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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...

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.

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

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.

Internet-based computer information systems play critical roles in many aspects of modern society. However, these systems are constantly under cyber attacks that can cause catastrophic consequences. To defend these systems effectively, it is necessary to measure and predict the effectiveness of cyber defense mechanisms. In this article, we investigate how to measure and predict the effectiveness of an important cyber defense mechanism that is known as early-warning. This turns out to be a challenging problem because we must accommodate the dependence among certain four-dimensional time series. In the course of using a dataset to demonstrate the prediction methodology, we discovered a new nonexchangeable and rotationally symmetric dependence structure, which may be of independent value. We propose a new vine copula model to accommodate the newly discovered dependence structure, and show that the new model can predict the effectiveness of early-warning more accurately than the others. We also discus...

In this paper we investigate a cybersecurity model: An attacker can launch multiple attacks against a target with a termination strategy that says that the attacker will stop after observing a number of successful attacks or when the attacker is out of attack resources. However, the attacker's observation of the attack outcomes (i.e., random variables indicating whether the target is compromised or not) has an observation error that is specified by both a false-negative and a false-positive probability. The novelty of the model we study is the accommodation of the dependence between the attack outcomes, because the dependence was assumed away in the literature. In this model, we characterize the monotonicity and bounds of the compromise probability (i.e., the probability that the target is compromised). In addition to extensively showing the impact of dependence on quantities such as compromise probability and attack cost, we give methods for finding the optimal strategy that leads to maximum compromise probability or minimum attack cost. This study highlights that the dependence between random variables cannot be assumed away, because the results will be misleading.

Cyber attacks have become a problem that is threatening the economy, human privacy, and even national security. Before we can adequately address the problem, we need to have a crystal clear understanding about cyber attacks from various perspectives. This is a challenge because the Internet is a large-scale complex system with humans in the loop. In this paper, we investigate a particular perspective of the problem, namely the extreme value phenomenon that is exhibited by cyber attack rates, which are the numbers of attacks against a system of interest per time unit. It is important to explore this perspective because understanding the statistical properties of extreme cyber attack rates will pave the way for cost-effective, if not optimal, allocation of resources in real-life cyber defense operations. Specifically, we propose modeling and predicting extreme cyber attack rates via marked point processes, while using the Value-at-Risk as a natural measure of intense cyber attacks. The point process...

Like how useful weather forecasting is, the capability of forecasting or predicting cyber threats can never be overestimated. Previous investigations show that cyber attack data exhibits interesting phenomena, such as long-range dependence and high nonlinearity, which impose a particular challenge on modeling and predicting cyber attack rates. Deviating from the statistical approach that is utilized in the literature, in this paper we develop a deep learning framework by utilizing the bi-directional recurrent neural networks with long short-term memory, dubbed BRNN-LSTM. Empirical study shows that BRNN-LSTM achieves a significantly higher prediction accuracy when compared with the statistical approach.

In this paper, we propose a novel method for constructing upper bounds of the quasi-stationary distribution of SIS processes. Using this method, we obtain an upper bound that is better than the state-of-the-art upper bound. Moreover, we prove that the fixed point map Φ [7] actually preserves the equilibrium reversed hazard rate order under a certain condition. This allows us to further improve the upper bound. Some numerical results are presented to illustrate the results.

Summary Discrete power-law distributions have significant consequences for understanding many phenomena in practice, and have attracted much attention in recent decades. However, in many practical applications, there exists a natural upper bound for the probability tail. In this paper, we develop maximum likelihood estimates for truncated discrete power-law distributions based on the upper order statistics, and large sample properties are mentioned as well. Monte Carlo simulation is carried out to examine the finite sample performance of the estimates. Applications in real cyber attack data and peak gamma-ray intensity of solar flares are highlighted.

The PMU network has been utilized to monitor the system operation of a smart grid in recent decades. This brings a significant cyber risk for the power grid as the attacker can manipulate the PMU network to introduce the false positive and false negative observation errors. A novel risk model is proposed for the PMU networks by considering the observation errors. It is discovered that ignoring the due observation errors can severely underestimate the PMU risks as shown by the theoretical and simulation studies. The risk mitigation strategies for improving the safety levels of PMUs with the observation errors are studied. It is found that the optimal mitigation strategy can be different when the observation errors are present. Numerical examples are presented for illustrations as well.

It is important to understand to what extent, and in what perspectives, cyber attacks can be predicted. Despite its evident importance, this problem was not investigated until very recently, when we proposed using the innovative methodology of {\em gray-box prediction}. This methodology advocates the use of gray-box models, which accommodate the statistical properties/phenomena exhibited by the data. Specifically, we showed that gray-box models that accommodate the Long-Range Dependence (LRD) phenomenon can predict the attack rate (i.e., the number of attacks per unit time) 1-hour ahead-of-time with an accuracy of 70.2-82.1\%. To the best of our knowledge, this is the first result showing the feasibility of prediction in this domain. We observe that the prediction errors are partly caused by the models' incapability in predicting the large attack rates, which are called {\em extreme values} in statistics. This motivates us to analyze the {\em extreme-value phenomenon}, by using two complementary approaches: the Extreme Value Theory (EVT) and the Time Series Theory (TST). In this paper, we show that EVT can offer long-term predictions (e.g., 24-hour ahead-of-time), while gray-box TST models can predict attack rates 1-hour ahead-of-time with an accuracy of 86.0-87.9\%. We explore connections between the two approaches, and point out future research directions. Although our prediction study is based on specific cyber attack data, our methodology can be equally applied to analyze any cyber attack data of its kind.

Data-driven understanding of cybersecurity posture is an important problem that has not been adequately explored. In this paper, we analyze some real data collected by CAIDA's network telescope during the month of March 2013. We propose to formalize the concept of cybersecurity posture from the perspectives of three kinds of time series: the number of victims i.e., telescope IP addresses that are attacked, the number of attackers that are observed by the telescope, and the number of attacks that are observed by the telescope. Characterizing cybersecurity posture therefore becomes investigating the phenomena and statistical properties exhibited by these time series, and explaining their cybersecurity meanings. For example, we propose the concept of sweep-time, and show that sweep-time should be modeled by stochastic process, rather than random variable. We report that the number of attackers and attacks from a certain country dominates the total number of attackers and attacks that are observed by the telescope. We also show that substantially smaller network telescopes might not be as useful as a large telescope.