# Investigating the Influence of Inverse Preferential Attachment on Network Development

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Overall Network Characteristics of Simulated Networks

#### 3.2. Degree Distributions

#### 3.2.1. Test for Power Law Fits via Bootstrapping

_{min}value (the minimum value for which the power law holds; see the x

_{min}and α columns in Table 2). Note that all exponents were <2, lower than what is usually observed in real-world networks, where 2 < α < 3 [19]. This may be due to the simplicity of the simulations conducted (i.e., only 1 node and 1 edge were added to the network at each iteration), which led to sparser networks.

#### 3.2.2. Statistical Comparison with Other Degree Distributions

## 4. Discussion

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Table A1.**Degree distributions of simulated networks were fitted to exponential, log-normal, and Poisson distributions and compared against the fitted power law distribution using Vuong’s test of mis-specified, non-nested hypotheses. Note that each set of model comparisons was conducted for each of the 100 simulated networks per condition or network type. The present table displays the mean and standard deviations of the test statistic and p-values.

Power Law Vs. Exponential | ||||||

V-statistic | 2-sided p | 1-sided p | ||||

Network | M | SD | M | SD | M | SD |

PATT | 2.387 | 0.614 | 0.046 | 0.080 | 0.023 | 0.040 |

Inverse PATT | −0.156 | 0.251 | 0.800 | 0.113 | 0.560 | 0.098 |

Random | 0.420 | 0.359 | 0.682 | 0.222 | 0.347 | 0.120 |

PATT/Inverse PATT | ||||||

200/800 | 1.614 | 0.399 | 0.135 | 0.111 | 0.067 | 0.056 |

400/600 | 2.320 | 0.483 | 0.037 | 0.048 | 0.018 | 0.024 |

600/400 | 2.597 | 0.578 | 0.025 | 0.041 | 0.012 | 0.021 |

200/800 | 2.626 | 0.511 | 0.021 | 0.039 | 0.011 | 0.020 |

Inverse PATT/PATT | ||||||

200/800 | 1.116 | 0.357 | 0.293 | 0.151 | 0.147 | 0.076 |

400/600 | 0.593 | 0.288 | 0.570 | 0.176 | 0.285 | 0.088 |

600/400 | 0.221 | 0.241 | 0.817 | 0.161 | 0.416 | 0.088 |

200/800 | 0.009 | 0.253 | 0.849 | 0.120 | 0.498 | 0.097 |

Power law vs. Log-normal | ||||||

V-statistic | 2-sided p | 1-sided p | ||||

Network | M | SD | M | SD | M | SD |

PATT | −0.735 | 0.350 | 0.488 | 0.171 | 0.756 | 0.085 |

Inverse PATT | −0.695 | 0.227 | 0.498 | 0.137 | 0.751 | 0.069 |

Random | −0.672 | 0.267 | 0.517 | 0.160 | 0.742 | 0.080 |

PATT/Inverse PATT | ||||||

200/800 | −0.777 | 0.201 | 0.446 | 0.091 | 0.777 | 0.045 |

400/600 | −0.809 | 0.391 | 0.448 | 0.132 | 0.776 | 0.066 |

600/400 | −0.798 | 0.340 | 0.450 | 0.154 | 0.775 | 0.077 |

200/800 | −0.774 | 0.354 | 0.465 | 0.138 | 0.768 | 0.069 |

Inverse PATT/PATT | ||||||

200/800 | −0.519 | 0.272 | 0.618 | 0.172 | 0.691 | 0.086 |

400/600 | −0.500 | 0.272 | 0.631 | 0.176 | 0.685 | 0.088 |

600/400 | −0.514 | 0.260 | 0.620 | 0.167 | 0.690 | 0.084 |

200/800 | −0.590 | 0.270 | 0.570 | 0.171 | 0.715 | 0.085 |

Power law vs. Possion | ||||||

V-statistic | 2-sided p | 1-sided p | ||||

Network | M | SD | M | SD | M | SD |

PATT | 1.860 | 0.066 | 0.063 | 0.009 | 0.032 | 0.005 |

Inverse PATT | 3.198 | 0.411 | 0.003 | 0.005 | 0.002 | 0.003 |

Random | 2.453 | 0.151 | 0.015 | 0.006 | 0.008 | 0.003 |

PATT/Inverse PATT | ||||||

200/800 | 3.390 | 0.161 | 0.001 | 0.001 | 0.000 | 0.000 |

400/600 | 2.926 | 0.138 | 0.004 | 0.002 | 0.002 | 0.001 |

600/400 | 2.554 | 0.118 | 0.011 | 0.003 | 0.006 | 0.002 |

200/800 | 2.184 | 0.075 | 0.029 | 0.005 | 0.015 | 0.003 |

Inverse PATT/PATT | ||||||

200/800 | 1.899 | 0.072 | 0.058 | 0.009 | 0.029 | 0.005 |

400/600 | 1.967 | 0.084 | 0.050 | 0.009 | 0.025 | 0.005 |

600/400 | 2.103 | 0.118 | 0.037 | 0.010 | 0.018 | 0.005 |

200/800 | 2.460 | 0.240 | 0.017 | 0.012 | 0.008 | 0.006 |

**Figure A1.**Boxplots of two-sided p-values from Vuong’s test of non-nested models comparing the fits of power law and log-normal distributions to the degree distributions of simulated networks. Panel (

