Spacing ratio characterization of the spectra of directed random networks

Thomas Peron, Bruno Messias F. de Resende, Francisco A. Rodrigues, Luciano da F. Costa, and J. A. Méndez-Bermúdez

Phys. Rev. E 102, 062305 – Published 10 December 2020

Abstract

Previous literature on random matrix and network science has traditionally employed measures derived from nearest-neighbor level spacing distributions to characterize the eigenvalue statistics of random matrices. This approach, however, depends crucially on eigenvalue unfolding procedures, which in many situations represent a major hindrance due to constraints in the calculation, especially in the case of complex spectra. Here we study the spectra of directed networks using the recently introduced ratios between nearest and next-to-nearest eigenvalue spacing, thus circumventing the shortcomings imposed by spectral unfolding. Specifically, we characterize the eigenvalue statistics of directed Erdős-Rényi (ER) random networks by means of two adjacency matrix representations, namely, (1) weighted non-Hermitian random matrices and (2) a transformation on non-Hermitian adjacency matrices which produces weighted Hermitian matrices. For both representations, we find that the distribution of spacing ratios becomes universal for a fixed average degree, in accordance with undirected random networks. Furthermore, by calculating the average spacing ratio as a function of the average degree, we show that the spectral statistics of directed ER random networks undergoes a transition from Poisson to Ginibre statistics for model 1 and from Poisson to Gaussian unitary ensemble statistics for model 2. Eigenvector delocalization effects of directed networks are also discussed.

Received 25 August 2020
Accepted 17 November 2020

DOI:https://doi.org/10.1103/PhysRevE.102.062305

Physics Subject Headings (PhySH)

Complex systems Many-body localization

Networks & random structures Random graphs

Random matrix theory Statistical methods

NetworksCondensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Thomas Peron¹, Bruno Messias F. de Resende², Francisco A. Rodrigues¹, Luciano da F. Costa², and J. A. Méndez-Bermúdez ^1,3

¹Institute of Mathematics and Computer Science, University of São Paulo, São Carlos 13566-590, São Paulo, Brazil
²São Carlos Institute of Physics, University of São Paulo, São Carlos 13566-590, São Paulo, Brazil
³Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado postal J-48, Puebla 72570, México

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Vol. 102, Iss. 6 — December 2020

