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Isotopy and energy of physical networks


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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Fig. 1: Graph linking number.
Fig. 2: Energy and GLN.
Fig. 3: Statistical physics of physical networks.
Fig. 4: Modelling brain layout.

Data availability

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 availability

Code is available for this paper at 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).

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Authors and Affiliations



Y.L. performed the mathematical modelling, developed the algorithm, ran and analysed the simulations, generated the figures, and contributed to writing the manuscript. N.D. contributed to the mathematical modelling, running the simulations and writing the manuscript. A.-L.B. contributed to the conceptual design of the study and was the lead writer of the manuscript.

Corresponding author

Correspondence to Albert-László Barabási.

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Competing interests

A.-L.B. is the founder of Scipher, Nomix and Foodome that bring network tools to health science.

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Supplementary information

Supplementary Information

Supplementary Figs. 1–16, discussion and Table 1.

Source data

Source data Fig. 2

Statistical data for Fig. 2c–f.

Source data Fig. 3

Statistical data for Fig. 3c,d.

Source data Fig. 4

Statistical data for Fig. 4c,d.

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Liu, Y., Dehmamy, N. & Barabási, AL. Isotopy and energy of physical networks. Nat. Phys. 17, 216–222 (2021).

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