Biblio

Network reliability studies properties of networks subjected to random failures of their components. It has been widely adopted to modeling and analyzing real-world problems across different domains, such as circuit design, genomics, databases, information propagation, network security, and many others. Two practical situations that usually arise from such problems are (i) the correlation between component failures and (ii) the uncertainty in failure probabilities. Previous work captured correlations by modeling component reliability using general Boolean expression of Bernoulli random variables. This paper extends such a model to address the second problem, where we investigate the use of Beta distributions to capture the variance of uncertainty. We call this new formalism the Beta uncertain graph. We study the reliability polynomials of Beta uncertain graphs as multivariate polynomials of Beta random variables and demonstrate the use of the model on two realistic examples. We also observe that the reliability distribution of a monotone Beta uncertain graph can be approximated by a Beta distribution, usually with high accuracy. Numerical results from Monte Carlo simulation of an approximation scheme and from two case studies strongly support this observation.

Attack graphs used in network security analysis are analyzed to determine sequences of exploits that lead to successful acquisition of privileges or data at critical assets. An attack graph edge corresponds to a vulnerability, tacitly assuming a connection exists and tacitly assuming the vulnerability is known to exist. In this paper we explore use of uncertain graphs to extend the paradigm to include lack of certainty in connection and/or existence of a vulnerability. We extend the standard notion of uncertain graph (where the existence of each edge is probabilistically independent) however, as significant correlations on edge existence probabilities exist in practice, owing to common underlying causes for dis-connectivity and/or presence of vulnerabilities. Our extension describes each edge probability as a Boolean expression of independent indicator random variables. This paper (i) shows that this formalism is maximally descriptive in the sense that it can describe any joint probability distribution function of edge existence, (ii) shows that when these Boolean expressions are monotone then we can easily perform uncertainty analysis of edge probabilities, and (iii) uses these results to model a partial attack graph of the Stuxnet worm and a small enterprise network and to answer important security-related questions in a probabilistic manner.