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Geometric renormalization unravels self-similarity of the multiscale human connectome

Structural connectivity in the brain is typically studied by reducing its observation to a single spatial resolution. However, the brain poses a rich architecture organized in multiple scales linked to one another. We explored the multiscale organization of the human connectome using a dataset of healthy subjects reconstructed at five different resolutions. We find that the structure of the human brain remains self-similar when the resolution length is progressively decreased by hierarchical coarse-graining of the anatomical regions. Strikingly, geometric renormalization of connectome maps in hyperbolic space, which decreases the resolution by coarse-graining and averaging over short similarity distances, predicts the properties of connectomes including self-similarity. Our results suggest that the same principles regulate connectivity between brain regions at different length scales and that the multiscale self-similarity of brain connectomes may offer an advantageous architecture for navigation purposes. The implications are varied and can affect fundamental debates, like whether the brain is working near a critical point, and lead to applications including advanced tools to simplify the digital reconstruction and simulation of the brain.


INTRODUCTION
Extensive study of the topology of the human connectome [1][2][3] revealed characteristic features of complex networks, including the small-world phenomenon [4][5][6], high levels of clustering [6], heterogeneous degree distributions (even though not scale-free) [7,8], rich club effect [9], and community structure [10,11]. These structural features have been typically observed at specific scales fixed by the resolution of the experimental imaging technique and the subsequent coarse-graining of the data into brain regions. Only very recently did the structure of the brain began to be considered at multiple resolutions simultaneously [12,13], which now calls for novel methodological advances to understand its multiscale nature and, in particular, how these different scales are interrelated.
In the context of complex networks, the study of the multiscale problem-and related concepts like scale invariance and self-similarity [14]-is built upon the renormalization technique of statistical physics [15,16], which successfully explained the universality of critical behavior in phase transitions [17] by recursive averaging over short-distance degrees of freedom [18]. The first efforts to renormalize complex networks were mainly based on a box-covering procedure where distances between nodes are measured in terms of the length of shortest paths [19]. However, shortest paths are a poor source of length- * marian.serrano@ub.edu based scaling in networks due to the small-world property [20] that introduces correlations between the coexisting scales.
A successful alternative [21] is based on a class of geometric network models in which the topology is coupled to a hidden metric space through a universal distancebased connectivity law [22]. This approach revealed that the effective geometry of real networks is hyperbolic [23], and explains universal features shared by many real networks [22]-including the small-world property, scale-free degree distributions, and clustering-as well as fundamental mechanisms, such as preferential attachment in growing networks [24] and the emergence of communities [25,26]. Effective distances in the geometric network model can be estimated by embedding networks in the hyperbolic disk [27]. These maps sustain efficient navigation [27], a remarkable finding that is also valid for the brain [28], and also provide a valid source of length-based scaling factors allowing to explore networks at different resolutions via a geometric renormalization (GR) technique [21]. This technique revealed that real scale-free networks show geometric scaling and can be unfolded into a shell of self-similar layers that distinguishes the coexisting scales and their interactions.
In this work, we focus on the multiscale nature of the human connectome and show that GR predicts its scale invariance with high fidelity. We use a high-quality cohort dataset of healthy subjects [12]. First, we investigate the properties of connectomes at different resolutions obtained by a hierarchical coarse-graining of the anatomical regions, and we identify scale invariant properties. Second, we apply GR to the hyperbolic map of the highest resolution layer of the connectomes to obtain a multiscale unfolding, or GR shell, and find a striking congruency between the empirical observations at each scale and the predictions given by the model. Third, we explore the effects of impairing the geometric properties of connectomes on self-similarity and navigation. Altogether, our results indicate that the same rules explain the formation of short and long range connections in the brain-within the range of length scales covered by the dataset-, and support GR as a valid archetypical model for the multiscale structure of the human brain.

SELF-SIMILARITY OF THE MULTISCALE HUMAN (MH) CONNECTOME
We analyze the multiscale organization of the connectomes [12] of 40 healthy human subjects. For each subject, the multiscale connectome is organized in five layers with different resolutions. Nodes in each layer correspond to parcels in the cortical and subcortical regions (the brainstem is excluded). We enumerate the layers starting from the highest resolution (HR) connectome at layer l = 0, containing N = 1014 nodes, while networks at lower resolutions have been defined with 462, 233, 128, and 82 nodes (different subjects may present negligible fluctuations, see Table S1 in SI). The parcellations at different resolutions are spatially hierarchical, with a correspondence between the nodes at different length scales as defined by the coarse-grained regions. The hierarchical decomposition was obtained by grouping sets of 2 or 3 neighboring brain regions to build a partition with decreased resolution where connection densities were recomputed between each pair of resulting parcels. The operation was repeated several times until the 82 parcels at the lower resolution scale were recovered. See also Supplementary Information (SI).
