  Spreading broadly refers to the notion of an entity propagating throughout a
networked system via its interacting components. Evidence of its ubiquity and
severity can be seen in a range of phenomena, from disease epidemics to
financial systemic risk. In order to understand the dynamics of these critical
phenomena, computational models map the probability of propagation as a
function of direct exposure, typically in the form of pairwise interactions
between components. By doing so, the important role of indirect interactions
remains unexplored. In response, we develop a simple model that accounts for
the effect of both direct and subsequent exposure, which we deploy in the novel
context of failure propagation within a real-world engineering project. We show
that subsequent exposure has a significant effect in key aspects, including
the: (a) final spreading event size, (b) propagation rate, and (c) spreading
event structure. In addition, we demonstrate the existence of hidden
influentials in large-scale spreading events, and evaluate the role of direct
and subsequent exposure in their emergence. Given the evidence of the
importance of subsequent exposure, our findings offer new insight on particular
aspects that need to be included when modelling network dynamics in general,
and spreading processes specifically.
