To demonstrate the effectiveness of GRID in quantifying structural similarity with EMD, the following results demonstrate its use in predicting bulk modulus (K) from crystal structures. Bulk modulus is an important parameter related to many technologically important properties such as lattice thermal conductivity and mechanical deformation. There are several existing literature reports of ML-based prediction of K (Table ) using different datasets, input features and machine learning algorithms. Comparison between these studies is challenging due to the different data sets and metrics used and whether the model is fitted on a linear or logarithmic scale, but the current state of the art methods using data from the Materials Project achieve a MAE of around 10 GPa (or 0.05 using log10(K / GPa) as input). Using DFT-simulated bulk moduli obtained from the materials project (12,178 entries after cleaning, see methods) we have computed the GRID for each material and determined the EMD between all structure pairs (the 'dissimilarity matrix'). Such a dissimilarity matrix can be used directly to train a number of ML models; in this case we have used the simplest approach of interpolating between the values of the k-nearest neighbouring points (kNN). A kNN approach relies on the similarity matrix accurately representing proximity within S-C space, and also that the available data effectively sample this S-C space. For comparison, we have also trained a kernel ridge regression (KRR) model using a linear kernel (see methods); for well-sampled data these two approaches should replicate each other. Based on GRID alone, the model approaches a minimum MAE of 33 GPa. Given that this model contains no composition information it is remarkably accurate, but has a fundamental limit; two materials with the same structure but different composition will necessarily have the same predicted bulk modulus. It is therefore informative to consider how many materials within our data possess the same GRID; by comparing pairs of GRIDs we find that there are 279 non-unique GRIDs, shared across 582 materials (i.e. two or three materials have the same GRID). As an example, ScGaNi2 and PtAlLi2 (Fig. ) have identical GRIDs (EMD = 0) but have bulk moduli of 130 GPa and 88 GPa, respectively. Thus, structurally identical but compositionally distinct structures represent roughly 5% of the total dataset. It is worth noting that the number of structures with the same GRID will decrease with a greater distance cut off and/or smaller exponential smoothing of distances, at the expense of larger feature spaces (requiring more computational resources).