GRAFENE: Graphlet-based alignment-free network approach integrates 3D structural and sequence (residue order) data to improve protein structural comparison

Initial protein structural comparisons were sequence-based. Since amino acids that are distant in the sequence can be close in the 3-dimensional (3D) structure, 3D contact approaches can complement sequence approaches. Traditional 3D contact approaches study 3D structures directly and are alignment-based. Instead, 3D structures can be modeled as protein structure networks (PSNs). Then, network approaches can compare proteins by comparing their PSNs. These can be alignment-based or alignment-free. We focus on the latter. Existing network alignment-free approaches have drawbacks: 1) They rely on naive measures of network topology. 2) They are not robust to PSN size. They cannot integrate 3) multiple PSN measures or 4) PSN data with sequence data, although this could improve comparison because the different data types capture complementary aspects of the protein structure. We address this by: 1) exploiting well-established graphlet measures via a new network alignment-free approach, 2) introducing normalized graphlet measures to remove the bias of PSN size, 3) allowing for integrating multiple PSN measures, and 4) using ordered graphlets to combine the complementary PSN data and sequence (specifically, residue order) data. We compare synthetic networks and real-world PSNs more accurately and faster than existing network (alignment-free and alignment-based), 3D contact, or sequence approaches.


S4 Real-world PSNs with CATH categorization
ProteinPDB contains 17,884 protein domains that have CATH categorization, which for a given PSN construction strategy results in 17,884 PSNs. Of these PSNs, to ensure that PSNs are of reasonable "confidence", we focus for further analyses on those PSNs that meet all of the following criteria: 1) the given network has more than 100 nodes, 2) the maximum diameter of the network is more than five, and 3) the network is composed of a single connected component. For different PSN construction strategies, the above criteria can result in different numbers of PSNs. For the first PSN construction strategy (any heavy atom type, 4Å distance cut-off), this results in 9,509 such PSNs. In the main paper (also, see Supplementary Table S2), we report the number of PSNs with respect to this PSN construction strategy. The number of PSNs resulting from using one of the other three PSN construction strategies is of the similar order.
First, we test how well the considered PC approaches can compare PSNs between the top hierarchical categories (i.e., labels) of CATH: alpha (α), beta (β ), alpha/beta (α/β ), and few secondary structures. Only for few secondary structures, none of the domains in ProteinPDB belong to this category, and so we remove the few secondary structures category from further consideration. Of the 9,509 PSNs, 2,628, 3,085, and 3,796 PSNs belong to (i.e., are labeled with) α, β , and α/β categories, respectively. We denote this PSN set as CATH-primary (Fig. 2 in the main paper). The set contains a large enough number of PSNs in each category, which ensures enough statistical power for further analyses.
Second, we test how well the PC approaches can compare PSNs between the second-level hierarchical categories of CATH. That is, within each of the top-level categories of CATH, we compare PSNs belonging to their sub-categories, i.e., second-level categories of CATH. To ensure enough statistical power for further analyses, we focus only on those top-level categories that have at least two sub-categories with at least 30 PSNs each. Each of the three top-level CATH categories satisfies this, and hence, for each of them, we analyze all of their sub-categories that each contain at least 30 PSNs. This results in three PSN sets, denoted as α, β , and α/β (Fig. 2 in the main paper).
Third, we test how well the PC approaches can compare PSNs between the third-level hierarchical categories of CATH. That is, within each of the second-level categories of CATH, we compare the PSNs belonging to their sub-categories, i.e., third-level categories of CATH. To ensure enough statistical power for further analyses, we focus only on those second-level categories that have at least two sub-categories with at least 30 PSNs each. This results in nine PSN sets, denoted as 1.10, 1.20, 2.30, 2.40, 2.60, 2.160, 3.10, 3.30, and 3.40 (Fig. 2 in the main paper).
Fourth, we test how well the PC approaches can compare PSNs between the fourth-level hierarchical categories of CATH. That is, within each of the third-level categories of CATH, we compare PSNs belonging to their sub-categories, i.e., fourth-level categories of CATH. To ensure enough statistical power for further analyses, we focus only on those third-level categories that

