# Diffusion dynamics and synchronizability of hierarchical products of networks

## TL;DR

This paper investigates how hierarchical network structures influence diffusion and synchronization, revealing distinct regimes and optimal coupling conditions that depend on the subnetworks' properties.

## Contribution

It introduces a method to analyze eigenvalues of the Laplacian for hierarchical networks, elucidating how primary and secondary subnetworks affect dynamics.

## Key findings

- Diffusion rate depends on secondary network at small coupling.
- Diffusion saturates at large coupling, dominated by primary network.
- Synchronization is optimized at an intermediate coupling value.

## Abstract

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.