  The hierarchical product of networks represents a natural tool for building
large networks out of two smaller subnetworks: a primary subnetwork and a
secondary subnetwork. Here we study the dynamics of diffusion and
synchronization processes on hierarchical products. We apply techniques
previously used for approximating the eigenvalues of the adjacency matrix to
the Laplacian matrix, allowing us to quantify the effects that the primary and
secondary subnetworks have on diffusion and synchronization in terms of a
coupling parameter that weighs the secondary subnetwork relative to the primary
subnetwork. Diffusion processes are separated into two regimes: for small
coupling the diffusion rate is determined by the structure of the secondary
network, scaling with the coupling parameter, while for large coupling it is
determined by the primary network and saturates. Synchronization, on the other
hand, is separated into three regimes: for both small and large coupling
hierarchical products have poorly synchronization properties, but is optimized
at an intermediate value. Moreover, the critical coupling value that optimizes
synchronization is shaped by the relative connectivities of the primary and
secondary subnetworks, compensating for significant differences between the two
subnetworks.
