and industries need to plan to respond to the incident and restore capability as quickly as possible, although, Alikhan remarked, “The number of potential scenarios is infinite, and we have neither the time nor resources to plan against them.” Alikhan suggested that risk managers conceptualize the scope of critical assets in need of protecting through a resilience and risk analysis framework. He recommended that decision makers take an inventory of all functions provided for by their institutions“ use of GPS. Next, through risk-informed prioritization, planners can ascertain whether a backup is really needed for any services provided by GPS. “In this fiscal environment,” Alikhan said, “we will need to shrink the haystack, or size of the problem, in order to be effective.” Because the interdependencies of critical infrastructures are not fully appreciated by the public and private sectors, he stressed that a mix of governmental, commercial, and international stakeholders should develop plans and procedures to make systems more resilient. The remainder of the workshop was organized around discussions in three sessions: (1) identifying the vulnerability of GPS technologies and services to space weather, (2) identifying how this vulnerability affects users who depend on GPS, and (3) developing opportunities and policies for building resilience. In the first session, participants agreed that the vulnerability of GPS needs to be defined for different segments (e.g., satellites, signals, users). While some understanding of user needs exists, the definition of user requirements needs more clarification because space weather events will affect some technologies more than others. Also, a threat assessment would be valuable to best understand how severe the impacts might be and what mitigation options are available. With the recent news that Google™ is building self-driving robotic cars that depend on GPS, trying to capture possible future vulnerabilities is extremely challenging. During a debate on how much to stress the impact of a solar flare as massive as the 1859 Carrington event, many warned that while the public should be made aware of such a potential occurrence, it should not govern planning, since such a doomsday scenario is uncertain and expected to happen only very infrequently. When discussing in the second session how users of GPS are affected, participants discovered that communicating with GPS stakeholders and users is a challenge because many do not understand the impacts of adverse space weather. In particular, many of the service providers and end users of GPS were not in business during the previous solar maximum and therefore are not familiar with space weather hazards. To help remedy this, outreach approaches were discussed, and some participants argued that the space weather community should anticipate the needs of the users. Others instead urged the space weather community to deal only with the service providers—not end users—and ask them rather than tell them what they need. Despite this debate, there was broad agreement that space weather researchers should develop better measurements and predictions of the ionosphere. In addition, participants agreed that scintillation specifications and predictions would be useful to many users of GPS. Improved predictions would be possible by installing more dual-frequency stations and scintillation monitors, they noted. The workshop concluded with a discussion of policy options for building resilience. Participants debated what measures could be taken, such as developing new policies, investing in research and development, changing industry practices, and public and private sector collaboration. Scott Pace, director of George Washington University“s Space Policy Institute, stressed in his presentation that stakeholders and disaster planners should focus on three aspects: prediction, detection, and mitigation. Better space physics/heliophysics science is needed, and it was recommended that this be a top priority for the Solar and Space Physics decadal survey, which sets future nationwide research targets and is currently under review by the U.S. National Research Council. Findings and recommendations from the workshop will be published in a report by AMS. The results will be widely distributed to various stakeholders, including policy makers, GPS service providers, federal agencies, research communities, and international organizations. Genene Fisher is a senior policy fellow in the American Meteorological Society's Policy Program. E-mail: [email protected]

The presented ongoing research project focuses on the challenges concerning the protection of critical infrastructures. Since critical infrastructures become more and more cross-linked a single breakdown can cause serious cascading effects on other infrastructures. Interdependencies of critical infrastructures are boundaryspanning and therefore public authorities, operating companies and other stakeholders (e.g. German Federal Agency for technical relief, Red Cross) as well as stakeholders of neighbouring states must have access to a solution which enables the urgently needed cooperation of all parties involved. The aim of the project is to develop an online platform to facilitate knowledge exchange among stakeholders, an online tool to analyse the interdependencies and to design a sustainable and effective knowledge management. This platform will allow registered members to find and make use of the information about the stakeholders and their competencies concerning the protection of critical infrastructures.

This research addresses the limitations of traditional network models in capturing the complexity and dynamics of real-world social networks. Motivated by the need for a more comprehensive and flexible framework, the study introduces the Hybrid Modern Network Model (HMNM). The HMNM integrates foundational models like the Stochastic Block Model (SBM) and Preferential Attachment with advanced machine learning techniques, including Graph Neural Networks (GNNs), Reinforcement Learning (RL), Hierarchical Random Graphs (HRGs), Generative Adversarial Networks (GANs), and Variational Autoencoders (VAEs). The methods employed involve constructing initial network structures using SBM, simulating network growth through preferential Attachment, learning node embeddings with GNNs, dynamically optimizing network properties using RL, capturing hierarchical community structures with HRGs, controlling degree distributions using GANs, and uncovering latent patterns with VAEs. The empirical illustration of HMNM highlights its effectiveness in providing a more realistic, scalable, and comprehensive analysis of social networks compared to traditional models. Integrating diverse methodologies allows for accurately modeling of network structures, dynamic processes, and latent patterns. In conclusion, the HMNM offers significant advancements in network modeling, providing a robust and flexible framework for analyzing social networks. This model overcomes the limitations of traditional models and delivers deeper insights into the complexities and dynamics of social structures. Future research will optimize the HMNM and explore its applications across various domains. The R programming code used for the network simulations and visualizations is conceptual and demonstrates the HMNM framework. The results and metrics are illustrative placeholders, emphasizing the methodology rather than empirical validation.

A probabilistic graphical model approach to stochastic multiscale partial differential equations

This paper presents first an ontology of risk for interdependent and heterogeneous Critical Infrastructures (CIs). It defines a data structure called Service Quality Descriptor (SQD) specifying the degradation of QoS over time, which should be shared between interconnected CI. SQD are shared in real time and contain a precise prediction of the future quality of service, so that this sharing can be useful to avoid failures, identify interdependencies, or accelerate and coordinate power failure recoveries and service restoration. Finally, the paper proposes a simplified method to determine the SQD of a delivery service as a linear function of the SQD of supporting services and parameters depending on the service delivery infrastructure. The approach has been defined within the European FP-7 project MICIE, in line with the EU initiative to establish a Critical Infrastructure Warning Information Network (CIWIN).KeywordsRisk analysisSecurity Quality DescriptorQuality of ServiceRisk ontologyInterdependency riskCritical infrastructure ProtectionRisk Level

Nowadays the simulation of scenarios composed by interdependent critical infrastructures is a challenge. One of the main challenge is how to manage into a single framework heterogeneous and interdependent infrastructures characterized by very different time scales, modeling paradigms and functional behavior. In this paper we propose a two-stage approach, where several sector specific simulators are integrated within a common framework by means of multi-domain simulation platforms. We describe the rational behind our approach and its advantages with reference to two complementary implementations: CISIA and Federated ABMS.

The failure of urban Electrical Distribution Systems (EDSs) can lead to direct or indirect consequences for their interdependent critical infrastructures, for example, the hospitals, the fire department, etc., negatively affecting the well-being of citizens in the city. To improve the resilience of the coupled power network, the Operational Interdependence Simulator (OIS) is proposed to help prioritize the queue of failure recovery and Minimum Spanning Tree and Shortest Path Tree topology reconfiguration to minimize critical load downtime. Additionally, AI agents are introduced to assist decision-making processes. By utilizing machine learning and training in thousands of offline scenarios, the system can be used for real-time calculation of the optimal recovery paths when given the number and location of electrical faults. The IEEE 70-node distribution system case validated the high prediction accuracy and post-disaster topology reconfiguration while considering the interdependencies.