For each layer l of each subject, we measured the following properties: the complementary cumulative degree distribution P (l) c (k (l) res ), degree-degree correlations using the normalized average nearest-neighbour degreek res ) [29], the community structure detected by the Louvain method [30], the average degree and the average clustering coefficient. These quantities were calculated as a function of the rescaled degree k (l) res = k (l) / k (l) to account for the variation of the average degree across layers. We found that these measures yield similar results across all subjects; Fig. 1 provides the results for a typical individual. Results for all subjects can be found in SI Figs. S1-S6.
We observe a clear self-similar behavior across layers for the degree distributions, degree-degree correlations, and clustering spectra (see Fig. 1a-c). The rich club effect in Fig. 1d also shows significant self-similarity, although its intensity is progressively diluted as the resolution is decreased. Figure 1e shows the modularity Q (l) emp of the detected community partition, and the adjusted mutual information AMI (l) emp between the community partition directly detected in layer 0 and the community partition induced in layer l = 0 by that in layer l -with modularity Q (l,0) emp -(see Methods). The overlap between communities at different resolutions remains important even if the modularity is slightly weakened, especially in the last two layers. The average degree k (l) , shown in Fig. 1f, increases very weakly at l = 1, but this trend changes to a clear decrease in the last two layers 4 and 5. This points to finite size effects, also noticeable in the behavior of the average clustering which increases mildly from layer 0 until layer 3 and more pronouncedly in the last two layers (causing the shift observed in the correspondinḡ c (l) (k (l) res ) curves in Fig. 1b). The mild dilution effect in the intensity of rich-club behavior and modularity could be related to an increased sensitivity of mesoscopic metrics (as compared to local measurements like degree or clustering) to finite size effects.

GEOMETRIC RENORMALIZATON OF THE HUMAN CONNECTOME
We now show that the scale invariance of the MH connectome, as revealed by hierarchical coarse-graining of anatomical regions, is fully consistent with the multiscale unfolding obtained by successive applications of GR [21] to the HR connectome (i.e., layer 0 of the MH connectome). To do so, we first obtained a hyperbolic map of the HR connectome in which its 1014 brain regions are represented as nodes and are assigned two coordinates. The first coordinate, the hidden degree, is related to the observed degree of a given node in the network and thus to its popularity. The second coordinate corresponds to the angular position on a circle and encodes the similarity between nodes by aggregating all other attributes that modulate the likelihood of connections. These coordinates are inferred by maximizing the congruency of the HR topology with the hidden metric space network model [27,31], where the likelihood of connections depends on effective distances in the latent geometry (see Methods). Critically, the similarity dimension provides an adequate source of length scales to apply the renormalization transformation [21].
New maps at lower resolutions are then obtained by applying GR iteratively. Doing so allows to zoom out from the high-resolution layer and, thus, to capture progressively longer range connections. This technique proceeds by forming non-overlapping blocks of consecutive pairs of nodes along the similarity dimension, independently of whether they are connected or not, thereby coarsegraining the blocks into supernodes. Each supernode is then located in a new map within the angular region defined by its corresponding block in the original map so that the angular ordering of nodes is preserved. The links between nodes belonging to two different blocks, if any, are renormalized into one single link between the two corresponding supernodes. We repeated this procedure 4 times to generate a 5-layer multiscale shell of the original network. The first layer (l = 0), the highest resolution (HR) layer of the MH connectome, contains 1014 nodes. Each subsequent layer generated by GR has 2 l times fewer nodes (508, 254, 127, and 64 nodes with l = 1, 2, 3 and 4), whereas the corresponding layers in the MH connectome have typically 462, 233, 128, and 82 nodes respectively.
