# Relating Vertex and Global Graph Entropy in Randomly Generated Graphs

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## Abstract

**:**

## 1. Introduction and Background

#### 1.1. Overview

#### 1.2. Global Graph Entropy

**Definition**

**1.**

**Definition**

**2.**

#### 1.3. Local Entropy Measures

**Inverse Degree:**In this case, we denote the vertex probability as$$P({v}_{i})=\frac{Z}{{k}_{i}^{\gamma}},\mathrm{with}$$$${Z}^{-1}=\sum _{j}{k}_{j}^{-\gamma}\mathrm{to}\mathrm{ensure}\mathrm{normalization}.$$**Fractional Degree:**We use in this case the following for vertex probability:$$P({v}_{i})=\frac{{k}_{i}}{2\left|E\right|}.$$**Clustering Coefficient:**The clustering coefficient measures the probability of an edge existing between the neighbors of a particular vertex. However, its use in the context of a vertex entropy needs to be adjusted by a normalization constant $Z={\sum}_{i}{C}_{1}^{i}$ to be a well behaved probability measure and sum to unity. For simplicity, we omit this constant and assert the following:$$P({v}_{i})={C}_{1}^{i}.$$

#### 1.4. Alternate Formulations of Entropy

## 2. Experimental Analysis

#### 2.1. Method and Objectives

#### 2.2. Scale-Free Graphs

#### 2.3. Gilbert Random Graphs $G(n,p)$

#### 2.4. Alternate Entropy Formulations

## 3. Theoretical Discussion of the Results

#### 3.1. Gilbert Random Graphs

#### 3.2. Scale-Free Graphs

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Sum of vertex entropies for whole graph vs. chromatic information for Barabási–Albert scale-free graphs of constant $\left|V\right|$.

**Figure 2.**Sum of vertex entropies for whole graph vs. chromatic information for Gilbert graphs $G(n,p)$ for $p\in [0.31,0.7]$.

**Figure 3.**Sum of collision vertex entropies for whole graph vs. chromatic information for Gilbert graphs $G(n,p)$ for $p\in [0.31,0.7]$ and scale-free graphs of constant $\left|V\right|$.

**Figure 4.**Calculated $\chi $ versus measured $\chi $, for Gilbert graphs $G(n,p)$ with $n=300$ and $p\in [0.3,1.0]$. Overlaid is the least squares fit for ${H}_{3}$.

**Table 1.**Model selection analysis for inverse degree entropy for scale-free graphs of constant $\left|V\right|$.

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −734.33 | 0.00 | −740.33 | 0.00 |

${\mathbf{H}}_{\mathbf{2}}$ | $\mathbf{-}\mathbf{830.12}$ | $\mathbf{-}\mathbf{95.80}$ | $\mathbf{-}\mathbf{839.14}$ | $\mathbf{-}\mathbf{98.80}$ |

${H}_{3}$ | −825.16 | −90.84 | −837.18 | −96.85 |

${H}_{4}$ | −821.95 | −87.62 | −836.97 | −96.63 |

${H}_{5}$ | −818.63 | −84.31 | −836.66 | −96.32 |

**Table 2.**Model selection analysis for fractional degree entropy for scale-free graphs of constant $\left|V\right|$.

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −708.84 | 0.00 | −714.84 | 0.00 |

${H}_{2}$ | −715.24 | −6.41 | −724.25 | −9.41 |

${H}_{3}$ | −719.49 | −10.66 | −731.51 | −16.66 |

${\mathbf{H}}_{\mathbf{4}}$ | $\mathbf{-}\mathbf{719.62}$ | $\mathbf{-}\mathbf{10.79}$ | $\mathbf{-}\mathbf{734.64}$ | $\mathbf{-}\mathbf{19.80}$ |

${H}_{5}$ | −715.31 | −6.47 | −733.33 | −18.49 |

**Table 3.**Model selection analysis for cluster entropy for scale-free graphs of constant $\left|V\right|$.

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −728.34 | 0.00 | −734.35 | 0.00 |

${H}_{2}$ | −796.47 | −68.12 | −805.48 | −71.13 |

${\mathbf{H}}_{\mathbf{3}}$ | $\mathbf{-}\mathbf{798.84}$ | $\mathbf{-}\mathbf{70.50}$ | $\mathbf{-}\mathbf{810.86}$ | $\mathbf{-}\mathbf{76.51}$ |

