  The dynamics of cascading activation, such as rapid changes in public opinion
and the outbreak of disease epidemics, have a crucial dependence on the
connectivity patterns among the agents. We study cascading dynamics in modular,
degree-heterogeneous networks, and consider the impact of intra-module seeding
strategy on inter-module spread. Specifically, we establish that although
activating the highest-degree nodes is more effective than random selection at
growing a cascade locally, there is a critical level of inter-module
connectivity required for a cascade to cross from one module to another,
irrespective of the seeding strategy. We present an analytical proof of this
statement for the case that each module has the same degree distribution and
all module pairs have the same inter-module connectivity, while our simulation
results suggest its validity for more general situations, including a ring of
modules. Interestingly, we find that on a network comprised of two modules,
this critical level is primarily determined by the degree distribution of the
\emph{alter} module, as opposed to the seed module. Our analytical approach
extends a method developed by Gleeson, but is able to capture different seeding
strategies using only one dynamical variable per module, namely the conditional
exposure probability. Our work shows that the possibility of a global cascade
depends sensitively on inter-module connectivity, and less on the intra-module
seeding strategy. This suggests, for example, that slight changes to
inter-module connectivity can be a feasible intervention strategy to promote or
inhibit global cascades.