S6 Real-world PSNs of the same size
To benchmark PSN-based approaches for protein comparison in a way that the comparison cannot be biased by PSN size, we need PSN data of the same (or at least similar) network size (analogous to the synthetic network data sets). For this analysis, we focus on PSNs of α and β labels from the CATH-primary data set. First, within this data set, we aim to identify PSNs that are of reasonable size, i.e., that have ∼100 nodes. We further filter the resulting PSN set according to the following rules: 1) the number of nodes in all α and β PSNs is the same, 2) the number of edges in all α and β PSNs is statistically significantly similar (Mann-Whitney U test; p-value < 0.05), and 3) there are at least six PSNs in each of the two label categories. We end up with two such PSN sets. The first set is comprised of 24 PSNs having 95 nodes and 343-362 edges, where 12 PSNs are from α and 12 PSNs are from β . We denote this PSN set as CATH-95. The second set is comprised of 28 PSNs having 99 nodes and 347-374 edges, where 12 PSNs are from α and 16 PSNs are from β . We denote this PSN set as CATH-99. Second, within the CATH-primary data set, we aim to identify even larger PSNs, i.e., PSNs that have ∼250 nodes. We again further filter the resulting PSN set according to the same three rules as above, except that in rule 1, we do not force the number of nodes of all PSNs to match (as we could not identify multiple PSNs that satisfy this constraint) but instead it is sufficient that the PSNs are of statistically significantly similar size in terms of the number of nodes (Mann-Whitney U test; p-value < 0.05). This results in another PSN set, which is comprised of 16 PSNs having 251-265 nodes and 1,003-1,076 edges, where nine PSNs are from α and seven PSNs are from β . We denote this PSN set as CATH-251-265. Note that the reported numbers of PSNs in these three "equal size" PSN sets are with respect to the first PSN construction strategy (any heavy atom type, 4Å distance cut-off). Yet, the numbers remain the same for the other three PSN construction strategies.
Two alternative graphlet approaches were used in the context of PSNs 11,12 , but they were used to predict (classify in a supervised manner) functional residues in PSNs (where residues are nodes in PSNs) and not for PSN comparison. Since these approaches compare nodes rather than networks, and since they are supervised (while our study is unsupervised, per our discussion in Section "Evaluation of PC accuracy" of the main manuscript), the approaches do not fit the context of our study. As such, we do not consider them further.
Existing non-graphlet approaches. Several PSN measures have already been used for PC: average degree, average distance, maximum distance, average closeness centrality, average clustering coefficient, intra-hub connectivity, and assortativity [13][14][15][16][17] . For each measure, for each pair of networks, we compute Euclidean distance between the networks' vectors (because all vectors are 1-dimensional, here we cannot use cosine similarity as for our GRAFENE approach). We describe these measures below. Average degree. The average degree of a network can be interpreted as a measure of the overall connectivity of the network. The degree of a node is the number of its network neighbors. The average degree of a network is the average of degrees of all nodes in the network. This measure has been used for analyzing protein structures by [13][14][15][16][17] . Average distance. The distance between two nodes in a network is the length of the shortest path between the nodes. The average distance of a network is the average of distances over all pairs of nodes in the network. This measure has been used for analyzing protein structures by 16,17 . Maximum distance. The maximum distance of a network is the largest of all distances in the network. This measure has been used for analyzing protein structures by 16 . Average closeness centrality. The closeness centrality of a node in a network can be interpreted to be the nearness of the node to all other nodes in the network. The closeness centrality cl(v) of a node v ∈ V is computed as cl(v) = 1 ∑ u∈V d (u,v) , where d(v, u) is the distance between nodes v and u. The average closeness centrality of a network is the average of the closeness centrality values of all nodes in the network. This measure has been used for analyzing protein structures by 16,17 . Average clustering coefficient. The clustering coefficient of a node in a network can be interpreted as a measure of the connectivity between the neighbors of the node. Given a node v with m neighbors, the clustering coefficient cc(v) of the node v is computed as cc , where b is the number of edges in the network connecting the m neighbors of v. The average clustering coefficient of a network is the average of clustering coefficient values of all nodes in the network. This measure has been used for analyzing protein structures by 16,17 . Intra-hub connectivity. The intra-hub connectivity of a network can be interpreted as the overall connectivity of the hub nodes within the network. 14 defined a node to be a hub in a PSN if the degree of the node is at least three. We adopt the same strategy to define a hub node in this study. Given k such hub nodes in a network, the intra-hub connectivity of the network is computed as m , where m is the number of connections between the hub nodes and k(k−1) 2 is the maximum possible number of connections between the hub nodes. This measure has been used for analyzing protein structures by 14 .
Assortativity. The assortativity of a network can be interpreted as the tendency of the high degree nodes to be connected with other high degree nodes (see 18 for details). This measure has been used for analyzing protein structures by 16 .
We combine the seven measures into an eighth measure, Existing-all, to investigate whether the integration of different and complementary topological measures helps PC. We use Existing-all within our PCA framework. This way, we can fairly compare our graphlet measures (i.e., different versions of our GRAFENE approach) and the existing non-graphlet measures within the same framework.

