  We develop a new ensemble of modular random graphs in which degree-degree
correlations can be different in each module and the inter-module connections
are defined by the joint degree-degree distribution of nodes for each pair of
modules. We present an analytical approach that allows one to analyze several
types of binary dynamics operating on such networks, and we illustrate our
approach using bond percolation, site percolation, and the Watts threshold
model. The new network ensemble generalizes existing models (e.g., the
well-known configuration model and LFR networks) by allowing a heterogeneous
distribution of degree-degree correlations across modules, which is important
for the consideration of nonidentical interacting networks.
