) †The first two authors should be regarded as joint First Authors The human brain displays a complex network topology, whose structural organization is widely studied using diffusion tensor imaging. The original geometry from which emerges the network topology is known, as well as the localization of the network nodes in respect to the brain morphology and anatomy. One of the most challenging problems of current network science is to infer the latent geometry from the mere topology of a complex network. The human brain structural connectome represents the perfect benchmark to test algorithms aimed to solve this problem. Studying the brain as a network of interconnected nodes and the recent developments of network theory, contributed to unveil the key structural principles underlying the topology of the healthy human brain 1,2 . One of the peculiar rules on which brain topology relies is the tendency of the network nodes to cluster into modules with high efficiency and short path length, thus reflecting an intrinsic small-world architecture, functionally segregated (local clustering) and integrated (global efficiency) [3][4][5] . Indeed, structural magnetic resonance (MR) studies based on diffusion tensor imaging (DTI, an imaging technique that allows to perform tractography and connectome reconstruction) have demonstrated that the human brain shows a modularity structure, consisting of around five to six modules, corresponding to known functional subsystems 3,6 . In addition to the existence of a structural core 3 , the brain seems to exhibit a rich-club organization, with highly connected and central nodes (hubs) having a strong tendency to be mutually interconnected, thus constituting a focal point for whole-brain Such topological patterns of connectivity are often intricately related with the physical distances between elements in brain networks. On one hand, brain regions that are spatially close have a relatively high probability of being interconnected, on the other hand, longer white matter projections are more expensive in terms of their material and energy costs, thus making connections between spatially far brain structures less likely 8 . Interestingly, the topology of many real networks seems to be characterized by a latent hyperbolic geometry and the hyperbolic space is a promising universal space of representation for real networks, preserving many of their fundamental topological properties 9 . However, mapping a given real network to its hyperbolic space remains still an open challenge. The Popularity Similarity Optimization (PSO) model proposed that the trade-off between node popularity and similarity contributes to establish new connections and the hyperbolic space offers a congruous geometrical representation for this mechanism of self-organization 9 . In particular, according to the PSO model, the radial coordinates and the angular distances of the nodes in the hyperbolic disk respectively represent the node popularity and similarity. To the best of our knowledge, the few attempts that have been made to reveal the brain connectivity's intrinsic geometry in the Euclidean space failed 10 , and the embedding of brain connectomes is still an unexplored field. In this regard, by using game theory and threedimensional Euclidean embedding, Gulyás et al. 11 showed that the brain has a highly navigable skeleton, which reflects that the spatial organization of the brain is nearly optimal for information transfer. Ye et al. (2015) , using dimensionality reduction techniques, one linear (multidimensional scaling) and one nonlinear (Isomap), introduced a new mathematical framework that, as anticipated above, failed to represent the three-dimensional intrinsic geometry of the human brain connectome 10 , concluding that such "intrinsic geometry only minimally relates to neuroanatomy" 10 . However, the fact that the intrinsic geometry they inferred did not relate to neuroanatomy could be imputed to embedding limitations of the employed algorithms. In fact, the brain networks are physically expensive systems and it is likely that several features of real brain anatomy have been structured to control such wiring costs 8 . It is well known that human brain networks show both anatomically short-distance connections and long-distance pathways between several spatially remote modules and anatomical areas. Network geometry is a promising field and has the potential to promote a significant improvement in revealing and understanding the hidden geometry from which emerges a complex network structure in healthy and pathological conditions. Nevertheless, as we mentioned above, mapping real networks into the hyperbolic space remains an open problem. Recently, Thomas et al. 12 and Muscoloni et al. (article under revision) 13 proposed coalescent embedding, a class of topological-based unsupervised nonlinear dimension reduction machine learning able to perform efficient mapping of complex networks in the 2D hyperbolic disk, the 3D hyperbolic sphere, and potentially also in higher-dimensions. Applying this new class of algorithms, we investigated whether we were able to demonstrate unsupervisedly -exploiting only the mere topology of the brain networks -that it is possible to disclose in the hyperbolic space the fundamental properties of the hidden brain geometry, which in our case is inferred from the MR-DTI structural connectivity. We considered three different brain weighted networks, each network was obtained from a different dataset (Table 1) as the mean or median of all the weighted networks that characterized the healthy individuals included in the dataset. The network weights of the first and second datasets present a value that relates to the number of streamlines (NOS) between brain regions, while in the third dataset the streamlines distance (SD) is reported. In general, we selected datasets that differ for typology of weights and the technical details are given in the Methods and Suppl. Information. Coalescent embedding in the 2D hyperbolic disk perfectly segregates the structural networks into two distinct sections, corresponding to the left and right hemispheres ( (Table 1) , but for simplicity of narration hereafter we will show the figures of the dataset that offered the best results. In this case, we selected Dataset I (Nigro et al. and the median-network is represented in Fig. 1 . It seems that the first rule of organization of brain networks that emerges in the