In contrast, a hyperbolic plane would need to be compressed in order to fit onto a flat plane. To simulate the effect of compressing hyperbolic space onto a flat surface, the researchers used a special type of resonator called a coplanar waveguide resonator. When microwave photons pass through this resonator, they behave in the same way whether their path is straight or meandering.
Looking at the chip's central heptagon is akin to looking through a fisheye camera lens, in which objects at the edge of the field of view appear smaller than in the center -- the heptagons look smaller the farther they are from the center.
This arrangement allows microwave photons that move through the resonator circuit to behave like particles in a hyperbolic space.
The chip's ability to simulate curved space could enable new investigations in quantum mechanics, including properties of energy and matter in the warped space-time around black holes. The material could also be useful for understanding complex webs of relationships in mathematical graph theory and communication networks. Most applications of the material would require "doing something to make it so that they can tell there's another photon there.
Mattias Fitzpatrick, who graduated with a Ph. Journal Nature. DOI In order to treat discrete objects like graphs, mathematicians have developed several alternative and somehow simplified versions of the Ricci curvature, each highlighting and emphasizing different aspects of the underlying network. This freedom of choice has lent urgency to a rigorous study on how the different curvatures actually compare when put to test on specific examples like road networks, biological networks, and collaboration networks.
Euro road network where nodes are cities and edges are roads connecting different European cities. Size of nodes and width of edges are proportional to the absolute value of the corresponding Forman-Ricci curvature.
This work constitutes a major first step in this direction, providing not only an empirical comparison of the behavior of the most used discrete versions of the Ricci curvature, namely the Ollivier—Ricci and the Forman—Ricci curvature, but also an affirmative clear-cut answer.
After analyzing several different networks and data sets, coming from various fields including the social sciences and biology, the researchers in this interdisciplinary team were able to deduce that the two notions, despite being different in spirit, yield very similar results. Interestingly, the comparison works especially well on real existing networks, like the European road network depicted in the figure.
This opens an avenue for the analysis of large networks. This paper is a collaborative project of mathematicians in cooperation with computer scientists and biologists. Homepage Newsroom From the Institutes Meaningful relationships - mathematical insights into the geometry of complex networks. Meaningful relationships - mathematical insights into the geometry of complex networks.
June 07, Networks of very different type — from social to biological, from economical to technical — are shaping our modern world, and for many years, scientists are investigating their structures and function. Their recent promising results are now published in Scientific Reports.
Areejit Samal, R.
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