S7.2 Existing 3D contact approaches
These include DaliLite 19 and TM-align 20 . Given two proteins (i.e., 3D co-ordinates of their residues), each of DaliLite and TM-align outputs the proteins' structural similarity score: z-score in the case of DaliLite and TM-score in the case of TM-align. In our evaluation framework, we sort all protein pairs in terms of their increasing distance, i.e., decreasing z-scores for DaliLite and decreasing TM-scores for TM-Align, and then we evaluate DaliLite and TM-Align as discussed in Section "Evaluation of PC accuracy" of the main manuscript.

S7.3 Existing sequence approach
The sequence-based approach that we use, which we call AAComposition, works as follows. For a given protein, for each amino acid type i (out of 20 possible types), we divide the number of amino acids of type i by the total number of amino acids in the protein sequence. We use the resulting 20 values, along with the length of the protein sequence, as the protein's sequence-based measure (i.e., feature vector). Then, we use this measure within our PCA framework. This way, we can fairly compare network-and sequence-based measures within the same framework.

S8 Performance trends of different PC approaches on same PSN sets and of same PC approaches on different PSN sets
Performance trends of different PC approaches on same PSN sets. We sometimes observe a difference in trends between different PC approaches for same PSN sets. Specifically, in the case of the CATH database, all approaches result in a consistent trend that their accuracy for CATH-α is higher than their accuracy for CATH-β . Similarly, in the case of the SCOP database, the majority of the approaches show a consistent trend that their accuracy for SCOP-β is higher than their accuracy for SCOP-α, except the GDDA, GCD, and AAComposition PC approaches, whose accuracy for SCOP-α is higher than their accuracy for SCOP-β . This difference in the trends between the different approaches (GDDA, GCD, and AAComposition versus all others) for SCOP is an approach-specific issue, meaning that some approaches might simply work better for (i.e., better capture patterns in) data of type 1 (e.g., α) than for data of type 2 (e.g., β ), while other approaches might show the opposite trend (i.e., work better for data of type 2 than for data of type 1). It is hard to explain why this is, especially for the network-based approaches, because these approaches are heuristics (due to the computational intractability, i.e., NP-hardness, of the network comparison problem) without a theoretic guarantee on their accuracy (and especially on their accuracy on certain data types as opposed to other data types).
Performance trends of same PC approaches on different PSN sets. Additionally, we observe a difference in the performance of same PC approaches on different PSN sets. Specifically, a given approach might have higher accuracy for CATH-α than for CATH-β , but the same approach might have lower accuracy for SCOP-α than for SCOP-β . This trend inconsistency holds for all considered PC approaches except GDDA, GCD, and AAComposition; for both CATH and SCOP, the accuracy of these three approaches is higher for α than for β . This trend inconsistency is likely a data-specific issue: 1) CATH and SCOP do not necessarily contain the exact same PSNs (meaning that some PSNs that are in CATH might be missing from SCOP, and vice versa), and 2) for those PSNs that are in both CATH and SCOP, the PSNs might be categorized into some protein domain group (e.g., α) in CATH but to a different protein domain group (e.g., α/β ) in SCOP, because the methodologies that CATH and SCOP use to categorize proteins into domain groups are not identical. If any of these two conditions is met, this could explain the observed trend inconsistency. Indeed, we find that: • 8% of the PSNs that are labeled as α in CATH are labeled as β , α/β , or α+β in SCOP.
