New method to determine the dimensionality of complex networks through hyperbolic geometry
Oct 2022, phys.org
"The intrinsic geometry of data sets or complex networks is not obvious"Infer the dimensionality of complex networks with hyperbolic geometrics, which captures the complexity of relational structures of the real world.The study provides a multidimensional hyperbolic model of complex networks that reproduces its connectivity, with an ultra-low and customizable dimensionality for each specific network. This enables a better characterization of its structure—e.g., at a community scale—and the improvement of its predictive capability.
(configurational geometric model or SD model)
The study reveals unexpected regularities:
- extremely low dimensions of molecular networks associated with biological tissues
- slightly higher dimensionality required by social networks and the Internet
- brain connectomes are close to three dimensions in their automatic organization
The model postulates a law of interconnection of the network elements (or nodes) that is gravitational, so nodes that are closer in a similarity space —of spherical geometry in D dimensions—and with more popularity—an extra dimension corresponding to the importance of the node—are more likely to establish connections."In the study, the similarity and popularity variables are combined to give rise to the hyperbolic geometry of the model, which emerges as the natural geometry representing the hierarchical architecture of complex networks.
(It collapses the dimensions.)
"The Internet only requires D = 7 dimensions to be mapped into the hyperbolic space of our model, whereas this name is multiplied by six and scales to D = 47 in one of the most recent techniques using Euclidean space," says Professor Marián Boguñá.
via University of Barcelona: Pedro Almagro et al, Detecting the ultra low dimensionality of real networks, Nature Communications (2022). DOI: 10.1038/s41467-022-33685-z
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