Abstract
Network geometries are typically characterized by having a finite spectral dimension (SD), \(d_{s}\) that characterizes the return time distribution of a random walk on a graph. The main purpose of this work is to determine the SD of a variety of random graphs called random geometric graphs (RGGs) in the thermodynamic regime, in which the average vertex degree is constant. The spectral dimension depends on the eigenvalue density (ED) of the RGG normalized Laplacian in the neighborhood of the minimum eigenvalues. In fact, the behavior of the ED in such a neighborhood characterizes the random walk. Therefore, we first provide an analytical approximation for the eigenvalues of the regularized normalized Laplacian matrix of RGGs in the thermodynamic regime. Then, we show that the smallest non zero eigenvalue converges to zero in the large graph limit. Based on the analytical expression of the eigenvalues, we show that the eigenvalue distribution in a neighborhood of the minimum value follows a power-law tail. Using this result, we find that the SD of RGGs is approximated by the space dimension d in the thermodynamic regime.
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Notes
- 1.
The notation \(f(n) =\mathrm {\Omega }(g(n))\) indicates that f(n) is bounded below by g(n) asymptotically, i.e., \(\exists K>0\) and \( n_{o} \in \mathbb {N}\) such that \(\forall n > n_{0}\) \(f(n) \ge K g(n)\).
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Acknowledgement
This research was funded by the French Government through the Investments for the Future Program with Reference: Labex UCN@Sophia-UDCBWN.
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Avrachenkov, K., Cottatellucci, L., Hamidouche, M. (2020). Eigenvalues and Spectral Dimension of Random Geometric Graphs in Thermodynamic Regime. In: Cherifi, H., Gaito, S., Mendes, J., Moro, E., Rocha, L. (eds) Complex Networks and Their Applications VIII. COMPLEX NETWORKS 2019. Studies in Computational Intelligence, vol 881. Springer, Cham. https://doi.org/10.1007/978-3-030-36687-2_80
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