# Information Entropy of Tight-Binding Random Networks with Losses and Gain: Scaling and Universality

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Network Model with Losses and Gain

#### 1.2. Previous Work

## 2. Results

#### 2.1. Scaling of Information Entropy

#### 2.2. Eigenvalue Properties

## 3. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Average information entropy $\u2329S\u232a$ normalized to ${S}_{\mathrm{GOE}}\approx \mathrm{ln}(N/2.07)$ as a function of the connectivity $\alpha $ of Erdős–Rényi tight-binding random networks (of sizes ranging from $N=250$–2000) with balanced losses and gain with strength $\gamma $. (

**a**) $\gamma =0.01$, (

**b**) $\gamma =1$, and (

**c**) $\gamma =2$. Each symbol was computed by averaging over ${10}^{6}$ eigenvectors.

**Figure 2.**Average information entropy $\u2329S\u232a$ normalized to ${S}_{\mathrm{GOE}}\approx \mathrm{ln}(N/2.07)$ as a function of the connectivity $\alpha $ of Erdős–Rényi tight-binding random networks of size N with different loss-and-gain strengths $\gamma $. (

**a**) $N=250$, (

**b**) $N=1000$, and (

**c**) $N=4000$. Insets: enlargements of the boxes around the localization-to-delocalization transition point in main panels. Each symbol was computed by averaging over ${10}^{6}$ eigenvectors.

**Figure 3.**Localization-to-delocalization transition point ${\alpha}^{\ast}$ (defined as the value of $\alpha $ for which $\u2329\phantom{\rule{3.33333pt}{0ex}}S\phantom{\rule{3.33333pt}{0ex}}\u232a/{S}_{\mathrm{GOE}}\phantom{\rule{3.33333pt}{0ex}}\approx \phantom{\rule{3.33333pt}{0ex}}0.5$) as a function of (

**a**) the network size N (for several values of $\gamma $) and (

**b**) the loss-and-gain strength $\gamma $ (for several values of N). In (

**b**), we set $\delta $ to −0.98. Dashed lines in (

**a**) and (

**b**) proportional to ${N}^{-0.98}$ and $\gamma $, respectively, are plotted to guide the eye; see Equations (5) and (6).

**Figure 4.**Average Shannon entropy $\u2329S\u232a$ normalized to ${S}_{\mathrm{GOE}}$ as a function of the scaling parameter $\xi $ (see Equation (7)) of Erdős–Rényi tight-binding random networks with losses and gain. (

**a**) $N=250$ for different values of loss-and-gain strength $\gamma $, (

**b**) $\gamma =1$ for different network sizes N, and (

**c**) different combinations of N and $\gamma $.

**Figure 5.**Density plots of eigenvalues $\lambda $ in the complex plane for several combinations of sparsity $\alpha $ and loss-and-gain strengths $\gamma $. The network size was set to $N=1000$. The sparsity increases from left to right, while the loss-and-gain strength from top to bottom. To construct each density plot, ${10}^{6}$ eigenvalues were used.

**Figure 6.**Density plots of eigenvalues $\lambda $ in the complex plane for three network sizes N (500, 1000, and 2000) and increasing values of $\xi $ (from top to bottom). To construct each density plot, ${10}^{6}$ eigenvalues were used.

$\mathit{\gamma}$ | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1 | 1.2 | 1.4 | 1.6 | 1.8 | 2 |
---|---|---|---|---|---|---|---|---|---|---|---|

$\mathcal{C}$ | 2.18 | 2.06 | 2.09 | 2.11 | 2.17 | 2.27 | 2.28 | 2.33 | 2.4 | 2.46 | 2.5 |

$\delta $ | −0.997 | −0.982 | −0.979 | −0.976 | −0.976 | −0.976 | −0.978 | −0.977 | −0.979 | −0.979 | −0.979 |

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**MDPI and ACS Style**

Martínez-Martínez, C.T.; Méndez-Bermúdez, J.A.
Information Entropy of Tight-Binding Random Networks with Losses and Gain: Scaling and Universality. *Entropy* **2019**, *21*, 86.
https://doi.org/10.3390/e21010086

**AMA Style**

Martínez-Martínez CT, Méndez-Bermúdez JA.
Information Entropy of Tight-Binding Random Networks with Losses and Gain: Scaling and Universality. *Entropy*. 2019; 21(1):86.
https://doi.org/10.3390/e21010086

**Chicago/Turabian Style**

Martínez-Martínez, C. T., and J. A. Méndez-Bermúdez.
2019. "Information Entropy of Tight-Binding Random Networks with Losses and Gain: Scaling and Universality" *Entropy* 21, no. 1: 86.
https://doi.org/10.3390/e21010086