${H}_{4}$ | −794.89 | −66.54 | −809.90 | −75.55 |

${H}_{5}$ | −793.77 | −65.43 | −811.80 | −77.44 |

**Table 4.**Model selection analysis for edge density for scale-free graphs of constant $\left|V\right|$.

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −777.77 | 0.00 | −783.78 | 0.00 |

${\mathbf{H}}_{\mathbf{2}}$ | $\mathbf{-}\mathbf{844.94}$ | $\mathbf{-}\mathbf{67.17}$ | $\mathbf{-}\mathbf{853.96}$ | $\mathbf{-}\mathbf{70.18}$ |

${H}_{3}$ | −842.39 | −64.62 | −854.40 | −70.63 |

${H}_{4}$ | −839.21 | −61.43 | −854.23 | −70.45 |

${H}_{5}$ | −836.87 | −59.10 | −854.89 | −71.11 |

**Table 5.**Model selection analysis for inverse degree entropy for random graphs of constant $\left|V\right|$.

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −2004.92 | 0.00 | −2008.75 | 0.00 |

${H}_{2}$ | −2181.68 | −176.76 | −2189.33 | −180.59 |

${\mathbf{H}}_{\mathbf{3}}$ | $\mathbf{-}\mathbf{2182.62}$ | $\mathbf{-}\mathbf{177.70}$ | $\mathbf{-}\mathbf{2194.10}$ | $\mathbf{-}\mathbf{185.35}$ |

${H}_{4}$ | −2176.82 | −171.90 | −2192.12 | −183.37 |

${H}_{5}$ | −2171.14 | −166.22 | −2190.27 | −181.52 |

**Table 6.**Model selection analysis for fractional degree entropy for random graphs of constant $\left|V\right|$.

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −1806.14 | 0.00 | −1809.96 | 0.00 |

${\mathbf{H}}_{\mathbf{2}}$ | $\mathbf{-}\mathbf{1874.29}$ | $\mathbf{-}\mathbf{68.15}$ | $\mathbf{-}\mathbf{1881.94}$ | $\mathbf{-}\mathbf{71.98}$ |

${H}_{3}$ | −1868.70 | −62.56 | −1880.17 | −70.21 |

${H}_{4}$ | −1859.34 | −53.20 | −1874.64 | −64.68 |

${H}_{5}$ | −1856.25 | −50.11 | −1875.38 | −65.42 |

**Table 7.**Model selection analysis for cluster entropy for random graphs of constant $\left|V\right|$.

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −2109.61 | 0.00 | −2113.44 | 0.00 |

${\mathbf{H}}_{\mathbf{2}}$ | $\mathbf{-}\mathbf{2146.19}$ | $\mathbf{-}\mathbf{36.58}$ | $\mathbf{-}\mathbf{2153.84}$ | $\mathbf{-}\mathbf{40.40}$ |

${H}_{3}$ | −2140.43 | −30.82 | −2151.91 | −38.47 |

${H}_{4}$ | −2134.61 | −25.00 | −2149.92 | −36.48 |

${H}_{5}$ | −2128.86 | −19.25 | −2147.99 | −34.56 |

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −951.67 | 0.00 | −955.49 | 0.00 |

${\mathbf{H}}_{\mathbf{2}}$ | $\mathbf{-}\mathbf{985.93}$ | $\mathbf{-}\mathbf{34.26}$ | $\mathbf{-}\mathbf{993.58}$ | $\mathbf{-}\mathbf{38.08}$ |

${H}_{3}$ | −980.15 | −28.48 | −991.63 | −36.13 |

${H}_{4}$ | −974.37 | −22.71 | −989.68 | −34.18 |

${H}_{5}$ | −969.80 | −18.13 | −988.93 | −33.43 |

**Table 9.**Model selection analysis for inverse degree Renyi entropy for random graphs of constant $\left|V\right|$.

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −1801.25 | 0.00 | −1805.08 | 0.00 |

${\mathbf{H}}_{\mathbf{2}}$ | $\mathbf{-}\mathbf{1886.55}$ | $\mathbf{-}\mathbf{85.30}$ | $\mathbf{-}\mathbf{1894.20}$ | $\mathbf{-}\mathbf{89.12}$ |

${H}_{3}$ | −1880.76 | −79.51 | −1892.23 | −87.16 |

${H}_{4}$ | −1875.15 | −73.90 | −1890.46 | −85.38 |

${H}_{5}$ | −1869.02 | −67.77 | −1888.15 | −83.08 |

**Table 10.**Model selection analysis for fractional degree Renyi entropy for random graphs of constant $\left|V\right|$.

