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The fractal-hyperbolic geometry of networks

The fractal-hyperbolic geometry of networks

Network geometry helps us better understand complex systems at all sizes of organizations, as well as the collective phenomena that emerge from their information flow. Being useful in a wide range of applications, from understanding how the brain functions to Internet routing, a variety of approaches have been employed to study complex networks from different perspectives, leading to novel fundamental insights. One such approach is geometry.

A multidisciplinary team examines the most important theoretical and practical breakthroughs in network geometry techniques, as well as future research directions and problems in this frontier of complexity research:

We review three major research directions in network geometry: the self-similar fractal geometry of network structure, the hyperbolic geometry of networks’ latent spaces, and the geometry induced by dynamic processes, such as diffusion, in networks.

In the study of network geometry, understanding how the dynamical features of network growth processes influence their asymptotic self-similar patterns is an exciting and yet unexplored topic. The process of zooming-out used in the study finds the (statistically equivalent) inverse in the dynamics of the explored network so that the varying structures observed at increasing length scales correspond to the evolution of certain dynamical variables. For example:

In the simple case of growing trees, it has further been proved that this connection is a consequence of the explicit dependence of the fractal dimension on the growth rates ruling the system’s evolution.

Zooming out on complex networks: Shortest-path-distance renormalization. a) Demonstration of the box covering technique. b) Demonstration of the shortest-path-distance.
Zooming out on complex networks: Geometric renormalization. a) Similarly to A), the renormalization transformation zooms out by changing the minimum length scale from that of the original network to a larger value, this time in the similarity space. First, non-overlapping blocks of consecutive nodes are defined along the similarity circle. Second, the blocks are coarse-grained into supernodes. Each supernode is then placed within the angular region defined by the corresponding block so that the order of nodes is preserved. Finally, two supernodes are connected if any of their constituents were in the precursor layer. b) Hyperbolic embedding of the human metabolic network and its renormalized layer l = 2. The colours of the nodes correspond to the community structure detected by the Louvain algorithm. Notice how the renormalized network preserves the original community structure despite being four times smaller.

The finding of complex networks’ self-similarity under adequate scale transformations provided the first proof that they had some nontrivial geometric features. The fractal geometry of networks enables casting the self-similar symmetries underlying the organization of complex systems under the three pillars of scaling, universality, and renormalization.

The hidden geometry can be explained at a more fundamental level in terms of “latent hyperbolic geometry,” which has applications in a variety of areas. Small-worldness, degree heterogeneity, clustering, community structure, symmetries, and navigability are all logical explanations for the construction of genuine complex networks in these latent hyperbolic spaces.

Furthermore, hyperbolic geometry has strong ties to self-similar metric spaces, implying that the fractal exponents observed in self-similar networks may have acceptable counterparts in their corresponding latent spaces and that this theory might be applied to pure small-world structures.

See Also

Networks in latent geometry. a) One dimensional sphere b) Two-dimensional hyperbolic disk c) Critical Gap Method d) Metabolic map of bacterium E.coli e) Internet at the Autonomous Systems level f) World trade map in 2013 g) Finding a path from A to F h) Proximity of shortest paths in hyperbolic networks to hyperbolic geodesics.

This approach has impacted areas as diverse as mathematics, neuroscience, and machine learning. In neuroscience, geometric navigation offers a possible explanation and a mechanism for the routing of information in the brain. For example, this model has been recently used to create a hyperbolic map of the human olfactory space.

While network geometry is still a developing field, it provides a novel theoretical framework for gaining profound insights into the underlying principles of complex systems and, more broadly, physical reality.

Network geometry, Marián Boguñá, Ivan Bonamassa, Manlio De Domenico, Shlomo Havlin, Dmitri Krioukov & M. Ángeles Serrano

Published: January 2021
DOI: 10.1038/s42254-020-00264-4

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