Many materials that are out of equilibrium can learn one or more inputs that are repeatedly applied. Yet, a common framework for understanding such memories is lacking. Here we construct minimal representations of cyclic memory behaviors as directed graphs, and we construct simple physically-motivated models that produce the same graph structures. We show how a model of worn grass between park benches can produce multiple transient memories---a behavior previously observed in dilute suspensions of particles and charge-density-wave conductors---and the Mullins effect. Isolating these behaviors in our simple model allows us to assess the necessary ingredients for these kinds of memory, and to quantify memory capacity. We contrast these behaviors with a simple Preisach model that produces return-point memory. Our analysis provides a unified method for comparing and diagnosing cyclic memory behaviors across different materials.
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