  Networks facilitate the spread of cascades, allowing a local perturbation to
percolate via interactions between nodes and their neighbors. We investigate
how network structure affects the dynamics of a spreading cascade. By
accounting for the joint degree distribution of a network within a generating
function framework, we can quantify how degree correlations affect both the
onset of global cascades and the propensity of nodes of specific degree class
to trigger large cascades. However, not all degree correlations are equally
important in a spreading process. We introduce a new measure of degree
assortativity that accounts for correlations among nodes relevant to a
spreading cascade. We show that the critical point defining the onset of global
cascades has a monotone relationship to this new assortativity measure. In
addition, we show that the choice of nodes to seed the largest cascades is
strongly affected by degree correlations. Contrary to traditional wisdom, when
degree assortativity is positive, low degree nodes are more likely to generate
largest cascades. Our work suggests that it may be possible to tailor spreading
processes by manipulating the higher-order structure of networks.
