While the structural characteristics of a network are uniquely determined by its adjacency matrix1,2,3, in physical networks, such as the brain or the vascular system, the network’s three-dimensional layout also affects the system’s structure and function. We lack, however, the tools to distinguish physical networks with identical wiring but different geometrical layouts. To address this need, here we introduce the concept of network isotopy, representing different network layouts that can be transformed into one another without link crossings, and show that a single quantity, the graph linking number, captures the entangledness of a layout, defining distinct isotopy classes. We find that a network’s elastic energy depends linearly on the graph linking number, indicating that each local tangle offers an independent contribution to the total energy. This finding allows us to formulate a statistical model for the formation of tangles in physical networks. We apply the developed framework to a diverse set of real physical networks, finding that the mouse connectome is more entangled than expected based on optimal wiring.
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Source data are provided with this paper. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
Code is available for this paper at https://github.com/YanchenLiu1/GLN. All other code that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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We thank J. A. Brum and E. Towlson for useful discussions and for providing the processed Allen Institute mouse brain data, S. Cook for providing the C. elegans data, M. Viana for providing the mitochondrial network data and A. Grishchenko for 3D and data visualizations. We were supported by grants from the NSF (grant nos. 1735505 and 1734821), ERC (grant no. 810115 - DYNASNET) and John Templeton Foundation (grant no. 61006). N.D. was also supported by the Office of Naval Research (grant no. 00014-18-9-001).
A.-L.B. is the founder of Scipher, Nomix and Foodome that bring network tools to health science.
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Liu, Y., Dehmamy, N. & Barabási, AL. Isotopy and energy of physical networks. Nat. Phys. 17, 216–222 (2021). https://doi.org/10.1038/s41567-020-1029-z
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