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Figure 1
(a–c) Ensemble average of the ratio $r_{C}$ for the adjacency matrices represented by the diluted real Ginibre ensemble, $〈r_{C} (A_{dRGE})〉$ , as a function of the probability $p$ for several network sizes $n$ . The horizontal dashed line in panel (c) indicates $〈r_{C} (A_{dRGE})〉 = 0.6188$ . (b) $〈r_{C} (A_{dRGE})〉$ as a function of $n$ for $p = 1$ and $p = 0$ . The horizontal dashed lines at $〈r_{C} (A_{dRGE})〉 = 0.7370$ and 0.5006 indicate the values of $〈r_{C} (A_{dRGE})〉$ for $p = 1$ and $p = 0$ , respectively, at $n = 1000$ . (d) Same curves of panel (c) but as a function of the average degree $〈k〉$ . Vertical dashed lines mark the values of $〈k〉$ (0.5, 1.5, 2, 3, and 10) chosen to report the probability density functions (PDFs) of $r_{C}, P (r_{C})$ , in Fig. 2. Inset: $p^{*}$ as a function of $n$ . The dashed line is the fitting of Eq. (5) to the data with fitting parameters $C = 2.1949$ and $δ = - 1.0235$ . Each symbol was computed from the ratios of $10^{6} / n$ directed random networks $G (n, p)$ .
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Figure 2
Probability density function of the ratio $r_{C}, P (r_{C})$ , for the adjacency matrix represented by the diluted real Ginibre ensemble. Each panel, corresponding to different values of the average degree $〈k〉$ , contains histograms of four different network sizes $n$ . The values of $〈k〉 = 0.5$ , 1.5, 2, 3, and 10 are marked as vertical dashed lines in Fig. 1. Each histogram was constructed from the ratios of $10^{6} / n$ directed random networks $G (n, p)$ .
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Figure 3
(a–c) Ensemble average of the ratio $r_{R}$ for the magnetic adjacency matrices of Eq. (2), $〈r_{R} (A_{M})〉$ , as a function of the probability $p$ for several network sizes $n$ . The inset in panel (a) is an enlargement in the interval $p \in [0.8, 1)$ . The horizontal dashed line in panel (c) indicates $〈r_{R} (A_{M})〉 = 0.4932$ . (b) $〈r_{R} (A_{M})〉$ as a function of $n$ for $p = 0$ , 0.8, and 1. The horizontal dashed lines at $〈r_{R} (A_{M})〉 = 0.5995$ , 0.5307, and 0.3867 indicate the values of $〈r_{R} (A_{M})〉$ for $p = 0.8$ , 1, and 0, respectively, at $n = 1000$ . (d) Same curves of panel (c) but as a function of the average degree $〈k〉$ . Vertical dashed lines mark the values of $〈k〉$ (0.5, 1.2, 1.5, 2, and 10) chosen to report the PDFs of $r_{R}, P (r_{R})$ , in Fig. 5. Inset: $p^{*}$ as a function of $n$ . The dashed line is the fitting of Eq. (5) to the data with fitting parameters $C = 1.9563$ and $δ = - 1.0584$ . Each symbol was computed from the ratios of $10^{6} / n$ directed random networks $G (n, p)$ .
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Figure 4
(a–c) Ensemble average of the ratio $r_{C}$ for the magnetic adjacency matrices of Eq. (2), $〈r_{C} (A_{M})〉$ , as a function of the probability $p$ for several network sizes $n$ . The inset in panel (a) is an enlargement in the interval $p \in [0.8, 1)$ . The horizontal dashed line in panel (c) indicates $〈r_{C} (A_{M})〉 = 0.559$ . (b) $〈r_{C} (A_{M})〉$ as a function of $n$ for $p = 0$ , 0.8, and 1. The horizontal dashed lines at $〈r_{R} (A_{M})〉 = 0.6175$ , 0.5688, and 0.5006 indicate the values of $〈r_{R} (A_{M})〉$ for $p = 0.8$ , 1, and 0, respectively, at $n = 1000$ . (d) Same curves of panel (c) but as a function of the average degree $〈k〉$ . Vertical dashed lines mark the values of $〈k〉$ (0.5, 1.2, 1.5, 2, and 10) chosen to report the PDFs of $r_{C}, P (r_{C})$ , in Fig. 5. Inset: $p^{*}$ as a function of $n$ . The dashed line is the fitting of Eq. (5) to the data with fitting parameters $C = 1.9721$ and $δ = - 1.0581$ . Each symbol was computed from the ratios of $10^{6} / n$ directed random networks $G (n, p)$ .
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Figure 5
Probability density function of the ratios $r_{R}$ (upper panels) and $r_{C}$ (lower panels), $P (r_{R})$ and $P (r_{C})$ , for the magnetic adjacency matrix of Eq. (2). Each panel, corresponding to different values of the average degree $〈k〉$ , contains histograms of four different network sizes $n$ . The values of $〈k〉 = 0.5$ , 1.2, 1.5, 2, and 10 are marked as vertical dashed lines in Figs. 3 and 4. In the rightmost panels the case $p = 1$ is also reported. Cyan lines in the upper left and upper right panels are $P_{PE} (r_{R})$ and $P_{GOE} (r_{R})$ , from Eqs. (8) and (10), respectively. The orange line in the upper right panel is $P_{GUE} (r_{R})$ from Eq. (9). Each histogram was constructed from the ratios of $10^{6} / n$ directed random networks $G (n, p)$ .
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Figure 6
Normalized average Shannon entropy $〈S〉$ as a function of the average degree $〈k〉$ for the adjacency matrices (a) $A_{dRGE}$ and (b) $A_{M}$ , corresponding to directed networks of size $n$ . In (a) [(b)] we are normalizing $〈S (A_{dRGE})〉$ [ $〈S (A_{M})〉$ ] to $S_{RGE}$ [ $S_{GUE}$ ]. The inset in (a) shows the numerically computed $S_{RGE}$ as a function of $n$ . The dashed line is a fitting to the data that provides $S_{RGE} \approx ln (n / 1.56)$ . Vertical dashed lines indicate the transition regime as deduced from $〈r_{C}〉$ : (a) $1 < 〈k〉 < 7$ and (b) $0.7 < 〈k〉 < 4$ . Each symbol was computed by averaging over $10^{6}$ eigenvectors.
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