  While many materials exhibit a complex, hysteretic response to external
driving, there has been a surge of interest in how the complex dynamics of
internal materials states can be understood and designed to process and store
information. We consider a system of connected rubber balloons that can be
described by a Preisach model of non-interacting hysterons under pressure
control, but for which the hysterons become coupled under volume control. We
study this system by exploring the possible transition graphs, as well as by
introducing a configuration space approach which tracks the volumes of each
balloon. Changes in the transition graphs turn out to be related to changes in
the topology of the configuration space of the balloons, providing a
particularly geometric view of how transition graphs can be designed, as well
as additional information on the existence of hidden metastable states. This
class of systems is more general than just balloons.
