The fingerprint distance introduced by is purely geometric and combines statistics of block faces and shape factors in computing a probability distribution of a spatial graph. formulated the measure in the context of quantifying differences in street patterns. A “block” denotes the 2-D region enclosed by graph edges. For any given spatial graph, this corresponds to the number of bounded subgraphs or primary cycles. We neglect isolated fractures and those having dead ends when computing these blocks. Given the network intensity in our dataset, such isolated fractures are minimal. Every block has an associated shape factor, “ϕ” which is expressed in terms of block area “A” and circumscribing circle area, “Ac”: 1ϕ=AAc.

The value of ϕ is always smaller than 1, with larger values meaning that the block-face shape is closer to that of a regular polygon. Figure a depicts shape factors of regular polygons versus that of polygons derived from spatial networks in Fig. b. No unique correspondence exists between a particular shape and a magnitude of ϕ; however, the overall distribution of ϕ indicates reveals block shape distribution patterns and highlights differences between spatial graphs. The shape factor alone does not fully serve as a similarity measure as blocks can have similar shapes but different face areas. The distribution of the block-face areas is binned logarithmically to integrate information from the shape factor and block area distributions. A conditional probability distribution, P(ϕ|A)P(A), is then defined representing the contribution of P(ϕ) for each area bin and the summation of which yields the fingerprint curve, P(ϕ): 2P(ϕ)=∑AP(ϕ|A)P(A).

An example of a “fingerprint”, so named by , is depicted in Fig. e and j, with the distribution curves for three area bins, for two fracture networks derived from image tiles (see Fig. b, c, f, and g) corresponding to Region 1 (Fig. a). The curves in Fig. e and j encapsulate information based on shape factors and block areas (see Fig. d and h), including the proportional contribution from all logarithmic area bins considered.

Denoting fα(ϕ) as the ratio of the number of faces with a shape factor “ϕ” that lie in a bin “α” over the total number of faces for that graph, a distance dα between two graphs Ga and Gb is computed by integrating over fα(ϕ) for the two different graphs. The distance based on fα(ϕ) of the two graphs for a single area bin is defined as 3dα(Ga,Gb)=∫01fαa(ϕ)-fαb(ϕ)ndϕ.

As per , the value of n can either be 1 or 2. We choose n=1 in our computation. The global fingerprint distance DFP between Ga and Gb can then be computed summing over all area bins α: 4DFP(Ga,Gb)=∑αdα(Ga,Gb)2.

We have attached our MATLAB implementation of the fingerprint distance in the code Supplement. We computed the distance matrix for all subgraphs corresponding to the three regions using this implementation.

The D-measure introduced by is a three-component distance metric with weighting constants for each component. The three properties of graphs compared are the network node dispersion (NND), node distance distribution (μ), and the alpha (α) centrality. The dissimilarity measure, DDM, is the weighted sum: DDM(g,h)=w1J(μg,μh)log⁡2+w2NND(g)-NND(h)+w32J(Pα(g),Pα(h))log⁡25+J(Pα(gc),Pα(hc′))log⁡2, where J indicates the Jensen–Shannon divergence. The constants w1, w2, and w3 in Eq. () are real and non-negative weights such that w1+w3+w3=1.

As per , the first term in Eq. () compares averaged connectivity node's patterns as per node distance distribution. define NND, within the second term, as a measure of the heterogeneity of a graph with respect to connectivity distances that capture global topological differences. The NND is computed as 6NND(G)=J(P1,…,PN)log⁡(d+1), where the numerator in Eq. () is the Jensen–Shannon divergence of N connectivity distance distributions [P1,P2…PN]. Pi is constructed as Pi=pi(j), where pi(j) is the fraction of nodes connected to node i at distance j. The Jensen–Shannon divergence of [P1,P2…PN] is expressed as 7J(P1,…,PN)=1N∑i,jpilog⁡pi(j)μj.

μj in Eq. () is the average of N distributions and can be written as 8μj=1N∑i=1Npi(j).

