# Comparison of Simulations with a Mean-Field Approach vs. Synthetic Correlated Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Measuring Assortative and Disassortative Mixing

## 3. The Function ${\mathit{k}}_{\mathit{n}\mathit{n}}$ in the Mean-Field Approach and for Concrete Networks

## 4. The Peak Time in Innovation Diffusion Dynamics

## 5. Further Remarks on the Function ${\mathit{k}}_{\mathit{nn}}$ and Its Average

**Proof.**

**Proof.**

- (i)
- $\frac{1}{{k}_{m}}{\sum}_{h,k}hkP(h\mid k)kP\left(k\right)$,
- (ii)
- $-\frac{1}{{k}_{m}}{\sum}_{h,k}hP(h\mid k)kP\left(k\right)$,
- (iii)
- $-\frac{1}{{k}_{m}}{\sum}_{h,k}kP(h\mid k)kP\left(k\right)$,
- (iv)
- $\frac{1}{{k}_{m}}{\sum}_{h,k}P(h\mid k)kP\left(k\right)$.

- (i)
- $\frac{1}{{k}_{m}}{\sum}_{k}{k}^{2}P\left(k\right){k}_{nn}\left(k\right)$,
- (ii)
- $-\frac{1}{{k}_{m}}\langle {k}^{2}\rangle $,
- (iii)
- $-\frac{1}{{k}_{m}}\langle {k}^{2}\rangle $,
- (iv)
- $\frac{{k}_{m}}{{k}_{m}}$.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 1.**The nine panels display the graph of the function ${k}_{nn}\left(k\right)$ for the cases of maximal degree $n=9,18,27,36,45,54,63,72,81$ for networks whose correlation matrices $P(h\mid k)$ are described in $\left(1\right)$ (a recipe for assortative networks from the work in [8] by Newman). Here, $\gamma =2.5$. Notice the quasi-invariance of the shape with respect to the maximal degree.

**Figure 2.**The nine panels display the graph of the function ${k}_{nn}\left(k\right)$ for the cases of maximal degree $n=9,18,27,36,45,54,63,72,81$ for networks whose correlation matrices $P(h\mid k)$ are described in $\left(2\right)$ (a recipe for assortative networks from the work in [17] by Vázquez et Al and the work in [18] by Nevokee et Al). Here, $\gamma =2.5$ and $\beta =0.5$. Moreover, here we have the quasi-invariance (actually, the invariance) of the shape with respect to the maximal degree.

**Figure 3.**The nine panels display the graph of the function ${k}_{nn}\left(k\right)$ for the cases of maximal degree $n=9,18,27,36,45,54,63,72,81$ for networks whose correlation matrices $P(h\mid k)$ are described in $\left(3\right)$ (a recipe for assortative networks from the work in [16]). Here, $\gamma =2.5$ and $\lambda =1$. Notice the quasi-invariance of the shape with respect to the maximal degree.

**Figure 4.**The nine panels display the graph of the function ${k}_{nn}\left(k\right)$ for the cases of maximal degree $n=9,18,27,36,45,54,63,72,81$ for networks whose correlation matrices $P(h\mid k)$ are described in $\left(4\right)$ (a recipe for assortative networks from the work in [16]). Here, $\gamma =2.5$ and $\mu =1$. Notice the quasi-invariance of the shape with respect to the maximal degree.

**Figure 5.**The nine panels display the graph of the function ${k}_{nn}\left(k\right)$ for the cases of maximal degree $n=9,18,27,36,45,54,63,72,81$ for networks whose correlation matrices $P(h\mid k)$ are described in $\left(5\right)$ (a recipe for assortative networks from the work in [16]). Here, $\gamma =2.5$ and $\nu =1$. Notice the quasi-invariance of the shape with respect to the maximal degree.

**Figure 6.**The nine panels display the graph of the function ${k}_{nn}\left(k\right)$ for the cases of maximal degree $n=9,18,27,36,45,54,63,72,81$ for networks whose correlation matrices $P(h\mid k)$ are described in $\left(6\right)$ (a recipe for disassortative networks from the work in [8] by Newman). Here, $\gamma =2.5$. Notice the quasi-invariance of the shape with respect to the maximal degree.

