Figure 3.

Multi-scale analysis. Here, we can see how to construct equivalence class microscales from a given macroscale such that mutual information is fixed. Left: a three-element system (top), and its associated TPM. We select a single node ($A$) to expand. Middle: expanding node $A$ into nodes $\alpha$ and $\beta$ results in a four-element system, which crucially preserves the mutual information from the joint-past to the joint future. We can select another node ($\alpha$) to expand again, resulting in Right: the final microscale expansion of our system. Note that the original node $A$ has been expanded twice, while the overall mutual information dynamics are preserved in all cases. (Online version in colour.)