The third term in Eq. () is based on probability density functions associated with α centrality of graph Pα(g) and α centrality of the graph complement Pα(gc). The value of weights was suggested by as w1=w2=0.45 and w3=0.1. We use the implementation provided by with these sets of weights to build the distance matrices for all subgraphs within the three regions of interest. We depict in Fig. , for the two example fracture networks, the three properties that are used in computing the D-measure.

The portrait divergence similarity score derives from network portraits introduced by for unweighted graphs and extended to weighted graphs by . For a graph g with N nodes, the network portrait is defined as a matrix Blk, where each entry is the number of nodes with k nodes at l distance. The limits of l and k are 0≤l≤d and 0≤k≤N-1, with d being the diameter of the graph. The row entries of the network matrix Blk are probability distributions of a random node having k nodes at a distance l: 9P(k|l)=BlkN. For a second graph h, if the network matrix is Blk′ with a corresponding probability distribution of Q(k|l) and diameter d′, the Kullback–Leibler (KL) divergence between P(k|l) and Q(k|l) is expressed as 10KL(P(k|l)||P(k|l))=∑l=0max⁡(d,d′)∑k=0NP(k,l)log⁡P(k,l)Q(k,l). The portrait divergence DPD(g,h) is computed by the Jensen–Shannon divergence between P(k|l) and Q(k|l): 11DPD(g,h)=JSD(P(k|l),Q(k|l). This can be expressed in terms of Kullback–Leibler divergences and mixture distributions as 12DPD(g,h)=12(KL(P||M)+KL(Q||M)), where the mixture distribution M of P(k|l) and Q(k|l) is given by 13M=12(P(k|l)+Q(k|l)).

The portrait divergence measure provides a single value 0≤DPD(g,h)≤1 for any pair of graphs. applied the portrait divergence measure to both synthetic and real-world networks. The code implementation of portrait divergence attached with is used to construct the distance matrices for all subgraphs within the three regions of interest. The network portrait or the Blk matrix for the example fracture graphs are depicted as heatmaps in Fig. .

The NetLSD distance was introduced by . It is based on a Frobenius norm computed between heat trace signatures of normalized Laplacian matrices of two graphs. For a graph g with a normalized Laplacians L and n nodes, a heat kernel matrix is defined as 14Ht=e-tL=∑j=1ne-tλjϕjϕjT. Using the heat kernel matrix Ht, a heat trace ht is defined as 15ht=∑j=1ne-tλj. For a second graph g′ with a heat trace signature of ht′, the NetLSD distance DLSD is then the Frobenius norm of the two heat signatures as 16DLSD=||ht,ht′||Frobenius.

Figure  depicts heat trace signatures computed using the NetLSD Python package implemented by for the two example fracture graphs. We use this package to populate the distance matrices associated with subgraphs from each region.

The values of graph similarity computed using the four metrics described by Eqs. (), (), (), and () for the two example fracture graphs depicted in Fig. c and g are summarized in Table .

The spatial distribution of clusters pertaining to the four distance metrics overlain over the network is shown in Fig. a–d along with the associated dendrograms for the top 10 clusters. The subgraphs are represented by coloured discs that follow a diverging colour scheme. The number of subgraphs within each of the top 10 clusters is also listed under the dendrogram branches. It may be noted that the top 10 clusters are shown to depict, analyse, and compare the spatial variation across distance measures. A complete, uncut dendrogram and associated heatmaps of the similarity measures are depicted in Fig.  in the Appendix. We can cut the dendrogram at different heights guided by slope changes in the weighted sum of squares plots shown in Fig. . The boundaries of spatial clusters vary with the dendrogram cut height, with subregions emerging by traversing deeper into the dendrogram. This variation is depicted in Figs. – in the Appendix for a range of clusters varying from 5–10. The number of subsamples for the four similarity measures pertaining to a dendrogram cut of k=10 is tabulated in Table .