• 0.3% of the PSNs that are labeled as α in SCOP are labeled as β or α/β in CATH.
• 38% of the PSNs that are labeled as β in SCOP are labeled as α or α/β in CATH.
• 40% of the PSNs that are labeled as α/β in CATH are labeled as α or β in SCOP.
• 43% of the PSNs that are labeled as α/β or α+β in SCOP are labeled as α or β in CATH.
Clearly, both of the above conditions are met, and hence, the observed trend inconsistency is not surprising. Note that the above results are with respect to the first PSN construction strategy (any heavy atom, 4Å) and the performance evaluation using AUPR. Figure S2. The performance comparison of the 15 considered approaches on each of the four considered synthetic network sets, with respect to AUROC, in terms of: (A) the approaches' ranks compared to one another, and (B) the approaches' raw AUROC values. In panel (A), for a given synthetic network set, the 15 approaches are ranked from the best (rank 1) to the worst (rank 15). So, the lower the rank, the better the approach. In panel (B), for each approach, its raw AUPR value is shown for each of the four synthetic network sets. So, the higher the AUROC value, the better the approach. For equivalent results with respect to AUPR values, see Fig. 4 in the main manuscript.

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Supplementary Figure S3. The PSN set group-specific performance comparison of the 24 considered approaches, averaged over all PSN sets in the given PSN set group, with respect to AUROC, in terms of: (A) the approaches' ranks compared to one another, and (B) the approaches' raw AUROC values. In panel (A), for a given PSN set, the 24 approaches are ranked from the best (rank 1) to the worst (rank 24). Then, for a given approach, its ranks over all group-specific PSN sets are averaged (the average ranks are denoted by circles, and bars denote the corresponding standard deviations). So, the lower the average rank, the better the approach. In panel (B), for each approach, its group-specific raw AUROC scores are averaged (the average values are denoted by circles, and bars denote the corresponding standard deviations). So, the higher the average AUROC value, the better the approach. The trends are very similar with respect to AUPR as well (Fig. 7 in the main manuscript). These results are for the best PSN construction strategy. Equivalent results for each of the PSN construction strategies are shown in Supplementary Fig. S4-S7.

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Supplementary Figure S4. The PSN set group-specific rank performance comparison of the 24 considered approaches, averaged over all PSN sets in the given PSN set group, with respect to AUPR, corresponding to the (A) first (any heavy atom, 4 A), (B) second (any heavy atom, 5Å), (C) third (any heavy atom, 6Å), and (D) fourth (α-carbon heavy atom, 7.5Å) PSN construction strategy. For a given PSN set, the 24 approaches are ranked from the best (rank 1) to the worst (rank 24). Then, for a given approach, its ranks over all group-specific PSN sets are averaged (the average ranks are denoted by circles, and bars denote the corresponding standard deviations). So, the lower the average rank, the better the approach. The trends are very similar with respect to AUROC as well ( Supplementary Fig. S5).

