ABSTRACT
Multi-state coherent systems, such as networks, share properties with mononomial ideals which are a cornerstone of modern computational algebra. By exploiting this connection it is possible to obtain tight upper and lower bounds on network reliability which can be shown to dominate traditional Bonferroni bounds, at every truncation level. The key object in the algebra is the multigraded Hilbert series which can be constructed from multigraded Betti numbers. For networks, many of the metrics for reliability and robustness can be expressed via the Hilbert series. One advantage of the purely algebraic methods is that they are distribution-free. On the other hand they can be combined with distributional assumptions such as those from Bayesian graphical models.