We also embedded each layer of the MH connectomes separately for comparison purposes. Figure 2 shows both the collection of maps from individual embedding of the MH layers and the GR shell produced from the HR connectome for a typical subject. The colors are given according to the 82 neuroanatomical regions represented by the nodes in layer l = 4. Notice that nodes corresponding to the same original, lower-resolution region remain angularly close at all scales. This implies that the inferred coordinates in the individual hyperbolic maps (the MH maps) encode significant information on the anatomical structure of the connectomes [28]. In addition, the maps in the GR shell show very similar color patterns, which indicates that the GR flow preserves to a large extent the structure of the lower-resolution regions encoded in the HR layer, and so preserves the spatial hierarchical organization of the regions used in the construction of the empirical MH connectomes.
We compared the topological properties of the MH connectomes shown in Fig. 1 with those computed for each layer in the GR shell. The results for a typical subject correspond to the solid lines on Fig. 1; see Figs. S1-S6 in the SI for the results for the remaining subjects). Strikingly, we observe that the agreement between the curves for the degree distributions, degree-degree correlations, clustering spectrum, rich club, and average degree and clustering is excellent at every scales, despite that the layers in the MH connectome come from empirical data and those in the GR shell are predictions of the GR method (note that the perfect overlap for l = 0 is trivial since the two networks are the same). Additionally, Fig. 1e shows the modularities Q GR in the GR shell (see Methods). The community structure is preserved to a great extent in the flow with values for the adjusted mutual information similar to those measured in the MH connectome. We also report the overlap between topological communities in the MH connectomes at each layer and the GR flow measured in their projection over layer 0 (to avoid the problem due to the different number of nodes), AMI (emp,GR) 0 . As with the rest of features, the community structure observed in the empirical brain networks is also well approximated by the GR shell.
Finally, Fig. 3 shows the empirical connection probabilities as a function of Euclidean distance (3D separation between region centers) in the MH connectome, and as a function of the effective hyperbolic distance in the GR shell. Finite size effects aside, the curves show scale invariance in Euclidean and in hyperbolic spaces, as expected given the self-similarity of the topological features shown in Fig. 1a-d. In Euclidean space, the curves overlap only when distances are rescaled by specific values obtained ad hoc (reported in the caption of Fig. 3). Inter-estingly, the curves in the GR shell overlap naturally due to the renormalizability of the geometric network model on which the GR technique is based. Despite the scaling of the probability of connection in Fig. 3a-b, Euclidean distances alone do not contain enough information to explain the connectivity properties of the MH connectome, as expected [32]. Indeed, a geometric model purely based on Euclidean distances would produce geometric random graphs lacking key features of real complex networks such as the small-world property. Node degrees are also an important factor and, to take it into consideration, we used the connection probability as in the S 1 model (see Methods) but using Euclidean distances x ij instead of similarity distances ∆θ ij . We also approximated the hidden degrees κ by the actual degree k (the two are are very similar, see Fig. S7 in SI, so that the topological features of networks obtained by using one or the other are also very similar). The values of β and µ were adjusted to match the clustering and average degree of the empirical connectome. As shown in Fig. 3d-f, Euclidean distance is certainly an important factor but not the only one determining the similarity distance needed to reproduce reliably the topological features of the MH connectome. In contrast, the fit of the S 1 model based on similarity distance, underlying the geometric renormalization technique and where distances are effective and incorporate other factors than Euclidean distance, is very good.
These results indicate that GR predicts naturally the scale invariance and the self-similarity of the multiscale human connectome with high accuracy, even if it requires only the information measured at a single length scale to provide rescaled layers that mimic closely (at the statistical level) the structure of the brain at higher scales. Notice that we did not add any information about the anatomical coarse-graining of brain regions in the MH connectome when going from one resolution to another in the renormalization process, we just estimated the similarity space from the HR empirical data, and used consecutive nodes in this space to produce the structure of each renormalized layer.

SELF-SIMILARITY AND NAVIGABILITY
Hyperbolic network maps sustain efficient navigability [27], a remarkable finding that is also valid for the brain [28]. To check the navigability properties of connectomes at different resolutions, we implemented greedy routing, a decentralized communication protocol in which a source node transmits a message along to its neighbor that is the closest to a target node in the metric space [33]. The performance of greedy routing is measured by the success rate, p s , and the average stretch of successful greedy paths,s. The success rate consists in the fraction of successful greedy paths when considering the N (N − 1) possible pairs source-target. Note that the success rate is typically lower than one since greedy routing does not guarantee that a message will reach its Euclidean S1 Standard S1 target node; greedy routing may send the message to an already visited node and therefore may get trapped in a loop. The average stretch of successful paths consists in the ratio of the number of links in the successful greedy path, and of the number of links in the topological shortest path, averaged over all successful greedy paths. Navigation is considered maximally efficient if the success rate and the stretch are both equal to one, meaning that all messages reach destination following shortest paths.