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Supplementary Figure S5. The PSN set group-specific rank performance comparison of the 24 considered approaches, averaged over all PSN sets in the given PSN set group, with respect to AUROC, corresponding to the (A) first (any heavy atom, 4Å), (B) second (any heavy atom, 5Å), (C) third (any heavy atom, 6Å), and (D) fourth (α-carbon heavy atom, 7.5Å) PSN construction strategy. For a given PSN set, the 24 approaches are ranked from the best (rank 1) to the worst (rank 24). Then, for a given approach, its ranks over all group-specific PSN sets are averaged (the average ranks are denoted by circles, and bars denote the corresponding standard deviations). So, the lower the average rank, the better the approach. The trends are very similar with respect to AUPR as well ( Supplementary Fig. S4).

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Supplementary Figure S6. The PSN set group-specific performance comparison of the 24 considered approaches, averaged over all PSN sets in the given PSN set group, with respect to AUPR values (expressed as percentages), corresponding to the (A) first (any heavy atom, 4Å), (B) second (any heavy atom, 5Å), (C) third (any heavy atom, 6Å), and (D) fourth (α-carbon heavy atom, 7.5Å) PSN construction strategy. For each approach, its group-specific raw AUPR values are averaged (the average values are denoted by circles, and bars denote the corresponding standard deviations). So, the higher the average AUPR value, the better the approach. The trends are very similar with respect to AUROC as well ( Supplementary Fig. S7).

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Supplementary Figure S7. The PSN set group-specific performance comparison of the 24 considered approaches, averaged over all PSN sets in the given PSN set group, with respect to AUROC values (expressed as percentages), corresponding to the (A) first (any heavy atom, 4Å), (B) second (any heavy atom, 5Å), (C) third (any heavy atom, 6Å), and (D) fourth (α-carbon heavy atom, 7.5Å) PSN construction strategy. For each approach, its group-specific raw AUROC values are averaged (the average values are denoted by circles, and bars denote the corresponding standard deviations). So, the higher the average AUROC value, the better the approach. The trends are very similar with respect to AUPR as well ( Supplementary  Fig. S6).

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Supplementary Figure S8. Distribution of PSN sets across four PSN construction strategies: 1, 2, 3, and 4. The results are with respect to AUPR. Each panel (one panel per PC approach) shows the following: for each PSN construction strategy, we calculate the percentage of all 3 + 35 = 38 real-world PSN sets for which the given PSN construction strategy performs the best; this is what the height of the given bar shows. Then, within each bar, we label the PSN sets according to the PSN set groups to which they belong.

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Supplementary Figure S9. Distribution of PSN sets across four PSN construction strategies: 1, 2, 3, and 4. The results are with respect to AUROC. Each panel (one panel per PC approach) shows the following: for each PSN construction strategy, we calculate the percentage of all 3 + 35 = 38 real-world PSN sets for which the given PSN construction strategy performs the best; this is what the height of the given bar shows. Then, within each bar, we label the PSN sets according to the PSN set groups to which they belong. Figure S10. The ranking of the four PSN construction strategies: 1, 2, 3, and 4. The ranking is shown with respect to AUPR. Each panel (one panel per PC approach) shows the following: for each PSN construction strategy, we calculate the percentage of all 3 + 35 = 38 real-world PSN sets in which the given PSN construction strategy performs the best, the second best, the third best, and the fourth best.

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Supplementary Figure S11. The ranking of the four PSN construction strategies: 1, 2, 3, and 4. The ranking is shown with respect to AUPR. Each panel (one panel per PC approach) shows the following: for each PSN construction strategy, we calculate the percentage of all 3 + 35 = 38 real-world PSN sets in which the given PSN construction strategy performs the best, the second best, the third best, and the fourth best. Note that unlike in Supplementary Fig. S10, here we consider two AUPR values to be tied if the absolute difference between them is ≤ 5% of the maximum achievable AUPR value.

