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.