# Evolutionary Features for Dynamic Link Prediction in Social Networks

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## Abstract

**:**

## 1. Introduction

**dynamic link prediction**or link prediction in dynamic networks. Dynamic link prediction is the process of inferring the possibility of future links among the dynamic entities or network actors through exploring historical or temporal information [18]. Different dynamic link prediction methods explore a wide range of techniques. Most of the techniques used a wide variety of structural and network topological features to compute the likelihood of link formations. For example, Zhang et al. [19] used a node (i.e., actor) centrality-based temporal link prediction where the authors distinguished the contributions of common neighbors to connection likelihood by their eigenvector centralities. By considering the importance of nodes as the probability of attracting other nodes, Wu et al. [20] used an eigenvector-based node ranking strategy along with a forecasting method called Adaptive Weighted Moving (AWM) for dynamic link prediction. Chi et al. [21] categorized the nodes into different levels based on the influence strength of the node compared to its neighbors that change over time. The authors computed the connection probability between a pair of nodes using their corresponding levels of influence strength and the attraction force between them to predict the missing links in dynamic networks. Chen and Li [22] formulated the link prediction problem in dynamic networks as an optimization problem that not only collectively leveraged the structural and temporal information to better infer a low-rank representation for each node but also preserved the deep network structure via high-order proximity among nodes. The authors also used an efficient block coordinate gradient descent approach to address the optimization problem.

## 2. Dynamic Similarity Metrics

#### 2.1. Actor-Level Evolution in a Dynamic Network

#### 2.1.1. Structural Dynamicity

#### 2.1.2. Neighbourhood Dynamicity

#### 2.2. Dynamic Features

#### 2.2.1. Temporal Similarity

#### 2.2.2. Correlation-Based Similarity

#### 2.2.3. Bray–Curtis Similarity

## 3. Network Datasets and Experimental Settings

#### 3.1. Datasets

#### 3.1.1. Undirected Networks

#### 3.1.2. Directed Networks

#### 3.1.3. Co-Authorship Networks

#### 3.2. Supervised Link Prediction

#### 3.3. Performance Evaluation

## 4. Results

#### 4.1. Classification Performance

#### 4.2. Feature Importance

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Visualisation of a dynamic link prediction framework considering a series of evolutionary network snapshots at different discrete timestamps (t = 1, 2, 3, 4) using abstract data.

**Figure 2.**Visualizations of measuring similarity between two temporal sequences (

**a**) traditional approach; (

**b**) Dynamic Time Warping approach. Dashed lines represent the distance between corresponding points in both time series.

**Figure 3.**P–R curves (

**left column**) and ROC curves (

**right column**) for undirected network ${G}_{UCI}$ (

**top row**), co-authorship network ${G}_{th}$ (

**middle row**), and directed network ${G}_{Rtwt}$ (

**bottom row**).

**Table 1.**Five different values of $si{m}_{i}(a,b)$ computed by using five different dynamic similarity metrics. Each metric computes the similarity/proximity between non-connected actor pair $(a,b)$ by considering their structural (i.e., ${\delta}^{a},{\delta}^{b}$) and neighbourhood (i.e., ${\lambda}^{a},{\lambda}^{b}$) dynamicity computed in dynamic networks comprised of T SINs.

Metric | Equation | Description |
---|---|---|

$si{m}_{1}(a,b)$ | $min\left({\sum}_{l=1}^{L}d({\delta}_{nl}^{a},{\delta}_{ml}^{b})\right)$ | Temporal similarity of structural dynamicity measured using Dynamic Time Warping (DTW) Technique |

$si{m}_{2}(a,b)$ | $min\left({\sum}_{l=1}^{L}d({\delta}_{nl}^{a},{\delta}_{ml}^{b})\right)$ | Temporal similarity of neighbourhood dynamicity measured using Dynamic Time Warping (DTW) Technique |

