where p(x, y) represents the probability of observing a particular neighbor relationship (x = 1 for neighbors, x = 0 for non-neighbors) between one material and a pair of other materials y. The marginal probabilities p(x) and p(y) represent how often each type of relationship occurs independently. A high mutual information value indicates that knowing whether two materials are neighbors provides strong information about their relationships with other materials, suggesting a meaningful neighborhood structure at that distance threshold. The optimal cutoff τ * is identified at the point where adding more distant neighbors no longer provides significant new information about the material relationships.