**a**) compares the degree distributions of pure networks and panel (

**b**) compares the degree distributions of blended networks. Non-significant p-values (based on an alpha-level of 0.05) indicate that neither distribution is preferred.

**Figure A2.**Boxplots of two-sided p-values from Vuong’s test of non-nested models comparing the fits of power law and Poisson distributions to the degree distributions of simulated networks. Panel (

**a**) compares the degree distributions of pure networks and panel (

**b**) compares the degree distributions of blended networks. Significant p-values (based on an alpha-level of 0.05) indicate that one distribution fits the empirical data better. Based on the one-sided p-values (see Table A1), the power law distribution provides a better fit than the Poisson distribution.

**Figure A3.**Cumulative degree distributions of exemplar networks generated via pure preferential attachment (

**left**), pure inverse preferential attachment (

**center**), and preferential attachment followed by inverse preferential attachment (hybrid model;

**right**).

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**Figure 1.**A summary of the 11 network growth conditions simulated in the present study. Red cells indicate growth by standard preferential attachment, blue cells indicate growth by inverse preferential attachment. PATT, preferential attachment.

**Figure 2.**Boxplots of ASPL (

**a**) and network diameter (

**b**) values of networks grown via preferential attachment (PATT), inverse preferential attachment (Inverse PATT), and random attachment (Random).

**Figure 3.**Network visualizations of exemplar networks generated via pure preferential attachment (

**left**), pure inverse preferential attachment (

**center**), and preferential attachment followed by inverse preferential attachment (hybrid model;

**right**). Each network consisted of 100 nodes. The size of each node reflects its degree.

**Figure 4.**Boxplots of ASPL (

**a**) and network diameter (

**b**) values of networks grown via blends of preferential attachment and inverse preferential attachment. The x-axis indicates the percentage proportion of nodes added before the network algorithm was switched. Red bars indicate networks first grown by PATT followed by inverse PATT. Blue bars indicate networks first grown by inverse PATT followed by PATT.

**Figure 5.**Boxplots of two-sided p-values from Vuong’s test of non-nested models comparing the fits of power law and exponential distributions to the degree distributions of simulated networks. Panel (

**a**) compares the degree distributions of pure networks and panel (

**b**) compares the degree distributions of blended networks. Non-significant p-values (based on an alpha-level of 0.05) indicate that neither distribution is preferred. Significant p-values (based on an alpha-level of 0.05) indicate that one distribution fits the empirical data better. Based on the 1-sided p-values (see Table A1), the power law distribution provides a better fit than the exponential distribution for all networks grown by preferential attachment, except for when the switch to its inverse variant occurs early.

**Table 1.**Means and standard deviations of network measures of simulated networks, summarized by each of the 11 simulation conditions. Note that all simulated networks had the same number of nodes and edges (1000 nodes and 999 edges).

Network | Nodes | Edges | ASPL | Diameter | |
---|---|---|---|---|---|

PATT | M | 1000 | 999 | 8.34 | 11.55 |

SD | 0 | 0 | 0.50 | 1.42 | |

Inverse PATT | M | 1000 | 999 | 13.07 | 16.42 |

SD | 0 | 0 | 0.62 | 1.80 | |

Random | M | 1000 | 999 | 10.91 | 13.81 |

SD | 0 | 0 | 0.52 | 1.54 | |

PATT–Inverse PATT | |||||

200/800 | M | 1000 | 999 | 9.97 | 13.27 |

SD | 0 | 0 | 0.50 | 1.47 | |

400/600 | M | 1000 | 999 | 9.22 | 12.58 |

SD | 0 | 0 | 0.49 | 1.58 | |

600/400 | M | 1000 | 999 | 8.83 | 12.29 |

SD | 0 | 0 | 0.49 | 1.43 | |

800/200 | M | 1000 | 999 | 8.55 | 11.93 |

SD | 0 | 0 | 0.49 | 1.50 | |

Inverse PATT–PATT | |||||

200/800 | M | 1000 | 999 | 11.25 | 14.28 |

SD | 0 | 0 | 0.59 | 1.60 | |

400/600 | M | 1000 | 999 | 12.00 | 15.17 |

SD | 0 | 0 | 0.61 | 1.63 | |

600/400 | M | 1000 | 999 | 12.48 | 15.59 |

SD | 0 | 0 | 0.59 | 1.76 | |

800/200 | M | 1000 | 999 | 12.81 | 15.92 |

SD | 0 | 0 | 0.62 | 1.74 |

**Table 2.**Power law scaling parameter estimates and uncertainty estimates from the bootstrap procedure. Note that the bootstrap procedure was conducted for each simulated network; the means and standard deviations of estimates are shown in the table.