$si{m}_{3}(a,b)$ | $\frac{{\sum}_{t}^{}\left[({\delta}^{a}\left(t\right)-\overline{{\delta}^{a}})({\delta}^{b}\left(t\right)-\overline{{\delta}^{b}})\right]}{\sqrt{{\sum}_{t}{({\delta}^{a}\left(t\right)-\overline{{\delta}^{a}})}^{2}{\sum}_{t}{({\delta}^{b}\left(t\right)-\overline{{\delta}^{b}})}^{2}}}$ | Correlation between structural dynamicity of two non-connected actors computed using Pearson correlation |

$si{m}_{4}(a,b)$ | $\frac{{\sum}_{t}^{}\left[({\lambda}^{a}\left(t\right)-\overline{{\lambda}^{a}})({\lambda}^{b}\left(t\right)-\overline{{\lambda}^{b}})\right]}{\sqrt{{\sum}_{t}{({\lambda}^{a}\left(t\right)-\overline{{\lambda}^{a}})}^{2}{\sum}_{t}{({\lambda}^{b}\left(t\right)-\overline{{\lambda}^{b}})}^{2}}}$ | Correlation between neighbourhood dynamicity of two non-connected actors computed using Pearson correlation |

$si{m}_{5}(a,b)$ | $1-\frac{{\sum}_{t=1}^{T}[\left(\right)open="|"\; close="|">{\delta}^{a}\left(t\right)-{\delta}^{b}\left(t\right)+\left(\right)open="|"\; close="|">{\lambda}^{a}\left(t\right)-{\lambda}^{b}\left(t\right)}{]}$ | Similarity by the abundance of structural and neighbourhood dynamicity between two non-connected actors computed using Bray–Curtis dissimilarity measure |

**Table 2.**Basic statistics of network datasets used in this study. The training duration represents the interval used to generate temporal short-interval networks and the sampling interval denotes the sliding window sizes used to sample dynamic networks. SINs represent the number of short-interval networks or network snapshots generated using the corresponding window size.

Dataset | Actors | Links | Training Duration | Testing Duration | Sampling Interval | SINs | ||
---|---|---|---|---|---|---|---|---|

Start | End | Start | End | $\mathit{\tau}$ | ||||

${G}_{MIT}$ | 96 | 1,086,404 | 14 September 2004 | 31 January 2005 | 1 February 2005 | 5 May 2005 | 1 day | 140 |

7 days | 20 | |||||||

30 days | 5 | |||||||

${G}_{Email}$ | 167 | 82,927 | 2 January 2010 | 31 July 2010 | 1 August 2010 | 30 September 2010 | 1 day | 186 |

7 days | 31 | |||||||

30 days | 8 | |||||||

${G}_{UCI}$ | 1899 | 61,734 | 24 March 2004 | 31 May 2004 | 1 June 2004 | 26 October 2004 | 1 day | 45 |

7 days | 7 | |||||||

30 days | 3 | |||||||

${G}_{FF}$ | 11,715 | 42,698 | 1 January 2007 | 31 March 2007 | 1 April 2007 | 30 April 2007 | 1 day | 90 |

7 days | 13 | |||||||

30 days | 3 | |||||||

${G}_{retwt}$ | 14,370 | 39,124 | 14 September 2010 4 a.m. | 14 October 2010 4 a.m. | 14 October 2010 4 a.m. | 15 October 2010 4 a.m. | 6 h | 121 |

12 h | 61 | |||||||

24 h | 31 | |||||||

${G}_{th}$ | 6798 | 290,597 | 1 October 1993 | 31 December 1998 | 1 January 1999 | 10 December 1999 | 1 year | 6 |

${G}_{ph}$ | 16,959 | 2,322,259 | 15 March 1992 | 31 December 1998 | 1 January 1999 | 31 December 1999 | 1 year | 7 |

**Table 3.**Classification performances by three classifiers considering the classification datasets of undirected, directed networks and co-authorship networks considering three different window sizes used to sample dynamic networks. Both directed and undirected network datasets used three different sampling window sizes to generate SINs in the dynamic networks, whereas the co-authorship networks used only a yearly sliding window.