**Figure 7.**The nine panels display the graph of the function ${k}_{nn}\left(k\right)$ for the cases of maximal degree $n=9,18,27,36,45,54,63,72,81$ for networks with $P\left(k\right)={k}^{4}{e}^{-k}$ (non scale-free) and correlation matrices $P(h\mid k)$ as in $3)$ with $\lambda =1$.

**Figure 8.**Each of the six panels displays the graph of an averaged function ${k}_{nn}\left(k\right)$ obtained according to the procedure for the construction of an average of concrete networks. The panels (

**a**–

**f**) are, respectively, relative to networks related to the points (1), (2), (3), (4), (5), (6) of Section 3.

**Figure 9.**The panels (

**a**–

**f**), respectively, refer to networks related to the points (1)–(6) of Section 3. Each of them displays the evolution in time of the functions ${x}^{\prime}\left(t\right)={\sum}_{j=1}^{n}\frac{d}{dt}{x}_{j}\left(t\right)$ and ${X}^{\prime}={\sum}_{i=1}^{N}\frac{d}{dt}{X}_{i}\left(t\right)$, derivatives of the “cumulative” solutions of the systems (15) and (16), respectively. More precisely, in each panel the blue graph represents the numerical solution of system (15) for networks with correlation matrices $P(h\mid k)$ as in Section 3 and the yellow graph represents the solution of system (16) on top of a concrete network of the same family. The values of t in correspondence of which the two graphs exhibit a maximum are the peak times.

**Figure 10.**The two panels, respectively, display the dependence on n of ${K}_{n}$ (as defined in (17)) and ${min}_{k}F(n,k)$ (as defined in (18)) for networks with degree distribution $P\left(k\right)\propto 1/{k}^{\gamma}$ for five values of $\gamma $ and whose correlation matrices $P(h\mid k)$ are as in $\left(5\right)$ of Section 3.

**Table 1.**In the first line, the values of r are given for six Markovian networks as in the points (1)–(6) of Section 3 (with $P\left(k\right)=c/{k}^{2.5}$ and $n=9$). In the second line, for each of the cases (1)–(6), the averaged value is given of the assortativity coefficients of the networks corresponding to the thirty matrices ${e}_{k,h}^{\left[j\right]}$ (with $j\in \{1,\cdots ,30\}$) generated by a rewiring.

Assortativity Coefficient r | net 1 | net 2 | net 3 | net 4 | net 5 | net 6 |
---|---|---|---|---|---|---|

in the mean-field approach | 0.20 | 0.50 | 0.75 | 0.67 | 0.61 | −0.20 |

as an average for concrete networks | 0.15 | 0.11 | 0.32 | 0.30 | 0.30 | −0.13 |

with standard error of the mean | 0.01 | 0.01 | 0.02 | 0.02 | 0.02 | 0.01 |

**Table 2.**The values are here reported of the peak time for six networks with correlation matrices as in the points (1)–(6) of Section 3. The values in the first line are evaluated within a mean-field approach, those in the second line are obtained as averages of peak times over concrete networks.

Peak Time Evaluated | net 1 | net 2 | net 3 | net 4 | net 5 | net 6 |
---|---|---|---|---|---|---|

in the mean-field approach | 4.68 | 4.61 | 4.45 | 4.30 | 4.23 | 5.03 |

as an average for concrete networks | 4.28 | 4.28 | 3.94 | 3.93 | 4.00 | 4.68 |

with standard error of the mean | 0.04 | 0.03 | 0.04 | 0.04 | 0.03 | 0.03 |

**Table 3.**These three tables display values of ${K}_{n}$ and of the peak times ${t}_{max}$ in correspondence of different values of the parameters $\gamma $ for networks with $P\left(k\right)\propto 1/{k}^{\gamma}$ and correlation matrices as in $\left(5\right)$ of Section 3 with $\nu =1$. The tables are relative, respectively, to $n=9$ (table on the left), $n=45$ (central table), and $n=81$ (table on the right).