We studied navigation in the anatomical Euclidean embeddings, the collection of individual MH embeddings in the hyperbolic plane, and the GR shell for all the subjects in the cohort. Remarkably, the variability between subjects is very low, the results are shown in Fig. 4. In both geometries, there is a systematic trend towards a more efficient navigation, both in terms of success rate and stretch, as the resolution scale is decreased and longer range connections progressively dominate, and as the density of connections increases. However, the navigability of the hyperbolic maps is higher, as pointed before in [28], with larger success rates and lower stretch values. Navigability is the most efficient for the last two layers and independent of the underlying geometry due to the high density of connections. Interestingly, the efficiency of the navigation protocol in the GR shell is in perfect congruency with that in individual embeddings of the MH layers (see Fig. S8 in SI).
To understand the interplay between self-similarity and navigability, we compared the navigability of the GR connectome shells and of a null model multiscale network where self-similarity is destroyed while the topological properties of the HR layer are preserved. More specifically, we randomized the positions of nodes in layer 0 by swapping the coordinates θ i and θ j of pairs of nodes i and j selected at random. Then, we applied the geometric renormalization technique to the randomized network to obtain a randomized GR shell. Figures 4d-f show that this method destroys self-similarity in the renormalized layers. In addition, as shown in Figs. 4a and b, the efficiency of greedy routing in the randomized layers decreases dramatically (except in the last two layers due to the high density of connections as explained above), which suggests that multiscale self-similarity of brain connectomes may offer an advantageous organization for multiscale navigation purposes leveraging on the navigability properties at each scale.

DISCUSSION
The structure of the human brain displays widely different length scales, which magnifies its complexity, otherwise constrained by overarching patterns. We proved that the MH connectome is self-similar when the length resolution is progressively decreased by hierarchical coarse-graining of the anatomical regions. We used a high-quality dataset [12] consisting in the MH connectomes of 40 healthy subjects and displaying a remarkable level of homogeneity in the performed measurements. The connectomes are organized in five different layers, with decreasing spatial resolution, and whose size in terms of the number of regions reduces from about 1000 regions down to 128 and 82 nodes. The fact that topological observables on these last two layers have been shown to be affected by their finite size suggests an upper bound on the scale for which the MH connectome is selfsimilar. However, our results did not suggest any lower bound for the self-similarity of the MH connectome; a conclusion that will require higher resolution datasets of the MH connectome with more refined brain parcellations to be further investigated.
The observed scale invariance is unraveled by the application of GR to the highest resolution layer, which is based on hyperbolic space as the effective geometry of the brain. The high fidelity of the GR shell to the empirical data, including community structure, suggests that the same principles govern the connectivity between brain regions at different length scales, and that connectivity at lower resolutions can be inferred from observations at higher resolutions. Our findings call into question the principle that the brain works on the basis of independent centers. Its multiscale organization denotes instead a strong hierarchical ordering and integration at different length scales with a single connectivity law ruling short and long range connections. This solves the apparent tension between shortcuts providing global integration and the persistence of non-local features like modularity in brain networks [34].
One possible explanation for the observed selfsimilarity of the brain, and of complex networks in general, is that it offers an efficient way to organize their hierarchical structure for navigation purposes. While it has been modeled using different protocols, from shortest paths to random diffusion [35][36][37][38], control and regulation of communication in the brain is the subject of ongoing debates [37]. On the one hand, despite its elegant simplicity, shortest paths navigation relies on the unrealistic assumption that neural elements possess global knowledge of the network topology. On the other hand, random diffusion needs bias to travel via efficient routes. There is, however, evidence indicating that targeted information processing may play an important role in brain commu-nication dynamics. For instance, hippocampal neurons can transmit distinct behavior-contingent information selectively to different target areas [39]. Greedy routing protocols could, nonetheless, offer a simplified yet fundamental illustration of communication protocols in the brain [28,38,40].