Undirected Network | |||||||||
---|---|---|---|---|---|---|---|---|---|

RandomForest | |||||||||

Accuracy (%) | AUCROC | AUCPR | |||||||

Days | 1 | 7 | 30 | 1 | 7 | 30 | 1 | 7 | 30 |

${G}_{MIT}$ | 82.19 | 80.52 | 84.91 | 0.683 | 0.663 | 0.700 | 0.30 | 0.46 | 0.29 |

${G}_{Email}$ | 76.29 | 87.47 | 88.23 | 0.714 | 0.644 | 0.724 | 0.40 | 0.32 | 0.31 |

${G}_{UCI}$ | 89.46 | 84.95 | 84.67 | 0.764 | 0.713 | 0.654 | 0.34 | 0.29 | 0.29 |

${G}_{FF}$ | 85.03 | 84.98 | 85.33 | 0.687 | 0.636 | 0.773 | 0.39 | 0.36 | 0.43 |

Bagging | |||||||||

${G}_{MIT}$ | 70.69 | 71.71 | 71.77 | 0.611 | 0.614 | 0.671 | 0.33 | 0.44 | 0.31 |

${G}_{Email}$ | 77.22 | 77.69 | 75.96 | 0.656 | 0.594 | 0.603 | 0.34 | 0.27 | 0.33 |

${G}_{UCI}$ | 84.47 | 83.81 | 82.99 | 0.630 | 0.619 | 0.632 | 0.29 | 0.31 | 0.28 |

${G}_{FF}$ | 73.11 | 72.80 | 72.22 | 0.622 | 0.588 | 0.644 | 0.35 | 0.32 | 0.39 |

Logistic Regression | |||||||||

${G}_{MIT}$ | 73.30 | 72.22 | 72.68 | 0.536 | 0.613 | 0.590 | 0.26 | 0.38 | 0.22 |

${G}_{Email}$ | 78.23 | 77.91 | 78.13 | 0.654 | 0.637 | 0.563 | 0.36 | 0.30 | 0.25 |

${G}_{UCI}$ | 85.25 | 84.73 | 84.64 | 0.628 | 0.573 | 0.619 | 0.29 | 0.26 | 0.22 |

${G}_{FF}$ | 75.44 | 75.22 | 75.03 | 0.664 | 0.618 | 0.579 | 0.40 | 0.35 | 0.27 |

Directed Network | |||||||||

RandomForest | |||||||||

Hours | 6 | 12 | 24 | 6 | 12 | 24 | 6 | 12 | 24 |

${G}_{Retwt}$ | 87.87 | 87.59 | 87.03 | 0.739 | 0.712 | 0.720 | 0.36 | 0.26 | 0.26 |

Bagging | |||||||||

${G}_{Retwt}$ | 85.81 | 84.55 | 85.11 | 0.695 | 0.644 | 0.574 | 0.21 | 0.21 | 0.19 |

Logistic Regression | |||||||||

${G}_{Retwt}$ | 88.11 | 88.13 | 88.01 | 0.735 | 0.712 | 0.622 | 0.32 | 0.26 | 0.23 |

Co-Authorship Network (Window Size = 1 Year) | |||||||||

RandomForest | |||||||||

Accuracy | AUCROC | AUCPR | |||||||

${G}_{th}$ | 77.90 | 0.663 | 0.49 | ||||||

${G}_{ph}$ | 81.49 | 0.722 | 0.18 | ||||||

Bagging | |||||||||

${G}_{th}$ | 81.35 | 0.702 | 0.56 | ||||||

${G}_{ph}$ | 80.95 | 0.711 | 0.32 | ||||||

Logistic Regression | |||||||||

${G}_{th}$ | 66.45 | 0.593 | 0.43 | ||||||

${G}_{ph}$ | 70.90 | 0.581 | 0.11 |

**Table 4.**The rank of different dynamic features constructed in this study using different algorithms for directed, undirected and co-authorship networks. Ranks are in increasing order with number one denoting the highest ranking. The total column represents the aggregation of all ranking scores to generate the final ranking.

