We consider a dynamic network cascade process developed by Duncan Watts applied to a class of random networks, developed independently by Newman and Miller, which allows a specified amount of clustering (short loops). We adapt existing methods for locally tree-like networks to formulate an appropriate two-type branching process to describe the spread of a cascade started with a single active node and obtain a fixed-point equation to implicitly express the extinction probability of such a cascade. In so doing, we also recover a formula that has appeared in various forms in works by Hackett et al. and Miller which provides a threshold condition for certain extinction of the cascade. We find that clustering impedes cascade propagation for networks of low average degree by reducing the connectivity of the network, but for networks with high average degree, the presence of small cycles makes cascades more likely.

The paper discusses a new method for the propagation of risk levels. This discretization takes place by considering the phase space of the state and its subdivision into boxes. In each time step, the method computes the probability of the state being in one box. We start with a variety of technical and physical problems by showing how discrete modeling under uncertainties is technically meaningful and often even problem inherent. Then we select the technical problem of contaminant spread in water grids, to which we apply the method of risk-level propagation in depth. Extensions of the method such as modeling of contaminant mixing at junctions in the discretized phase space and the applicability of conservation laws arise naturally along these lines and are discussed in the context of the general theory.