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −1810.08 | 0.00 | −1813.90 | 0.00 |

${\mathbf{H}}_{\mathbf{2}}$ | $\mathbf{-}\mathbf{1891.42}$ | $\mathbf{-}\mathbf{81.34}$ | $\mathbf{-}\mathbf{1899.07}$ | $\mathbf{-}\mathbf{85.17}$ |

${H}_{3}$ | −1885.63 | −75.56 | −1897.11 | −83.21 |

${H}_{4}$ | −1880.33 | −70.25 | −1895.63 | −81.73 |

${H}_{5}$ | −1873.93 | −63.86 | −1893.06 | −79.16 |

**Table 11.**Model selection analysis for inverse degree Renyi entropy for scale-free graphs of constant $\left|V\right|$.

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${\mathbf{H}}_{\mathbf{1}}$ | $\mathbf{-}\mathbf{574.07}$ | 0.00 | $\mathbf{-}\mathbf{580.08}$ | 0.00 |

${H}_{2}$ | −569.17 | 4.90 | −578.18 | 1.90 |

${H}_{3}$ | −564.48 | 9.59 | −576.50 | 3.58 |

${H}_{4}$ | −559.90 | 14.17 | −574.92 | 5.16 |

${H}_{5}$ | −555.14 | 18.93 | −573.16 | 6.92 |

**Table 12.**Model selection analysis for fractional degree Renyi entropy for scale-free graphs of constant $\left|V\right|$.

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −694.87 | 0.00 | −700.88 | 0.00 |

${H}_{2}$ | −690.61 | 4.25 | −699.63 | 1.25 |

${\mathbf{H}}_{\mathbf{3}}$ | $\mathbf{-}\mathbf{698.94}$ | $\mathbf{-}\mathbf{4.07}$ | $\mathbf{-}\mathbf{710.96}$ | $\mathbf{-}\mathbf{10.08}$ |

${H}_{4}$ | −695.68 | −0.82 | −710.70 | −9.83 |

${H}_{5}$ | −690.75 | 4.11 | −708.78 | −7.90 |

Vertex Entropy Measure | Scale-Free Graphs | Random Graphs G(n, p) |
---|---|---|

Inverse Degree | $2{m}^{2}n/9\mathrm{ln}2$ | ${p}^{-1}{\mathrm{log}}_{2}(pn)$ |

Fractional Degree | $m{\mathrm{log}}_{2}(2mn)$ | ${\mathrm{log}}_{2}n$ |

Clustering Coefficient | $4m{\mathrm{log}}_{2}(n/4m)$ | $-np{\mathrm{log}}_{2}p$ |

Model | Bayesian Information Criteria | ${\mathbf{\Delta}}_{\mathit{BIC}}$ | Akaike Information Criteria | ${\mathbf{\Delta}}_{\mathit{AIC}}$ |
---|---|---|---|---|

${H}_{1}$ | −90.00 | 0.00 | −92.25 | 0.00 |

${H}_{2}$ | −138.79 | −48.79 | −143.28 | −51.04 |

${\mathbf{H}}_{\mathbf{3}}$ | $\mathbf{-}\mathbf{146.53}$ | $\mathbf{-}\mathbf{56.53}$ | $\mathbf{-}\mathbf{153.28}$ | $\mathbf{-}\mathbf{61.03}$ |

${H}_{4}$ | −142.72 | −52.72 | −151.71 | −59.47 |

${H}_{5}$ | −138.60 | −48.60 | −149.84 | −57.59 |

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**MDPI and ACS Style**

Tee, P.; Parisis, G.; Berthouze, L.; Wakeman, I.
Relating Vertex and Global Graph Entropy in Randomly Generated Graphs. *Entropy* **2018**, *20*, 481.
https://doi.org/10.3390/e20070481

**AMA Style**

Tee P, Parisis G, Berthouze L, Wakeman I.
Relating Vertex and Global Graph Entropy in Randomly Generated Graphs. *Entropy*. 2018; 20(7):481.
https://doi.org/10.3390/e20070481

**Chicago/Turabian Style**

Tee, Philip, George Parisis, Luc Berthouze, and Ian Wakeman.
2018. "Relating Vertex and Global Graph Entropy in Randomly Generated Graphs" *Entropy* 20, no. 7: 481.
https://doi.org/10.3390/e20070481