  Maxmin-$\omega$ is a new threshold model, where each node in a network waits
for the arrival of states from a fraction $\omega$ of neighborhood nodes before
processing its own state, and subsequently transmitting it to downstream nodes.
Repeating this sequence of events leads to periodic behavior. We show that
maxmin-$\omega$ reduces to a smaller, simpler, system represented by tropical
mathematics, which forges a useful link between the nodal state update times
and circuits in the network. Thus, the behavior of the system can be analyzed
directly from the smaller network structure and is computationally faster. We
further show that these reduced networks: (i) are not unique; they are
dependent on the initialisation time, (ii) tend towards periodic orbits of
networks -- "attractor networks." In light of these features, we vary the
initial condition and $\omega$ to obtain statistics on types of attractor
networks. The results suggest the case $\omega =0.5$ to give the most stable
system. We propose that the most prominent attractor networks of nodes and
edges may be seen as a 'backbone' of the original network. We subsequently
provide an algorithm to construct attractor networks, the main result being
that they must necessarily contain circuits that are deemed critical and that
can be identified without running the maxmin-$\omega$ system. Finally, we apply
this work to the C. elegans neuron network, giving insight into its function
and similar networks. This novel area of application for tropical mathematics
adds to its theory, and provides a new way to identify important edges and
circuits of a network.
