The emergence of detailed maps of physical networks, such as the brain connectome, vascular networks or composite networks in metamaterials, whose nodes and links are physical entities, has demonstrated the limits of the current network science toolset. Link physicality imposes a non-crossing condition that affects both the evolution and the structure of a network, in a way that the adjacency matrix alone—the starting point of all graph-based approaches—cannot capture. Here, we introduce a meta-graph that helps us to discover an exact mapping between linear physical networks and independent sets, which is a central concept in graph theory. The mapping allows us to analytically derive both the onset of physical effects and the emergence of a jamming transition, and to show that physicality affects the network structure even when the total volume of the links is negligible. Finally, we construct the meta-graphs of several real physical networks, which allows us to predict functional features, such as synapse formation in the brain connectome, that agree with empirical data. Overall, our results show that, to understand the evolution and behaviour of real complex networks, the role of physicality must be fully quantified.
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Data to generate random LPNs and reproduce the figures are available at https://github.com/posfaim/randLPN.
Code to generate random LPNs and reproduce the figures are available at https://github.com/posfaim/randLPN.
Kadic, M., Milton, G. W., van Hecke, M. & Wegener, M. 3D metamaterials. Nat. Rev. Phys. 1, 198–210 (2019).
Scheffer, L. K. et al. A connectome and analysis of the adult drosophila central brain. eLife 9, e57443 (2020).
Banavar, J. R., Maritan, A. & Rinaldo, A. Size and form in efficient transportation networks. Nature 399, 130 (1999).
Gagnon, L. et al. Quantifying the microvascular origin of bold-fMRI from first principles with two-photon microscopy and an oxygen-sensitive nanoprobe. J. Neurosci. 35, 3663 (2015).
Dehmamy, N., Milanlouei, S. & Barabási, A.-L. A structural transition in physical networks. Nature 563, 676 (2018).
Liu, Y., Dehmamy, N. & Barabási, A.-L. Isotopy and energy of physical networks. Nat. Phys. 17, 216 (2021).
Dorogovtsev S. N. & Mendes J. F. Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford Univ. Press, 2003).
Caldarelli G. Scale-free Networks: Complex Webs in Nature and Technology (Oxford Univ. Press, 2007).
Cohen R. & Havlin S. Complex Networks: Structure, Robustness and Function (Cambridge Univ, Press, 2010).
Newman M. Networks: An Introduction (Oxford Univ. Press, 2010).
Barabási A.-L. Network Science (Cambridge Univ. Press, 2016).
Barthélemy, M. Spatial networks. Phys. Rep. 499, 1 (2011).
Horvát, S. et al. Spatial embedding and wiring cost constrain the functional layout of the cortical network of rodents and primates. PLoS Biol. 14, e1002512 (2016).
Rubinstein M. et al. Polymer Physics, Vol. 23 (Oxford Univ. Press New York, 2003).
Song, C., Wang, P. & Makse, H. A. A phase diagram for jammed matter. Nature 453, 629 (2008).
West D. B. et al. Introduction to Graph Theory, Vol. 2 (Prentice Hall, 2001).
Tarjan, R. E. & Trojanowski, A. E. Finding a maximum independent set. SIAM J. Comput. 6, 537 (1977).
Flory, P. J. Intramolecular reaction between neighboring substituents of vinyl polymers. J. Am. Chem. Soc. 61, 1518 (1939).
Hartmann A. K. & Weigt M. Phase Transitions in Combinatorial Optimization Problems: Basics, Algorithms and Statistical Mechanics (Wiley, 2006).
Brightwell, G., Janson, S. & Luczak, M. The greedy independent set in a random graph with given degrees. Random Struct. Algorithms 51, 565 (2017).
Ercsey-Ravasz, M. et al. A predictive network model of cerebral cortical connectivity based on a distance rule. Neuron 80, 184 (2013).
Bassett, D. S. & Bullmore, E. T. Small-world brain networks revisited. Neuroscientist 23, 499 (2017).
Van Mieghem P. Graph Spectra for Complex Networks (Cambridge Univ. Press, 2010).
Abbe, E. Community detection and stochastic block models: recent developments. J. Mach. Learn. Res. 18, 6446 (2017).
Viana, M. P. et al. Mitochondrial fission and fusion dynamics generate efficient, robust, and evenly distributed network topologies in budding yeast cells. Cell Syst. 10, 287 (2020).
Rees, C. L., Moradi, K. & Ascoli, G. A. Weighing the evidence in Peters' rule: does neuronal morphology predict connectivity? Trends Neurosci. 40, 63 (2017).
Udvary, D. et al. The impact of neuron morphology on cortical network architecture. Cell Rep. 39, 110677 (2022).
Nicolaou, Z. G. & Motter, A. E. Mechanical metamaterials with negative compressibility transitions. Nat. Mater. 11, 608 (2012).
Kim, J. Z., Lu, Z., Blevins, A. S. & Bassett, D. S. Nonlinear dynamics and chaos in conformational changes of mechanical metamaterials. Phys. Rev. X 12, 011042 (2022).
D’Souza, R. M., Gómez-Gardenes, J., Nagler, J. & Arenas, A. Explosive phenomena in complex networks. Adv. Phys. 68, 123 (2019).
Heroy, S., Taylor, D., Shi, F., Forest, M. G. & Mucha, P. J. Rigidity percolation in disordered 3D rod systems. Multiscale Model. Sim. 20, 250 (2022).
Barrat, A., Barthelemy, M. & Vespignani A. Dynamical Processes on Complex Networks (Cambridge Univ. Press, 2008).
Ulloa Severino, F. P. et al. The role of dimensionality in neuronal network dynamics. Sci. Rep. 6, 1 (2016).
Bianconi G. Multilayer Networks: Structure and Function (Oxford Univ. Press, 2018).
Case, D. J., Liu, Y., Kiss, I. Z., Angilella, J.-R. & Motter, A. E. Braess’s paradox and programmable behaviour in microfluidic networks. Nature 574, 647 (2019).
Kim, J. Z., Lu, Z., Strogatz, S. H. & Bassett, D. S. Conformational control of mechanical networks. Nat. Phys. 15, 714 (2019).
Kovács, I. A., Barabási, D. L. & Barabási, A.-L. Uncovering the genetic blueprint of the C. elegans nervous system. Proc. Natl Acad. Sci. USA 117, 33570 (2020).
This research was funded by ERC grant no. 810115-DYNASNET.
A.-L.B. is the founder of Foodome and ScipherMedicine companies that explore the role of networks in health and urban environments. The other authors declare no competing interests.
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Pósfai, M., Szegedy, B., Bačić, I. et al. Impact of physicality on network structure. Nat. Phys. (2023). https://doi.org/10.1038/s41567-023-02267-1