Origins of fractality in the growth of complex networks

Abstract

Complex networks from such different fields as biology, technology or sociology share similar organization principles. The possibility of a unique growth mechanism promises to uncover universal origins of collective behaviour. In particular, the emergence of self-similarity in complex networks raises the fundamental question of the growth process according to which these structures evolve. Here we investigate the concept of renormalization as a mechanism for the growth of fractal and non-fractal modular networks. We show that the key principle that gives rise to the fractal architecture of networks is a strong effective ‘repulsion’ (or, disassortativity) between the most connected nodes (that is, the hubs) on all length scales, rendering them very dispersed. More importantly, we show that a robust network comprising functional modules, such as a cellular network, necessitates a fractal topology, suggestive of an evolutionary drive for their existence.

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Figure 1: Self-similar dynamical evolution of networks.
Figure 2: Empirical results on real complex networks.
Figure 3: Predictions of the renormalization growth mechanism of complex networks.
Figure 4: Practical implications of the renormalization growth approach and fractality.

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Acknowledgements

We would like to thank J. Brujić for illuminating discussions and E. Ravasz for providing the data on the metabolic network. S.H. wishes to thank the Israel Science Foundation, ONR and Dysonet for support. This work is supported by the National Science Foundation, DMR-0239504 to H.A.M.

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Correspondence to Hernán A. Makse.

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Song, C., Havlin, S. & Makse, H. Origins of fractality in the growth of complex networks. Nature Phys 2, 275–281 (2006). https://doi.org/10.1038/nphys266

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