Abstract
Complex systems must respond to external perturbations and, at the same time, internally distribute information to coordinate their components. Although networked backbones help with the latter, they limit the components’ individual degrees of freedom and reduce their collective dynamical range. Here we show that real-world networks balance the loss of response diversity with gain in information flow. Encoding network states as density matrices, we demonstrate that such a trade-off mathematically resembles the thermodynamic efficiency characterized by heat and work in physical systems, providing a variational principle to macroscopically explain the sparsity and empirical scaling law observed in hundreds of real-world networks across multiple domains, both analytically and numerically. We show that the emergence of topological features such as modularity, small-worldness and heterogeneity agrees with maximizing the trade-off between information exchange and response diversity from middle to large temporal scales. Our results suggest that the emergence of some of the most prevalent topological features of real-world networks might have a thermodynamic origin.
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The data necessary to reproduce the results of this work are available in Supplementary Data 1.
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Acknowledgements
M.D.D. acknowledges partial financial support from the Human Frontier Science Program Organization (HFSP ref. RGY0064/2022), from the University of Padua (PRD-BIRD 2022) and from the EU funding within the MUR PNRR ‘National Center for HPC, BIG DATA AND QUANTUM COMPUTING’ (project no. CN00000013 CN1).
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A.G. and M.D.D. designed the study, performed the theoretical analysis and wrote the manuscript. A.G. performed the numerical experiments.
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Supplementary Code 1
The necessary code to reproduce the results of this work.
Supplementary Data 1
Node and link numbers of the empirical networks relevant for the allometric scaling analysis.
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Ghavasieh, A., De Domenico, M. Diversity of information pathways drives sparsity in real-world networks. Nat. Phys. 20, 512–519 (2024). https://doi.org/10.1038/s41567-023-02330-x
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DOI: https://doi.org/10.1038/s41567-023-02330-x
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