x_{min} | α | Mean Bootstrapped α | SD Bootstrapped α | KS-Statistic | p-Value | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Network | M | SD | M | SD | M | SD | M | SD | M | SD | M | SD |

PATT | 2.09 | 2.27 | 1.45 | 0.06 | 1.55 | 0.08 | 0.36 | 0.14 | 0.11 | 0.02 | 0.74 | 0.16 |

Inverse PATT | 9.98 | 10.97 | 1.42 | 0.15 | 1.70 | 0.24 | 0.74 | 0.26 | 0.25 | 0.02 | 0.34 | 0.08 |

Random | 10.51 | 13.51 | 1.49 | 0.17 | 1.68 | 0.20 | 0.67 | 0.23 | 0.19 | 0.02 | 0.52 | 0.15 |

PATT/Inverse PATT | ||||||||||||

200/800 | 1.14 | 0.51 | 1.34 | 0.03 | 1.47 | 0.05 | 0.40 | 0.07 | 0.16 | 0.02 | 0.60 | 0.13 |

400/600 | 1.13 | 0.40 | 1.39 | 0.03 | 1.51 | 0.04 | 0.40 | 0.08 | 0.12 | 0.02 | 0.72 | 0.18 |

600/400 | 1.28 | 0.80 | 1.41 | 0.04 | 1.51 | 0.05 | 0.37 | 0.09 | 0.11 | 0.02 | 0.76 | 0.16 |

200/800 | 1.34 | 0.89 | 1.42 | 0.04 | 1.52 | 0.05 | 0.35 | 0.09 | 0.11 | 0.02 | 0.73 | 0.19 |

Inverse PATT/PATT | ||||||||||||

200/800 | 8.50 | 6.87 | 1.56 | 0.13 | 1.72 | 0.15 | 0.63 | 0.18 | 0.15 | 0.02 | 0.70 | 0.20 |

400/600 | 18.07 | 14.73 | 1.64 | 0.20 | 1.84 | 0.21 | 0.81 | 0.26 | 0.17 | 0.02 | 0.67 | 0.19 |

600/400 | 23.40 | 20.21 | 1.65 | 0.25 | 1.87 | 0.26 | 0.86 | 0.28 | 0.20 | 0.02 | 0.59 | 0.18 |

200/800 | 24.34 | 27.48 | 1.61 | 0.29 | 1.84 | 0.28 | 0.84 | 0.29 | 0.23 | 0.02 | 0.48 | 0.14 |

**Table 3.**Summary of Vuong’s tests of non-nested models comparing power law distributions to alternative distributions (exponential, log-normal, Poisson). The cell indicates the preferred distribution from the comparison; n.d. indicates that no distribution can be favored.

Network | PL vs. Exp | PL vs. LN | PL vs. Pos |
---|---|---|---|

PATT | PL | n.d. | PL |

Inverse PATT | n.d. | n.d. | PL |

Random | n.d. | n.d. | PL |

PATT/Inverse PATT | |||

200/800 | n.d. | n.d. | PL |

400/600 | PL | n.d. | PL |

600/400 | PL | n.d. | PL |

200/800 | PL | n.d. | PL |

Inverse PATT/PATT | |||

200/800 | n.d. | n.d. | PL |

400/600 | n.d. | n.d. | PL |

600/400 | n.d. | n.d. | PL |

200/800 | n.d. | n.d. | PL |

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**MDPI and ACS Style**

Siew, C.S.Q.; Vitevitch, M.S.
Investigating the Influence of Inverse Preferential Attachment on Network Development. *Entropy* **2020**, *22*, 1029.
https://doi.org/10.3390/e22091029

**AMA Style**

Siew CSQ, Vitevitch MS.
Investigating the Influence of Inverse Preferential Attachment on Network Development. *Entropy*. 2020; 22(9):1029.
https://doi.org/10.3390/e22091029

**Chicago/Turabian Style**

Siew, Cynthia S. Q., and Michael S. Vitevitch.
2020. "Investigating the Influence of Inverse Preferential Attachment on Network Development" *Entropy* 22, no. 9: 1029.
https://doi.org/10.3390/e22091029