$\mathit{\gamma}$ | ${\mathit{K}}_{\mathit{n}}$ | ${\mathit{t}}_{\mathit{max}}$ |
---|---|---|

n = 9 | ||

2.1 | 2.20 | 4.00 |

2.2 | 2.07 | 4.02 |

2.3 | 1.96 | 4.06 |

2.4 | 1.85 | 4.14 |

2.5 | 1.75 | 4.22 |

2.6 | 1.67 | 4.36 |

2.7 | 1.60 | 4.50 |

2.8 | 1.53 | 4.64 |

2.9 | 1.47 | 4.80 |

n = 45 | ||

2.1 | 3.30 | 2.00 |

2.2 | 2.88 | 2.32 |

2.3 | 2.54 | 3.08 |

2.4 | 2.26 | 3.76 |

2.5 | 2.04 | 4.24 |

2.6 | 1.87 | 4.58 |

2.7 | 1.73 | 4.84 |

2.8 | 1.61 | 5.02 |

2.9 | 1.52 | 5.18 |

n = 81 | ||

2.1 | 3.70 | 1.48 |

2.2 | 3.12 | 2.48 |

2.3 | 2.69 | 3.54 |

2.4 | 2.35 | 4.16 |

2.5 | 2.09 | 4.54 |

2.6 | 1.89 | 4.82 |

2.7 | 1.73 | 5.04 |

2.8 | 1.61 | 5.18 |

2.9 | 1.51 | 5.30 |

**Table 4.**These two tables display values of the peak times ${t}_{max}$ in correspondence of different values of the parameters $\gamma $ and $\nu $ for networks with $P\left(k\right)\propto 1/{k}^{\gamma}$ and correlation matrices as in $\left(5\right)$ of Section 3. The table above is relative to $n=9$, the table below is relative to $n=81$.

${\mathit{t}}_{\mathit{max}}$ | $\mathit{\nu}=0.2$ | $\mathit{\nu}=0.4$ | $\mathit{\nu}=0.6$ | $\mathit{\nu}=0.8$ | $\mathit{\nu}=1$ |
---|---|---|---|---|---|

n = 9 | |||||

$\gamma =2.1$ | 4.04 | 4.04 | 4.02 | 4.02 | 4.00 |

$\gamma =2.2$ | 4.04 | 4.04 | 4.04 | 4.04 | 4.02 |

$\gamma =2.3$ | 4.06 | 4.06 | 4.08 | 4.06 | 4.06 |

$\gamma =2.4$ | 4.12 | 4.12 | 4.12 | 4.14 | 4.14 |

$\gamma =2.5$ | 4.20 | 4.20 | 4.22 | 4.22 | 4.22 |

$\gamma =2.6$ | 4.30 | 4.32 | 4.32 | 4.34 | 4.36 |

$\gamma =2.7$ | 4.44 | 4.44 | 4.46 | 4.48 | 4.50 |

$\gamma =2.8$ | 4.58 | 4.60 | 4.62 | 4.62 | 4.64 |

$\gamma =2.9$ | 4.74 | 4.76 | 4.76 | 4.78 | 4.80 |

n = 81 | |||||

$\gamma =2.1$ | 1.38 | 1.40 | 1.42 | 1.46 | 1.48 |

$\gamma =2.2$ | 2.26 | 2.32 | 2.36 | 2.42 | 2.48 |

$\gamma =2.3$ | 3.56 | 3.54 | 3.54 | 3.54 | 3.54 |

$\gamma =2.4$ | 4.18 | 4.16 | 4.16 | 4.16 | 4.16 |

$\gamma =2.5$ | 4.58 | 4.56 | 4.56 | 4.56 | 4.54 |

$\gamma =2.6$ | 4.86 | 4.84 | 4.84 | 4.84 | 4.82 |

$\gamma =2.7$ | 5.06 | 5.04 | 5.04 | 5.04 | 5.04 |

$\gamma =2.8$ | 5.20 | 5.20 | 5.20 | 5.18 | 5.18 |

$\gamma =2.9$ | 5.32 | 5.32 | 5.30 | 5.30 | 5.30 |

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Bertotti, M.L.; Modanese, G.
Comparison of Simulations with a Mean-Field Approach vs. Synthetic Correlated Networks. *Symmetry* **2021**, *13*, 141.
https://doi.org/10.3390/sym13010141

**AMA Style**

Bertotti ML, Modanese G.
Comparison of Simulations with a Mean-Field Approach vs. Synthetic Correlated Networks. *Symmetry*. 2021; 13(1):141.
https://doi.org/10.3390/sym13010141

**Chicago/Turabian Style**

Bertotti, Maria Letizia, and Giovanni Modanese.
2021. "Comparison of Simulations with a Mean-Field Approach vs. Synthetic Correlated Networks" *Symmetry* 13, no. 1: 141.
https://doi.org/10.3390/sym13010141