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Supplementary Figure S12. The ranking of the four PSN construction strategies: 1, 2, 3, and 4. The ranking is shown with respect to AUROC. Each panel (one panel per PC approach) shows the following: for each PSN construction strategy, we calculate the percentage of all 3 + 35 = 38 real-world PSN sets in which the given PSN construction strategy performs the best, the second best, the third best, and the fourth best.

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Supplementary Figure S13. The ranking of the four PSN construction strategies: 1, 2, 3, and 4. The ranking is shown with respect to AUROC. Each panel (one panel per PC approach) shows the following: for each PSN construction strategy, we calculate the percentage of all 3 + 35 = 38 real-world PSN sets in which the given PSN construction strategy performs the best, the second best, the third best, and the fourth best. Note that unlike in Supplementary Fig. S12, here we consider two AUROC values to be tied if the absolute difference between them is ≤ 5% of the maximum achievable AUROC value.

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Supplementary Figure S14. The PSN construction strategy-specific performance comparison of the 24 considered PC approaches, with respect to AUROC, in terms of: (A) the approaches' ranks compared to one another, and (B) the approaches' raw AUROC values. In panel (A), for each PSN construction strategy, for a given PSN set, the 24 approaches are ranked from the best (rank 1) to the worst (rank 24). Then, for a given approach, its 35 ranks (corresponding to the 35 PSN sets) are averaged (the average ranks are denoted by circles, and bars denote the corresponding standard deviations). So, the lower the average rank, the better the approach. In panel (B), for each PSN construction strategy, for each approach, its 35 raw AUROC values (corresponding to the 35 PSN sets) are averaged (the average values are denoted by circles, and bars denote the corresponding standard deviations). So, the higher the average AUROC value, the better the approach. The trends are very similar with respect to AUPR as well (Fig. 8 in the main manuscript). These results are for the "all group" PSN set group that spans the 35 PSN sets of different sizes. Equivalent results for the individual groups 1-4 are shown in Supplementary Fig. S15-S18.

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Supplementary Figure S15. The PSN construction strategy-specific rank performance comparison of the 24 considered PC approaches, with respect to AUPR, corresponding to PSN set group : (A) 1, (B) 2, (C) 3, and (D) 4. For each PSN construction strategy, for a given PSN set, the 24 approaches are ranked from the best (rank 1) to the worst (rank 24). Then, for a given approach, its 35 ranks (corresponding to the 35 PSN sets) are averaged (the average ranks are denoted by circles, and bars denote the corresponding standard deviations). So, the lower the average rank, the better the approach. The trends are very similar with respect to AUROC as well ( Supplementary Fig. S16). Figure S20. Statistical significance of the difference between average raw values of the PC approaches, with respect to: (A) AUPR and (B) AUROC. For aesthetics, these results are only for the best approach in each category, namely: the best of our proposed PCA graphlet-based network approaches (GRAFENE version NormOrderedGraphlet-3-4(K)), the best of the existing non-PCA graphlet-based network approaches (GR-Align), the best of the existing non-graphlet network approaches (Existing-all), the best of the existing non-network 3D structural approaches (DaliLite), and the sequence-based approach (AAComposition). For each of the 35 PSN sets, raw AUPR/AUROC values for all five approaches are measured. Hence, for each approach, there are 35 raw AUPR/AUROC values (corresponding to the 35 PSN sets). For each pair of approaches, we compare the two given approaches' 35 raw AUPR/AUROC values using paired t-test. In the figure, every cell (i, j) indicates the statistical significance (in terms of p-value) of approach i being superior to approach j. The results are similar when we use ranks instead of raw AUPR/AUROC values ( Supplementary Fig. S19). Figure S21. The performance comparison of only the best PC approach in each category (for aesthetics purposes) on all three "equal size" PSN sets and all 35 PSN sets of different size, with respect to raw AUROC values. Namely, results are shown for: the best of our proposed PCA graphlet-based network approaches (GRAFENE version NormOrderedGraphlet-3-4(K)), the best of the existing non-PCA graphlet-based network approaches (GR-Align), the best of the existing non-graphlet network approaches (Existing-all), the best of the existing non-network 3D structural approaches (DaliLite), and the sequence-based approach (AAComposition). The vertical dotted lines separate the PSN sets into the five PSN set groups, namely (from left to right): "equal size", group 1, group 2, group 3, and group 4. For the equivalent results in terms of raw AUPR values, see Fig. 9 in the main manuscript.

