# Bounding Extremal Degrees of Edge-Independent Random Graphs Using Relative Entropy

## Abstract

**:**

## 1. Introduction

## 2. Bounds for Maximum and Minimum Degrees

**Theorem 1.**

**Proof.**

**Remark 1.**

**Remark 2.**

**Remark 3.**

**Remark 4.**

**Theorem 2.**

- (A)
- If $\underline{p}\gg (lnn)/n$, then $(1+o\left(1\right))n\underline{p}\le \delta \left(G\right)\le (1+o\left(1\right))n\overline{p}\phantom{\rule{1.em}{0ex}}a.a.s.;$
- (B)
- If $\Delta \left(A\right)\gg {ln}^{4}n$, then $\delta \left(G\right)\le n-{\lambda}_{1}\left(A\right)+(2+o\left(1\right))\sqrt{\Delta \left(A\right)}\phantom{\rule{1.em}{0ex}}a.a.s..$

**Proof.**

**Remark 5.**

**Remark 6.**

## 3. An Application to Random Graphs with Given Expected Degrees

**Corollary 1.**

**Proof.**

**Corollary 2.**

- (A)
- If ${w}_{min}^{2}\gg Vol\left(G\right)(lnn)/n$, then$$(1+o\left(1\right))\frac{n{w}_{min}^{2}}{Vol\left(G\right)}\le \delta \left(G\right)\le (1+o\left(1\right))\frac{n{w}_{max}^{2}}{Vol\left(G\right)}\phantom{\rule{1.em}{0ex}}a.a.s.;$$
- (B)
- If ${w}_{max}\gg {ln}^{4}n$, then$$\delta \left(G\right)\le n-\tilde{w}+(2+o\left(1\right))\sqrt{{w}_{max}}+o\left(1\right)\phantom{\rule{1.em}{0ex}}a.a.s..$$

**Example 1.**

**Table 1.**Maximum degree $\Delta \left(G\right)$ of $G\in G\left(w\right)$ with $w=({ln}^{4}n,\cdots ,{ln}^{4}n,{ln}^{5}n,\cdots ,{ln}^{5}n)$ (with half of the numbers being ${ln}^{4}n$). The theoretical upper and lower bounds are calculated from Corollary 1. Numerical results are based on average over 20 independent runs.

n | Theoretical Lower Bound | $\Delta \left(G\right)$ | Theoretical Upper Bound |
---|---|---|---|

$9.8\times {10}^{5}$ | $4.683\times {10}^{5}-(2+o\left(1\right))706.85+o\left(1\right)$ | $7.260\times {10}^{5}$ | $(1+o\left(1\right))9.317\times {10}^{5}$ |

$9.9\times {10}^{5}$ | $4.701\times {10}^{5}-(2+o\left(1\right))708.15+o\left(1\right)$ | $7.283\times {10}^{5}$ | $(1+o\left(1\right))9.352\times {10}^{5}$ |

$10.0\times {10}^{5}$ | $4.718\times {10}^{5}-(2+o\left(1\right))709.44+o\left(1\right)$ | $7.305\times {10}^{5}$ | $(1+o\left(1\right))9.387\times {10}^{5}$ |

$10.1\times {10}^{5}$ | $4.735\times {10}^{5}-(2+o\left(1\right))710.72+o\left(1\right)$ | $7.327\times {10}^{5}$ | $(1+o\left(1\right))9.421\times {10}^{5}$ |

$10.2\times {10}^{5}$ | $4.752\times {10}^{5}-(2+o\left(1\right))711.99+o\left(1\right)$ | $7.351\times {10}^{5}$ | $(1+o\left(1\right))9.455\times {10}^{5}$ |

**Table 2.**Minimum degree $\delta \left(G\right)$ of $G\in G\left(w\right)$ with $w=({ln}^{4}n,\cdots ,{ln}^{4}n,{ln}^{5}n,\cdots ,{ln}^{5}n)$ (with half of the numbers being ${ln}^{4}n$). The theoretical upper and lower bounds are calculated from Corollary 2. Numerical results are based on average over 20 independent runs.

n | Theoretical Lower Bound | $\delta \left(G\right)$ | Theoretical Upper Bound |
---|---|---|---|

$9.8\times {10}^{5}$ | $(1+o\left(1\right))4.896\times {10}^{3}$ | $1.826\times {10}^{4}$ | $5.117\times {10}^{5}+(2+o\left(1\right))706.85+o\left(1\right)$ |

$9.9\times {10}^{5}$ | $(1+o\left(1\right))4.907\times {10}^{3}$ | $1.829\times {10}^{4}$ | $5.200\times {10}^{5}+(2+o\left(1\right))708.15+o\left(1\right)$ |

$10.0\times {10}^{5}$ | $(1+o\left(1\right))4.918\times {10}^{3}$ | $1.832\times {10}^{4}$ | $5.282\times {10}^{5}+(2+o\left(1\right))709.44+o\left(1\right)$ |

$10.1\times {10}^{5}$ | $(1+o\left(1\right))4.929\times {10}^{3}$ | $1.834\times {10}^{4}$ | $5.365\times {10}^{5}+(2+o\left(1\right))710.72+o\left(1\right)$ |

$10.2\times {10}^{5}$ | $(1+o\left(1\right))4.940\times {10}^{3}$ | $1.837\times {10}^{4}$ | $5.448\times {10}^{5}+(2+o\left(1\right))711.99+o\left(1\right)$ |

**Example 2.**

**Figure 1.**Extremal degree versus the number of vertices n. The theoretical upper and lower bounds are from (17) and (18). Each data point is obtained by means of a mixed ensemble averaging of 30 independent runs of 10 graphs yielding a statistically ample sampling.

## Acknowledgments

## Conflicts of Interest

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Shang, Y.
Bounding Extremal Degrees of Edge-Independent Random Graphs Using Relative Entropy. *Entropy* **2016**, *18*, 53.
https://doi.org/10.3390/e18020053

**AMA Style**

Shang Y.
Bounding Extremal Degrees of Edge-Independent Random Graphs Using Relative Entropy. *Entropy*. 2016; 18(2):53.
https://doi.org/10.3390/e18020053

**Chicago/Turabian Style**

Shang, Yilun.
2016. "Bounding Extremal Degrees of Edge-Independent Random Graphs Using Relative Entropy" *Entropy* 18, no. 2: 53.
https://doi.org/10.3390/e18020053