Beyond technical considerations, the implications of our findings are varied and can affect general fundamental debates in neuroscience, such as whether the brain is a system working near a critical point. Recently, theoretical and empirical results have supported the hypothesis that the collective dynamics of large neuronal networks in the brain naturally operates near criticality to ensure optimal memory and information processing capability with fast susceptibility and adaptability to the state of the environment [41][42][43][44][45][46][47]. Our findings offer a complementary view from the perspective of brain structure, which presents signatures of a critical regime, like long range order across scales. It is then plausible that one of the causes of the scale-invariant dynamics observed in the brain is the scale invariance of the underlying connectome.
At the level of applications, both the scale invariance of the brain structure and the existence of a model that unravels its self-similarity may have an important impact in the development of advanced tools that simplify its digital reconstruction and simulation. At the same time, anatomical brain mapping techniques could also benefit from our results, suggesting that the number of regions in a brain atlas is an important question. Specific details at the smallest of the considered scales could be redundant when informing about the large-scale structural organization of the brain while an insufficient number of regions could bias the observations. Another potential advantage of the self-similarity of brain connectomes is that it can be used to detect possible biases, depending on length scales, associated with the different data preprocessing methods and brain mapping techniques. Finally, immediate follow-ups of our work include studies to assert the renormalizability of functional brain networks and alterations in renormalizability produced by normal aging or possible brain disorders.

Hidden metric space network models and embedding
In the S 1 model [22], the popularity of a node i is quantified by its hidden degree κ i , and its angular position θ i in a one-dimensional sphere (or circle) serves as a proxy for its similarity with the other nodes. The probability of connection between any pair of nodes takes the form of a gravity law, whose magnitude increases with the product of the hidden degrees (i.e., their combined popularities), and decreases with the angular distance between the two nodes. In other words, similar nodes are angularly closer and are therefore more likely to be connected. Specifi-cally, nodes i and j are connected with probability where N is the number of nodes, µ controls the average degree of the network, β controls the level of clustering, and ∆θ ij = π − |π − |θ i − θ j || is the angular distance between nodes i and j. Apart from minor corrections due to the finite size of the network, the hidden degrees κ i and κ j are proportional to the degrees of nodes i and j, respectively. There exists an equivalent purely geometric modelthe H 2 model [23]-in which the popularity and similarity dimensions are combined into a single distance in the hyperbolic plane such that closer nodes are more likely to be connected. In this representation, the angular coordinates remain as in the S 1 model, but the hidden degrees in the S 1 model are transformed into radial coordinates according to where the radius of the two-dimensional hyperbolic disk containing all nodes is with κ 0 = min({κ i }). Higher degree nodes are therefore located closer to the center of the H 2 disk. Substituting Eqs. (2) and (3) into Eq. (1) yields where x ij = r i + r j + 2 ln ∆θij 2 is a very good approximation of the hyperbolic distance between two points with coordinates (r i , θ i ) and (r j , θ j ) in the hyperbolic disk. The exact hyperbolic distance x ij is given by the hyperbolic law of cosines: x exact ij = acosh(cosh r i cosh r j − sinh r i sinh r j cos ∆θ ij ) .
The hyperbolic MH maps displayed on the left-hand side of Fig. 2 and, more importantly, used as the starting point of the GR process were obtained using the algorithm introduced in Ref. [48]. More precisely, these maps were inferred by finding the hidden degree and angular position of each node, {κ i } and {θ i }, that maximize the likelihood L that the structure of the network was generated by the S 1 model. In other words, they were obtained by maximizing where p ij is given at Eq. (1), and where {a ij } are the entries of the adjacency matrix of the network (a ij = a ji = 1 if nodes i and j are connected, a ij = a ji = 0 otherwise). In Fig. S9, we show the topological validation of the embedding of the HR network for subject No. 10.