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Supplementary Figure S22. (A) Precision-recall (PR) and (B) receiver operating characteristic (ROC) curves for the best approach in each category, namely: the best of our proposed PCA graphlet-based network approaches (GRAFENE version NormOrderedGraphlet-3-4(K)), the best of the existing non-PCA graphlet-based network approaches (GR-Align), the best of the existing non-graphlet network approaches (Existing-all), the best of the existing non-network 3D structural approaches (DaliLite), and the sequence-based approach (AAComposition). The results are for the three "equal-size" PSN sets. Also, these results are for the best PSN construction strategy. Figure S23. Precision-recall (PR) curves for the best approach in each category, namely: the best of our proposed PCA graphlet-based network approaches (GRAFENE version NormOrderedGraphlet-3-4(K)), the best of the existing non-PCA graphlet-based network approaches (GR-Align), the best of the existing non-graphlet network approaches (Existing-all), the best of the existing non-network 3D structural approaches (DaliLite), and the sequence-based approach (AAComposition). These results are for the 35 PSN sets of different size. Also, these results are for the best PSN construction strategy. Figure S24. Receiver operating characteristic (ROC) curves for the best approach in each category, namely: the best of our proposed PCA graphlet-based network approaches (GRAFENE version NormOrderedGraphlet-3-4(K)), the best of the existing non-PCA graphlet-based network approaches (GR-Align), the best of the existing non-graphlet network approaches (Existing-all), the best of the existing non-network 3D structural approaches (DaliLite), and the sequence-based approach (AAComposition). These results are for the 35 PSN sets of different size. Also, these results are for the best PSN construction strategy.

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Supplementary Figure S25. Ordered graphlets that are significantly represented in α (dark gray) or β (light gray) PSNs. Tables   Supplementary Table S1. Synthetic network sets that we use. For the given data set, the second column indicates whether its networks are of the same size or different sizes, and the last three columns indicate the number of its networks as well as their size(s) in terms of the number of nodes and edges. Data   Supplementary Table S10. Summary of method accuracy and running times. Accuracy of the given approach is shown with respect to its average ranking as well as its average raw score compared to all considered approaches across all 35 different-size PSN sets, and the results are shown based on AUPR as well as AUROC. We rank the approaches as follows. For the given PSN set, we determine which approach results in the highest accuracy (rank 1), the second highest accuracy (rank 2), etc. Then, we average the rankings of the given method over all PSN sets. So, the lower the average rank, the better the method. Since NormOrderedGraphlet-3-4(K) has the best average rank with respect to both AUPR and AUROC (shown in bold), we compute the statistical significance of the improvement of NormOrderedGraphlet-3-4(K) over each of the other approaches in terms of their ranks using paired t-test. We also do the same in terms of raw AUPR/AUROC values. Note that in the case of raw values, the higher the average AUPR/AUROC value, the better the approach. Running times of the approaches are shown when comparing proteins from the CATH-α set. Running times for the other data sets are qualitatively the same. Supplementary Table S13. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUPR values (expressed as percentages), corresponding to the PSN sets of "equal size", group 1, and group 2. Given a PSN set (within a given column), the AUPR for the "best" K is shown in bold. These results are with respect to the first PSN construction strategy, (any heavy atom type, 4Å distance cut-off). Supplementary Table S14. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUROC values (expressed as percentages), corresponding to the PSN sets of "equal size", group 1, and group 2.