We can observe that spatial autocorrelation exists for the FP (Fig. a), DM (Fig. b), and PD (Fig. d) similarity measures. The LSD yields a speckled pattern with no obvious spatial autocorrelation (Fig. c). In order to compare clustering results derived from the graph similarity measures, the spatial fracture persistence P20 and P21 computed using box counting (box size of 0.5×0.5 m) is depicted in Fig. e and f, respectively. Comparing clusters derived from graph similarity measures to the fracture persistence plots reveals boundaries within the network that are not easily discernable from the latter. Since LSD does not show spatial autocorrelation, we do not analyse it further.

Figure a–c depict topology histograms and rose plots of the clusters pertaining to the remaining three similarity measures. The orientation rose plots and topological summaries are generated by combining all circular samples identified under a cluster into 10 cluster subgraphs from the larger region fracture graph. It can be observed from the rose plots that the clusters have varying fracture orientations that transitions across the hierarchy identified by the dendrograms. The topological summaries of the clusters do not vary significantly. Figures – in the Appendix depict zoomed-in subgraphs corresponding to each of the top 10 clusters that visually confirm the intra-regional variation.

We briefly describe the characteristics of the clustering results prefixing “n” to the number of subsamples within a cluster to refer to a particular cluster at a k=10 dendrogram cut. From Fig. a and the zoomed-in archetypal examples in Fig. , the clustering derived from FP seems to have a N–S variation trend. The trend is corroborated by observing the dendrogram, which splits into a northern branch comprising of clusters n36, n24b, and n47 and a southern branch with clusters n6, n20, n41, n16, n2, and n24a. An outlier branch n3 exists at the boundary between northern and southern branches.

A similar variation is observable from the result of DM (see Fig. b). However, the cluster demarcations are less stark than with FP with a notable stippled pattern. A major dendrogram division is a branch consisting of a thin sliver in the NE (clusters n24, n25b, n5, and n15) which also include some boundary periphery samplings in the west and south of Region 1. The southwestern sliver is mainly contained in a branch containing cluster n17. The central parts of Region 1 fall under the dendrogram branch containing clusters n39, n10, and n25a. The remainder of the Region 1 is covered by branch containing clusters n19 and n40. Figure  depicts archetypal examples of subgraphs relevant to each cluster for DM.

The results of PD also depict N–S variation (see Fig. d) in the clustering. Similar to DM, PD is also sensitive to the subgraph completeness with peripheral clusters represented under n13. The branches comprising n13, n6, n9, n38, and n17 closely correspond to the trend of high fracture persistence (compare with Fig. e and f). Similar to results from FP and DM, the thin sliver in the NE of Region 1 is captured under the branch with clusters n3, n5, and n24. The remainder of Region 1 falls under clusters n25 and n79. Figure  depicts archetypal examples of subgraphs corresponding to each cluster for PD.

Spatial distribution along with dendrograms of top 10 clusters pertaining to the four graph similarity measures for Region 2 is depicted in Fig. . The full dendrograms and heatmaps are placed in Fig.  in the Appendix. The variation of spatial clusters with different choices of dendrogram cut heights is shown in Figs. – in the Appendix. The number of subsamples for the four similarity measures pertaining to a dendrogram cut of k=10 is tabulated in Table . Similar to Region 1, there is marked spatial autocorrelation with FP (Fig. a), DM (Fig. b), and PD (Fig. d), whereas the LSD (Fig. c) shows a speckled pattern. The spatial clustering results can be compared with the fracture persistence plots in Fig. e and f.

Node degree histograms and rose plots depict the differences in network topology and fracture orientations between the identified clusters pertaining to FP (Fig. a), DM (Fig. b), and PD (Fig. c). For all three measures, the shape of rose plots indicates a transition of principal orientations smoothly across clusters. For example, in Fig. a for FP, the more complex fracturing in the west of Region 2 is depicted by cluster n20 with a very diffuse rose plot, changing orientations to a predominantly orthogonal pattern in cluster n03. The DM (clusters n16 and n10 in Fig. b) and PD (clusters n08 and n17b in Fig. c) also identify this region of orthogonal fracturing. The corresponding topological summaries also depict an increased proportion of degree-4 nodes as compared to the histograms of other clusters.

