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Dependence among cyber risks has been an essential and challenging component of risk management. The current study characterizes cyber dependence from both qualitative and quantitative perspectives based on L-hop propagation model. From the qualitative side, it is shown that cyber risks always possess positive association based on the proposed risk propagation model. From the quantitative side, an explicit formula for computing the fundamental dependence measure of covariance is provided for an arbitrary network. In particular, we study the impacts of factors—especially external and internal compromise probabilities, propagation depth, and network topologies—on dependence among cyber risks. We conclude by presenting some examples and applications.

Data breaches cause millions of dollars in financial losses each year. The insurance industry has been exploring the ways to transfer such extreme risk. In this work, we investigate data breach cat...

Modeling cyber threats, such as the computer malicious software (malware) propagation dynamics in cyberspace, is an important research problem because models can deepen our understanding of dynamical cyber threats. In this paper, we study the statistical modeling of the macro-level evolution of dynamical cyber attacks. Specifically, we propose a Bayesian structural time series approach for modeling the computer malware propagation dynamics in cyberspace. Our model not only possesses the parsimony property (i.e. using few model parameters) but also can provide the predictive distribution of the dynamics by accommodating uncertainty. Our simulation study shows that the proposed model can fit and predict the computer malware propagation dynamics accurately, without requiring to know the information about the underlying attack-defense interaction mechanism and the underlying network topology. We use the model to study the propagation of two particular kinds of computer malware, namely the Conficker and Code Red worms, and show that our model has very satisfactory fitting and prediction accuracies.

Malicious hacking data breaches cause millions of dollars in financial losses each year, and more companies are seeking cyber insurance coverage. The lack of suitable statistical approaches for sco...

Modeling cyber risks has been an important but challenging task in the domain of cyber security. It is mainly because of the high dimensionality and heavy tails of risk patterns. Those obstacles have hindered the development of statistical modeling of the multivariate cyber risks. In this work, we propose a novel approach for modeling the multivariate cyber risks which relies on the deep learning and extreme value theory. The proposed model not only enjoys the high accurate point predictions via deep learning but also can provide the satisfactory high quantile prediction via extreme value theory. The simulation study shows that the proposed model can model the multivariate cyber risks very well and provide satisfactory prediction performances. The empirical evidence based on real honeypot attack data also shows that the proposed model has very satisfactory prediction performances., 25 pages

Cybersecurity risk has attracted considerable attention in recent decades. However, the modeling of cybersecurity risk is still in its infancy, mainly because of its unique characteristics. In this...

Analyzing cyber incident data sets is an important method for deepening our understanding of the evolution of the threat situation. This is a relatively new research topic, and many studies remain to be done. In this paper, we report a statistical analysis of a breach incident data set corresponding to 12 years (2005–2017) of cyber hacking activities that include malware attacks. We show that, in contrast to the findings reported in the literature, both hacking breach incident inter-arrival times and breach sizes should be modeled by stochastic processes, rather than by distributions because they exhibit autocorrelations. Then, we propose particular stochastic process models to, respectively, fit the inter-arrival times and the breach sizes. We also show that these models can predict the inter-arrival times and the breach sizes. In order to get deeper insights into the evolution of hacking breach incidents, we conduct both qualitative and quantitative trend analyses on the data set. We draw a set of cybersecurity insights, including that the threat of cyber hacks is indeed getting worse in terms of their frequency, but not in terms of the magnitude of their damage.

Computing the system signature is an attractive but challenging problem in system reliability. In this article, we propose a novel algorithm to compute the signature of a system with exchangeable c...

The computation of the signature of a complex system is often challenging, as it may involve a large number of components and a complex architecture. In this study, we propose a novel approach to computing the signature of a system consisting of subsystems with shared components. The proposed approach relies on a new concept called decomposed survival signatures. This new concept can be efficiently used to compute the signature of a modular system with an arbitrary organizing structure and to provide explicit formulas for computation. Applications in cyber systems, transportation systems, and branched oil pipeline systems are highlighted.

Modeling cybersecurity risks is an important, yet challenging, problem. In this paper, we initiate the study of modeling multivariate cybersecurity risks. We develop the first statistical a...

Due to the increasing reliance on networks, defending a cyber system is of vital importance. In this paper, we consider an important mechanism of early warning for defending a cyber system that has become a key component of constructing network defense in practice. We study the reliability of a system under attack from single or multiple sources. In particular, we discuss the effect of an early warning mechanism on the system reliability. We then propose the optimal strategy for defending a cyber system with early warning components in the worst attack scenario. The theoretical results are further validated by simulation evidence.

The online storage of sensitive data enjoys many benefits such as flexibility, cost-savings, scalability, and convenience but it also poses a big concern on the data confidentiality and availability. To ensure the confidentiality and availability of sensitive data over a network system, the data partition technique is often employed. We study the optimal data partition strategy over an arbitrary network under cyber threats. Both the outside attack and the risk propagation (i.e., inside attack) are considered for the data partition. The data breach probability and retrieve probability are discussed under both limited and unlimited risk propagation for various scenarios. It is discovered that the risk propagation can have much more impact on the optimal partition strategy than that of outside attacks, and the unlimited risk propagation leads to more severer cyber risk. The network topology significantly impacts the partition strategy which hints that the network topology should never be overlooked in practice. The corruption due to compromise can lead to different partition strategies. An optimal partition model is developed for determining the optimal strategy and the pareto non-dominated solutions are recommended for practical use.

Phasor measurement units (PMUs) have been integrated into the smart grid for monitoring the operational state of system and improving the reliability. Due to the high cost of PMU installment, the optimal placement strategies have attracted considerable attention in the literature. However, the impacts of cyber threats on the placement have been largely ignored owing to the cyber complexities. This paper initializes the study on the optimal PMU placement in a smart grid under the cyber threats. A probabilistic model is developed for assessing the unobservable risk of the power grid. We characterize the impacts of several cyber factors on the PMU placements including the number of directly attacked PMUs, the dependence among attack outcomes, and risk propagation. We further study the impacts of cyber attacks on the allocation strategies under a bi-level placement model. In particular, a novel ‘greedy’ algorithm for PMU placement is introduced with the presence of cyber risks. Our studies show that the cyber risk can significantly increase the unobservability risk of a power system which in turn requires additional PMU allocations, and the dependence among cyber attacks can lead to more unobservable risk.