Degree-targeted cascades in modular, degree-heterogeneous networks

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.