From FP clustering results (see Fig. a), the dendrogram identifies a western branch with clusters n20 and n22. The branch comprising of clusters n8, n3, and n4 correspond to the radial fracturing region identified by that originates from the fault in the SE of Region 2. Clusters n9, n17, and n36 all under a branch covering parts of Region 2 further away from the radial fracturing region. Clusters n41 and n52 originate under a branch forming the northern and eastern boundaries of Region 2. Figure  depicts archetypal subgraphs under each cluster in detail for FP. The clustering results of DM (Fig. b) and PD (Fig. d) appear to be similar and with dendrograms roughly splitting into three main branches that correspond to specific portions of Region 1. First is the radial fracturing area represented by branch-forming clusters n10, n16, and n23b for D-measure (Fig. b) and branch-forming clusters n17b and n8 for the portrait divergence (Fig. d). The area to the NW periphery of Region 2, farthest away from the fault, is represented by branch-forming clusters n19, n20, n14, and n25 for DM and by branch-forming clusters n24a, n15a, and n34 for PD. The transition region branch is represented within the DM dendrogram by clusters n38, n23a, and n24 and within the PD dendrogram by clusters n24b, n15b, and n28. Figures  and depict detailed subgraph examples for DM and PD, respectively.

The spatial distribution along with dendrograms of the top 10 clusters pertaining to the four graph similarity measures for Region 3 is depicted in Fig. . The full dendrograms and heatmaps are placed in Fig.  in the Appendix. The variation of spatial clusters with different choices of dendrogram cut heights (and number of clusters) is shown in Figs. – in the Appendix. Similar to the Region 1 and 2 results, there is marked spatial autocorrelation with FP (Fig. a), DM (Fig. b), and PD (Fig. d), whereas the LSD (Fig. c) shows a stippled pattern. The spatial clustering results can be compared with the fracture persistence plots in Fig. e and f. The number of subsamples for the four similarity measures associated with a dendrogram cut of k=10 is tabulated in Table .

Node degree histograms and rose plots depict the differences in network topology and fracture orientations between the identified clusters relating to FP (Fig. a), DM (Fig. b), and PD (Fig. c). For all three measures, the shape of rose plots indicates a transition of principal orientations smoothly across clusters. For example, in Fig. a for the fingerprint measure, the cluster n06 in the west of Region 3 has three main sets that become orthogonal in cluster n09, the nearest cluster eastwards. Cluster n05 at the eastern extremity of Region 3 has an orthogonal pattern that has rotated almost 80∘ clockwise compared to the western boundary. Orientations of fractures clusters between the eastern-most and western-most clusters show transitions between the extremal archetypes.

From the FP clustering results (Fig. a), the spatial variation appears to have an E–W trend. From the dendrogram, an eastern branch comprising clusters n6, n9, n4, n11a, and n14 and a western branch consisting of clusters n5, n28, n11b, and n27 can be identified. An outlier branch with cluster n2 appears at the interface between the eastern and western branches. Detailed visualization of archetypal subgraphs relating to each of the FP clusters is presented in Fig. . The dendrogram structure and spatial clustering for the DM (Fig. b) depicts a central region represented by a branch containing clusters n24, n15, n6, and n16. The eastern and western peripheries organize as clusters n1, n5a, n5b, n25, n3, and n17 under a second branch. Underneath this branch, clusters n1, n5a, and n5b correspond to extremities of the Region 3, which are not fully sampled. The dendrogram structure for the PD (Fig. d) is similar with clusters n7, n23, n25 organizing under the branch representing the central region and clusters n1, n5, n21, n4, n16, n3, and n12 forming the eastern and western peripheral regions. Clusters n1 and n5 pertain to extremities of Region 3, which are not fully sampled. Figures  and depict zoomed-in sections of the subgraphs relating to each of the top clusters that confirm the detected intra-regional variation for DM and PD, respectively.