Introduction

We explore network reliability primarily from the viewpoint of how the combinatorial structure of operating or failed states enables us to compute or bound the reliability. Many other viewpoints are possible, and are outlined in [7, 45, 147]. Combinatorial structure is most often reflected in the simplicial complex (hereditary set system) that represents all operating states of a network; most of the previous research has been developed in this vernacular. However, the language and theory of Boolean functions has played an essential role in this development; indeed these two languages are complementary in their utility to understand the combinatorial structure. As we survey exact computation and enumerative bounds for network reliability in the following, the interplay of complexes and Boolean functions is examined.

In classical reliability analysis, failure mechanisms and the causes of failure are relatively well understood. Some failure mechanisms associated with network reliability applications share these characteristics, but many do not. Typically, component failure rates are estimated based on historical data. Hence, time-independent, discrete probability models are often employed in network reliability analysis. In the most common model, network components (nodes and edges, for example) can take on one of two states: operative or failed. The state of a component is a random event that is independent of the states of other components. Similarly, the network itself is in one of two states, operative or failed.