  Networked systems have been used to model and investigate the dynamical
behavior of a variety of systems. For these systems, different levels of
complexity can be considered in the modeling procedure. On one hand, this can
offer a more realistic and rich modeling option. On the other hand, it can lead
to intrinsic difficulty in analyzing the system. Here, we present an approach
to investigate the dynamics of Kuramoto oscillators on networks with different
levels of connections: a network of networks. To do so, we utilize a
construction in network theory known as the join of networks, which represents
``intra-area" and ``inter-area" connections. This approach provides a reduced
representation of the original, multi-level system, where both systems have
equivalent dynamics. Then, we can find solutions for the reduced system and
broadcast them to the original network of networks. Moreover, using the same
idea we can investigate the stability of these states, where we can obtain
information on the Jacobian of the multi-level system by analyzing the reduced
one. This approach is general for arbitrary connection schemes between nodes
within the same area. Finally, our work opens the possibility of studying the
dynamics of networked systems using a simpler representation, thus leading to a
better understanding of the dynamical behavior of these systems.