Geometric renormalization
The geometric renormalization technique for complex networks was introduced on Ref. [21]. The approach relies on a geometric embedding of the networks to coarsegrain neighbouring nodes into supernodes. The renormalization transformation zooms out by changing the minimum length scale from that of the original network to a larger value. The process that we apply to the HR connectome is summarized as follows: First, starting from the original network in l = 0, the geometric map is obtained by embedding the nodes in the underlying geometry, identifying hidden degrees κ (0) i and angular coordinates in a one-dimensional sphere (circle) θ (0) i in the formulation of S 1 model. Second, we define non-overlapping blocks of consecutive nodes of size r = 2 along the similarity circle and apply the coarse-graining by merging the nodes in the blocks to supernodes. Each supernode is then placed within the angular region defined by the corresponding block so that the order of nodes is preserved. The renormalized networks remain maximally congruent with the hidden metric space model by assigning a hidden degree κ (l+1) i to supernode i in layer l + 1 as: as well as an angular coordinate θ (l+1) i given by Global parameters are rescaled as µ (l+1) = µ (l) /r, β (l+1) = β (l) , and N (l+1) = N (l) /r. This implies that the probability in Eq. (1) for two supernodes i and j to be connected in layer l + 1 maintains its original form independently of the layer. Third, we connect two supernodes i and j in layer l + 1 if and only if, in layer l, some node in i is connected to some node in j.
This operation is iterated starting from the original network at layer l = 0. Finally, the set of renormalized network layers l forms the multiscale shell of the network.

Renormalization flow of the community structure
To asses how the community structure of the empirical MH connectomes and of the GR unfolding change with the resolution scale, we obtained the community partitions P GR for every layer l using the Louvain method [30]. Notice that the community structure of layer 0 is the same in both cases. We also defined the partition induced by P Informed written consent in accordance with the Institutional guidelines (protocol approved by the Ethics Committee of Clinical Research of the Faculty of Biology and Medicine, University of Lausanne, Switzerland) was obtained for all subjects. Forty healthy subjects (16 females; 25.3 ± 4.9 years old) underwent an MRI session on a 3T Siemens Trio scanner with a 32-channel head coil. Magnetization prepared rapid acquisition with gradient echo (MPRAGE) sequence was 1-mm in-plane resolution and 1.2-mm slice thickness. DSI sequence included 128 diffusion weighted volumes + 1 reference b0 volume, maximum b value 8000 s/mm 2 , and 2.2 x 2.2 x 3.0 mm voxel size. EPI sequence was 3.3-mm in-plane resolution and 3.3-mm slice thickness with TR 1920 ms. DSI and MPRAGE data were processed using the Connectome Mapper Toolkit [49]. Each participant's gray and white matter compartments were segmented from the MPRAGE volume. The grey matter volume was subdivided into 68 cortical and 15 subcortical anatomical regions, according to the Desikan-Killiany atlas, defining 83 anatomical regions. These regions were hierarchically subdivided to obtain five parcellations, corresponding to five different scales [12]. Whole brain deterministic streamline tractography was performed on reconstructed DSI data, initiating 32 streamline propagations (seeds) per diffusion direction, per white matter voxel [50]. Within each voxel, seeds were randomly placed and for each seed, a fiber streamline was grown in two opposite directions with a 1mm fixed step. Fibers were stopped if a change in direction was greater than 60 degrees/mm. The process was complete when both ends of the fiber left the white matter mask. For each individual subject, connection weights between pairs of ROI are quantified as a fiber density [51]. Thus, the connection weight between the pair of brain regions {u, v} captures the average number of streamlines per unit surface between u and v, corrected by the average length of the streamlines connecting such brain regions. The aim of these corrections is to control for the variability in cortical region size and the linear bias toward longer streamlines introduced by the tractography algorithm. Fiber densities were used to construct individual subject structural connectivity matrices. Each matrix is then modeled as the weighted adjacency matrix W = w ij of a graph G = V,G with nodes V = v 1 , ..., v n representing ROIs, and weighted, undirected edges E = e 1 , ..., e m representing anatomical connections with their fiber densities.
The present study considers the unweighted version of the connectivity adjacency matrices and discards the brainstem (one node) for all subjects. We have also removed nodes that were isolated in the original dataset due to fluctuation in the data acquisition experiment, nodes that became isolated after the removal of the brainstem region, and nodes that were only connected to themselves by a self-loop. These adjustments cause some small variations in the number of nodes of the highest resolution layer from one subject to another. The highest resolution layer comprises typically 1014 equal sized regions of interest (ROI), which are then coarse-grained into 462, 233, 128, and 82 regions at higher resolutions.                S1. Overview of the considered connectomes. The number of nodes (N ), the number of links (L), the density of links (ρ = 2L/N (N − 1)), its average degree ( k = 2L/N ), the average local clustering coefficient ( c ), the assortativity coefficient (rc), the modularity (Q), the number of the communities (N c), and the hyperbolic embedding parameter β and µ.