2-dimensional homology classes and the diameters of the largest included sphere $D_i$ when using methane CH$_{4}$ as a probe molecule. The red line indicates the least squares regression line; Death time = 35.6$\times$$D_i$–100.8.](CH4_PH2.pdf "fig:") -3cm

To quantify the similarity between two materials in the barcode space, we combine as follows the $L^2$-distances between the persistence landcapes (see section \[theo\]) corresponding to their barcodes in the different dimensions. After testing landscape distances of different orders (i.e. $L^\infty$, $L^0$, $L^1$, $L^2$, and so on), $L^2$-distances were chosen because they gave the smallest errors in predicting global structural properties and performance properties for a test set of materials. Let $\Lambda_{d=1}$ (resp. $\Lambda_{d=2}$) be the $L^2$-landscape distance between the 1-dimensional (respectively, 2-dimensional) persistence barcodes, and let $L_{0}=\left|\frac{n_{1}}{V_{1}}-\frac{n_{2}}{V_{2}}\right|$ where $n_{i}$ is the number of points sampled on the pore surface of the $i^{\text{th}}$ material, and $V_{i}$ is the volume of the supercell. The distance between two materials in the barcode space is then $$D_{TS}:= \sqrt{\alpha_{0}L_{0}^{2}+\alpha_{1}\Lambda_{d=1}^{2}+\alpha_{2}\Lambda_{d=2}^{2}},$$ with coefficients $\alpha_{0}= 0.1, \; \alpha_{1}=0.45$, and $\alpha_{2}=0.45$, the values of which were chosen to minimize the error in predicting global structural properties and performance properties for a test set of materials. In dimension 0 the essential intervals are effectively 