Abstract
The dimension of the space underlying real-world networks has been shown to strongly influence the networks structural properties, from the degree distribution to the way the networks respond to diffusion and percolation processes. In this paper we propose a way to estimate the dimension of the manifold underlying a network that is based on Weyl’s law, a mathematical result that describes the asymptotic behaviour of the eigenvalues of the graph Laplacian. For the case of manifold graphs, the dimension we estimate is equivalent to the fractal dimension of the network, a measure of structural self-similarity. Through an extensive set of experiments on both synthetic and real-world networks we show that our approach is able to correctly estimate the manifold dimension. We compare this with alternative methods to compute the fractal dimension and we show that our approach yields a better estimate on both synthetic and real-world examples.
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References
Akiba, T., Nakamura, K., Takaguchi, T.: Fractality of massive graphs: scalable analysis with sketch-based box-covering algorithm. In: 2016 IEEE 16th International Conference on Data Mining (ICDM), pp. 769–774. IEEE (2016)
Chorley, M., Rossi, L., Tyson, G., Williams, M.: Pub crawling at scale: tapping untappd to explore social drinking. In: Proceedings of the International AAAI Conference on Web and Social Media, vol. 10 (2016)
Colizza, V., Pastor-Satorras, R., Vespignani, A.: Reaction-diffusion processes and metapopulation models in heterogeneous networks. Nat. Phys. 3(4), 276–282 (2007)
Crucitti, P., Latora, V., Porta, S.: Centrality measures in spatial networks of urban streets. Phys. Rev. E 73(3), 036125 (2006)
Daqing, L., Kosmidis, K., Bunde, A., Havlin, S.: Dimension of spatially embedded networks. Nat. Phys. 7(6), 481–484 (2011)
Erath, A., Löchl, M., Axhausen, K.W.: Graph-theoretical analysis of the swiss road and railway networks over time. Netw. Spat. Econ. 9(3), 379–400 (2009)
Gursoy, A., Keskin, O., Nussinov, R.: Topological properties of protein interaction networks from a structural perspective. Biochem. Soc. Trans. 36(Pt 6), 1398–1403 (2008)
Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., Boguná, M.: Hyperbolic geometry of complex networks. Phys. Rev. E 82(3), 036106 (2010)
Latora, V., Nicosia, V., Russo, G.: Complex Networks: Principles, Methods and Applications. Cambridge University Press, Cambridge (2017)
Lima, A., Rossi, L., Musolesi, M.: Coding together at scale: GitHub as a collaborative social network. In: Proceedings of 8th AAAI International Conference on Weblogs and Social Media (2014)
Minello, G., Rossi, L., Torsello, A.: On the von Neumann entropy of graphs. J. Complex Netw. 7(4), 491–514 (2019)
Newman, M.E.: Finding community structure in networks using the eigenvectors of matrices. Phys. Rev. E 74(3), 036104 (2006)
Rossi, L., Torsello, A., Hancock, E.R.: Node centrality for continuous-time quantum walks. In: Fränti, P., Brown, G., Loog, M., Escolano, F., Pelillo, M. (eds.) S+SSPR 2014. LNCS, vol. 8621, pp. 103–112. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44415-3_11
Rossi, L., Torsello, A., Hancock, E.R.: Measuring graph similarity through continuous-time quantum walks and the quantum Jensen-Shannon divergence. Phys. Rev. E 91(2), 022815 (2015)
Rossi, L., Torsello, A., Hancock, E.R., Wilson, R.C.: Characterizing graph symmetries through quantum Jensen-Shannon divergence. Phys. Rev. E 88(3), 032806 (2013)
Rozenfeld, H.D., Havlin, S., Ben-Avraham, D.: Fractal and transfractal recursive scale-free nets. New J. Phys. 9(6), 175 (2007)
Song, C., Gallos, L.K., Havlin, S., Makse, H.A.: How to calculate the fractal dimension of a complex network: the box covering algorithm. J. Stat. Mech. Theory Exp. 2007(03), P03006 (2007)
Song, C., Havlin, S., Makse, H.A.: Self-similarity of complex networks. Nature 433(7024), 392–395 (2005)
Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)
Wei, D.J., Liu, Q., Zhang, H.X., Hu, Y., Deng, Y., Mahadevan, S.: Box-covering algorithm for fractal dimension of weighted networks. Sci. Rep. 3(1), 1–8 (2013)
Weyl, H.: Über die asymptotische verteilung der eigenwerte. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-Physikalische Klasse 1911, 110–117 (1911)
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Rossi, L., Torsello, A. (2021). Estimating the Manifold Dimension of a Complex Network Using Weyl’s Law. In: Torsello, A., Rossi, L., Pelillo, M., Biggio, B., Robles-Kelly, A. (eds) Structural, Syntactic, and Statistical Pattern Recognition. S+SSPR 2021. Lecture Notes in Computer Science(), vol 12644. Springer, Cham. https://doi.org/10.1007/978-3-030-73973-7_16
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DOI: https://doi.org/10.1007/978-3-030-73973-7_16
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