III Supplementary
Given a network data set (within a given column), the AUROC for the "best" K is shown in bold. These results are with respect to the first PSN construction strategy, (any heavy atom type, 4Å distance cut-off). Supplementary Table S17. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUPR values (expressed as percentages), corresponding to the PSN sets of group 4. Given a fourth-level PSN data set (within a given column), the AUPR for the "best" K is shown in bold. These results are with respect to the first PSN construction strategy, (any heavy atom type, 4Å distance cut-off).  Supplementary Table S19. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUPR values (expressed as percentages), corresponding to the PSN sets of "equal size", group 1, and group 2. Given a PSN data set (within a given column), the AUPR for the "best" K is shown in bold. These results are with respect to the second PSN construction strategy, (any heavy atom type, 5Å distance cut-off). Given a PSN data set (within a given column), the AUROC for the "best" K is shown in bold. These results are with respect to the second PSN construction strategy, (any heavy atom type, 5Å distance cut-off).  Supplementary Table S21. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUPR values (expressed as percentages), corresponding to the PSN sets of group 3. Given a third-level PSN data set (within a given column), the AUPR for the "best" K is shown in bold. These results are with respect to the second PSN construction strategy, (any heavy atom type, 5Å distance cut-off).  Supplementary Table S23. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUPR values (expressed as percentages), corresponding to the PSN sets of group 4. Given a fourth-level PSN data set (within a given column), the AUPR for the "best" K is shown in bold. These results are with respect to the second PSN construction strategy, (any heavy atom type, 5Å distance cut-off).  Supplementary Table S24. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUROC values (expressed as percentages), corresponding to the PSN sets of group 4. Given a fourth-level PSN data set (within a given column), the AUROC for the "best" K is shown in bold. These results are with respect to the second PSN construction strategy, (any heavy atom type, 5Å distance cut-off).  Supplementary Table S25. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUPR values (expressed as percentages), corresponding to the PSN sets of "equal size", group 1, and group 2. Given a PSN data set (within a given column), the AUPR for the "best" K is shown in bold. These results are with respect to the third PSN construction strategy, (any heavy atom type, 6Å distance cut-off).  Supplementary Table S26. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUROC values (expressed as percentages), corresponding to the PSN sets of "equal size", group 1, and group 2.
Given a PSN data set (within a given column), the AUROC for the "best" K is shown in bold. These results are with respect to the third PSN construction strategy, (any heavy atom type, 6Å distance cut-off).  Supplementary Table S27. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUPR values (expressed as percentages), corresponding to the PSN sets of group 3. Given a third-level PSN data set (within a given column), the AUPR for the "best" K is shown in bold. These results are with respect to the third PSN construction strategy, (any heavy atom type, 6Å distance cut-off).  Supplementary Table S31. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUPR values (expressed as percentages), corresponding to the PSN sets of "equal size", group 1, and group 2. Given a PSN data set (within a given column), the AUPR for the "best" K is shown in bold. These results are with respect to the fourth PSN construction strategy, (α-carbon heavy atom type, 7.5Å distance cut-off).  Supplementary Table S34. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUROC values (expressed as percentages), corresponding to the PSN sets of group 3. Given a third-level PSN data set (within a given column), the AUROC for the "best" K is shown in bold. These results are with respect to the fourth PSN construction strategy, (α-carbon heavy atom type, 7.5Å distance cut-off).  Supplementary Table S35. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUPR values (expressed as percentages), corresponding to the PSN sets of group 4. Given a fourth-level PSN data set (within a given column), the AUPR for the "best" K is shown in bold. These results are with respect to the fourth PSN construction strategy, (α-carbon heavy atom type, 7.5Å distance cut-off).  Supplementary Table S36. Accuracy of the NormOrderedGraphlet-3-4(K) approach when varying the value of K, with respect to AUROC values (expressed as percentages), corresponding to the PSN sets of group 4. Given a fourth-level PSN data set (within a given column), the AUROC for the "best" K is shown in bold. These results are with respect to the fourth PSN construction strategy, (α-carbon heavy atom type, 7.5Å distance cut-off).