  Collections of bistable elements called hysterons provide a powerful model to
capture the sequential response and memory effects of frustrated, multistable
media in the athermal, quasistatic limit. While a century of work has
elucidated, in great detail, the properties of ensembles of non-interacting
hysterons - the so-called Preisach model - comparatively little is known about
the behavior and properties of interacting hysterons. Here we discuss a general
framework for interacting hysterons, focussing on the relation between the
design parameters that specify the hysterons and their interactions, and the
resulting transition graphs (t-graphs). We show how the structure of such
t-graphs can be thought of as being composed of a scaffold that is dressed by
(avalanche) transitions selected from finite binary trees. Moreover, we provide
a systematic framework to straightforwardly determine the design inequalities
for a given t-graph, and discuss an effective method to determine if a certain
t-graph topology is realizable by a set of interacting hysterons. Altogether,
our work builds on the Preisach model by generalizing the hysteron-dependent
switching fields to the state-dependent switching fields. As a result, while in
the Preisach model, the main loop identifies the critical hysterons that
trigger a transition, in the case of interacting hysterons, the scaffold
contains this critical information and assumes a central role in determining
permissible transitions. In addition, our work suggests strategies to deal with
the combinatorial explosion of the number and variety of t-graphs for
interacting hysterons. This approach paves the way for a deeper theoretical
understanding of the properties and statistics of t-graphs, and opens a route
to materializing complex pathways, memory effects and embodied computations in
(meta)materials based on interacting hysterons.