geometrical space is their structural segregation in two hemispheres, which is a simple concept yet quite neglected in previous studies on brain connectomics. Furthermore, we observed that also the anterior, central and posterior part of the brain were correctly allocated and segregated in the 2D hyperbolic disk (Fig. 2) . Circularseparation-scores -evaluating the matching between the angular organization of the nodes in the 2D embedding space and their real anatomical arrangement (according to node labels anterior-central-posterior) -are shown in Table 1 for the three datasets analysed. Interestingly, for the Dataset I, ncMCE offered also in this case a perfect angular node alignment (Fig. 2) . Hence, we can conclude that both left-right and frontal-central-back node arrangementobtained by coalescent embedding in the 2D hyperbolic disk -respected the brain morphology with an impressive level of accuracy, providing the first important result of this study. At this point, we were ready to investigate a more complicated hypothesis: if the structural brain networks hide a latent but clear anatomy-related geometry, the result of coalescent embedding in the 2D hyperbolic disk should respect also the traditional and well-known brain lobes organization. A previous study failed to prove this important correspondence 10 . Fig. 3 instead surprisingly confirms such hypothesis, and shows that the brain median-network of the Dataset I embedded by ncMCE segregates into spatially and anatomically distinct subnetworks corresponding to the brain lobes with almost perfect matching. Table 1 shows the circular separation scores for each of the three datasets, considering both mean and median networks for each dataset, and using all the coalescent embedding algorithms. Interestingly, the level of matching between embedding and anatomy is always high (accuracy>0.7) for at least one of the coalescent embedding algorithms in all the three datasets. In particular, we notice that ncMCE, which is a hierarchical embedding technique, offers top performance in the first dataset, while manifold techniques such as ISO and ncISO works better in the other two datasets. We speculate that these results might be related with the different strategies and linkweight-variables used to build the connectomes of the three different datasets, and we leave this technical topic open for future studies. As explained in 12,13 , among the unsupervised machine learning techniques adopted for coalescent embedding, the manifold-based (ISO, ncISO and LE) were the only ones that could be extended for mapping to the three-dimensional (3D) In addition, in Dataset III (van den Heuvel), we found that human structural brain networks exhibited a significant different geometry in two age range-specific groups (22-25 and 31-35 years old respectively). This latent geometry variation clearly emerged using coalescent embedding and considering as markers for the discrimination the average hyperbolic distance (HD) and hyperbolic shortest path (HSP) between the nodes 12,13 . On the contrary, the average of the original weights could not reveal any significant change between the two groups ( Table   2 ). We also investigated the possibility to detect age-related geometrical changes in the brain networks using the 3D hyperbolic sphere. Suppl. Table 1 confirms that all the methods and hyperbolic markers, apart from LE-3D-HSP, allow to discriminate between the two conditions using the three dimensions. However, there is not a significant improvement with respect to the 2D embedding, which appears to be enough to detect age-related between-groups variation in the latent geometry of brain networks contained in this dataset. If the age modifies the latent geometry of the healthy connectomes, pathological conditions related with brain degeneration should impair the latent geometry of patients' connectomes. Therefore, it was natural to investigate whether hyperbolic markers such as the ones described above, could be useful tools also to detect pathological-related between-groups variations in de novo drug naïve Parkinson's Disease (PD) patients in respect to healthy controls. The last interesting finding of our work is that our algorithms allow the detection of brain network geometrical pathological differences in the hyperbolic space. Indeed, we demonstrate that both the HD and HSP markers allow to uncover the altered latent geometry in the structural brain networks of the PD patients. On the contrary, also in this comparison, the mean of the weights in the original topology could not provide significant differences between the controls and PD groups (Table 3) . Then, we widened the framework from the 2D to the 3D hyperbolic space. The analysis of the 3D hyperbolic embedding revealed that all the coalescent embedding methods and all the hyperbolic markers could significantly identify the alteration in the intrinsic brain geometry of de novo drug naïve PD patients as shown in Suppl. Table 2 . However, again the 3D embedding did not offer anything more in comparison to the 2D. Before to conclude we took also in consideration the option to adopt HyperMap 15 and HyperMap-CN 16 (two alternative methods for mapping in the hyperbolic disk) for the groupcomparisons. It was not possible to run them on Dataset III (van den Heuvel) since the number of networks was too big. Given the high computational complexity of these methods, we could apply them on Dataset IV (Cacciola) only. Suppl. Table 3 shows that just HyperMap-HD allows to significantly detect latent geometry variations, but not the others HyperMap-based approaches. On one side this suggests that the HyperMap approaches generally offer a lower discriminative power, but on the other side confirms the latent geometry network modifications occurring in the 2D hyperbolic space because of the pathological condition. Taken together our results suggest that, although the human structural brain networks are weakly hyperbolic (Table 4) , the coalescent embedding algorithms still offer a powerful tool for revealing the latent brain geometry. We hope that these findings will represent a convincing starting point to bridge the gap between brain networks topology and latent geometry. Finally, we believe that the introduced methodology of connectomic investigation will open a new scenario for analysing brain disorders. The 2D geometrical space for structural brain connectome representation could be used for diagnostic and