On a framework for analysis and design of cascades on Boolean networks

We consider Boolean networks defined on directed graphs in which the state of every node belongs to the set {0; 1}. We think of a node in state 1 as having 'failed'. The state of every node at the next time instant is a function of the states of those nodes that link to it. Nodes fail according to a set of rules, and once a node fails it stays so forever. We develop a mathematical framework that allows us to find the smallest set of nodes whose failure at time zero causes the eventual failure of all nodes in a desired target set. Our methods are based on modeling network dynamics using Boolean polynomials and exploiting their properties. Rather than propagating the state forward using a nonlinear map, we characterize all possible steady-state configurations as the fixed points of the network's dynamics, and provide a simple algorithm for finding all such 'stable' configurations. We demonstrate the utility of our framework with the help of illustrative examples.