Feature Name | Information Gain | Chi-Square Attribute Evaluation | Support Vector Machine Evaluator | Random Forest Evaluator | Total |
---|---|---|---|---|---|

${G}_{MIT}$ | |||||

$si{m}_{1}(a,b)$ | 5 | 5 | 1 | 4 | 15 |

$si{m}_{2}(a,b)$ | 2 | 2 | 5 | 1 | 10 |

$si{m}_{3}(a,b)$ | 3 | 3 | 3 | 5 | 14 |

$si{m}_{4}(a,b)$ | 4 | 4 | 4 | 3 | 15 |

$si{m}_{5}(a,b)$ | 1 | 1 | 2 | 2 | 6 |

${G}_{Email}$ | |||||

$si{m}_{1}(a,b)$ | 5 | 5 | 3 | 3 | 16 |

$si{m}_{2}(a,b)$ | 2 | 2 | 1 | 2 | 7 |

$si{m}_{3}(a,b)$ | 3 | 3 | 5 | 1 | 12 |

$si{m}_{4}(a,b)$ | 4 | 4 | 2 | 4 | 14 |

$si{m}_{5}(a,b)$ | 1 | 1 | 4 | 5 | 11 |

${G}_{UCI}$ | |||||

$si{m}_{1}(a,b)$ | 2 | 2 | 1 | 2 | 7 |

$si{m}_{2}(a,b)$ | 1 | 1 | 2 | 1 | 5 |

$si{m}_{3}(a,b)$ | 5 | 5 | 3 | 4 | 17 |

$si{m}_{4}(a,b)$ | 4 | 4 | 5 | 5 | 18 |

$si{m}_{5}(a,b)$ | 3 | 3 | 4 | 3 | 13 |

${G}_{FF}$ | |||||

$si{m}_{1}(a,b)$ | 3 | 3 | 4 | 3 | 13 |

$si{m}_{2}(a,b)$ | 1 | 1 | 3 | 1 | 6 |

$si{m}_{3}(a,b)$ | 5 | 5 | 5 | 5 | 20 |

$si{m}_{4}(a,b)$ | 2 | 2 | 1 | 2 | 7 |

$si{m}_{5}(a,b)$ | 4 | 4 | 2 | 4 | 14 |

${G}_{Rtwt}$ | |||||

$si{m}_{1}(a,b)$ | 2 | 2 | 1 | 2 | 7 |

$si{m}_{2}(a,b)$ | 1 | 1 | 2 | 1 | 5 |

$si{m}_{3}(a,b)$ | 5 | 5 | 5 | 3 | 18 |

$si{m}_{4}(a,b)$ | 4 | 4 | 4 | 5 | 17 |

$si{m}_{5}(a,b)$ | 3 | 3 | 3 | 4 | 13 |

${G}_{th}$ | |||||

$si{m}_{1}(a,b)$ | 2 | 2 | 3 | 1 | 8 |

$si{m}_{2}(a,b)$ | 1 | 1 | 5 | 3 | 10 |

$si{m}_{3}(a,b)$ | 5 | 5 | 1 | 5 | 16 |

$si{m}_{4}(a,b)$ | 3 | 3 | 2 | 2 | 13 |

$si{m}_{5}(a,b)$ | 4 | 4 | 4 | 4 | 16 |

${G}_{ph}$ | |||||

$si{m}_{1}(a,b)$ | 5 | 5 | 2 | 4 | 16 |

$si{m}_{2}(a,b)$ | 3 | 3 | 1 | 1 | 8 |

$si{m}_{3}(a,b)$ | 2 | 2 | 3 | 2 | 9 |

$si{m}_{4}(a,b)$ | 1 | 1 | 5 | 5 | 12 |

$si{m}_{5}(a,b)$ | 4 | 4 | 4 | 3 | 15 |

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**MDPI and ACS Style**

Choudhury, N.; Uddin, S.
Evolutionary Features for Dynamic Link Prediction in Social Networks. *Appl. Sci.* **2023**, *13*, 2913.
https://doi.org/10.3390/app13052913

**AMA Style**

Choudhury N, Uddin S.
Evolutionary Features for Dynamic Link Prediction in Social Networks. *Applied Sciences*. 2023; 13(5):2913.
https://doi.org/10.3390/app13052913

**Chicago/Turabian Style**

Choudhury, Nazim, and Shahadat Uddin.
2023. "Evolutionary Features for Dynamic Link Prediction in Social Networks" *Applied Sciences* 13, no. 5: 2913.
https://doi.org/10.3390/app13052913