prognostic purpose and for therapeutic treatment evaluation, with possible impact across many domains of neurology and psychiatry. Methods The coalescent embedding methods have been tested on 4 structural human brain networks datasets. The first healthy controls dataset was taken from a study on structural network connectivity in Parkinson's Disease patients 14 , from which we included tractography-based networks of 30 healthy controls. For description and details of the construction of the structural connectivity matrices we refer to previous work of 14 . In 14 a thresholding procedure has been applied to retain only the pairwise connections with more than 3 streamlines. Each edge of the final matrices represented the product of the thresholded number of streamlines (NOS) and mean fractional anisotropy (FA) normalized by dividing each element by the maximum value of the matrix (NOS x FA / max value of the matrix). Table 4 summarizes some characteristics of the brain connectomes used in the present paper. In order to evaluate the latent geometry of brain networks, each network node was assigned to different anatomical classes according to three anatomical arrangements. For each dataset and every anatomical arrangement, the evaluation has been performed according to the following procedure: 1. Given a set of connectivity matrices, an average connectivity matrix is generated using either the mean or the median operator. 2. The accuracy of the coalescent embedding techniques can be improved if the network links are weighted using values that suggest the connectivity geometry 12,13 . Since the weights in the given connectivity matrices can indicate either connection strength (i.e. in I and II) or distances between the adjacent nodes (i.e. Dataset III), in the first case the values need to be reversed. The assumption is that the higher the strength the higher the similarity between the adjacent nodes, therefore the lower their distance. For every edge ( , ) the weight is reversed according to the following formula: Where is the average connectivity matrix computed at the previous point, ′ is the reversed average connectivity matrix, x and x are respectively the maximum and minimum edge weights in . The average connectivity matrix (reversed or not according to the previous point) is embedded in the two-dimensional hyperbolic space using the coalescent embedding techniques (MCE, ncMCE, ISO, ncISO and LE), which give as output the polar coordinates ( , ) of the nodes in the hyperbolic disk (for details on the methods please refer to the original publication 12,13 ). 3. The arrangement of the nodes over the angular coordinate space is compared to the annotated anatomical arrangement in order to evaluate the extent to which nodes belonging to the same anatomical class are close to each other in the angular coordinate  In order to evaluate the geometrical modifications of the brain networks corresponding to subjects in two different conditions (i.e. two age ranges in Dataset III and healthy versus pathological in Dataset IV), we assigned a geometrical marker to every network in the dataset and then we performed tests for assessing the discrimination between the two groups. For each connectivity matrix in the dataset, the geometrical marker has been assigned according to different methods. 1. The matrix is embedded in the 2-dimensional hyperbolic space using the coalescent embedding techniques (MCE, ncMCE, ISO, ncISO and LE as dimension reduction methods, with and without equidistant adjustment). Note that if the weights of the connectivity matrices indicate connection strength and not distances, they need to be reversed as described in the previous section. The marker is computed with two different options: a. Mean among all the pairwise hyperbolic distances of the nodes in the hyperbolic disk. b. Mean among all the pairwise hyperbolic shortest paths of the nodes in the hyperbolic disk, where the hyperbolic shortest path is defined as the sum of the hyperbolic distances over the shortest path. 2. The matrix is embedded in the 3-dimensional hyperbolic space using the coalescent embedding techniques (ISO, ncISO and LE as dimension reduction methods). Note that if the weights of the connectivity matrices indicate connection strength and not distances, they need to be reversed as described in the previous section. The marker is computed with the same two different options described in 1.a and 1.b, but considering distances in the hyperbolic sphere rather than in the hyperbolic disk. 3. The matrix is embedded in the 2-dimensional hyperbolic space using the HyperMap 15 and HyperMap-CN 16 techniques. Note that in this case the matrix is treated as unweighted (for details on the methods please refer to the original publications). The marker is computed with the same two different options described in 1.a and 1.b. 4. The marker is computed as the mean of all the edge weights in the original matrix. Once assigned for each method a geometrical marker to every connectivity matrix, three different tests are applied to the two populations of markers associated to the two groups in order to assess the ability of every method to discriminate the brain networks in the two conditions. The first test is the Wilkoxon rank-sum test, equivalent to the Mann-Whitney Utest, which is a nonparametric test for equality of population medians of two independent samples X and Y 23 . The other measures reported are the area under the receiver operating characteristic curve (AUROC or AUC for brevity) 24 and the area under the precision-recall curve (AUPR) 25 , two performance scores usually adopted in classification tasks. MATLAB code was used for all the methods and simulations, except for HyperMap-CN whose C code has been released by the authors at https://bitbucket.org/dk-lab/2015_code_hypermap. The simulations were carried out on a workstation under Windows 8.1 Pro with 512 GB of RAM and 2 Intel(R) Xenon(R) CPU E5-2687W v3 processors with 3.10 GHz.
REF demonstrate that the intrinsic geometry of a structural brain connectome relates to its Intrinsic geometry: the very high dimensional nonlinear geometric space natively induced by the brain's interconnectivity as encoded by the connectome, either structural or functional, instead of the neuroanatomical space.
Coalescent embedding in the hyperbolic space unsupervisedly discloses the hidden geometry of the brain
arXiv: Neurons and Cognition
Mathematics Biology Physics