The inverse structures at the start of each set of runs have much higher energies than the ground state (38.7 meV atom -1 for MnFe 2 O 4 and 45.2 meV atom -1 for ZnFe 2 O 4 ). As the run progresses, the energy of the structure decreases quite rapidly until only the lowest energy structures are sampled. In Figures and, the trajectory clearly shows that for each run the energy of the system decreases as the run progresses and after 20 steps does not increase to an energy above 30 meV atom -1 with structures where Fe tet > 4 no longer sampled. Across all 6 runs conducted starting from inverse structures, no substitution events are accepted directly from the initial structure. For MnFe 2 O 4 and ZnFe 2 O 4 starting runs, substitution steps are not accepted until cations are swapped to lower the energy to below 20 and 30 meV atom -1 respectively (relative to the ground state end members).