# Link Prediction in Complex Networks Using Average Centrality-Based Similarity Score

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## Abstract

**:**

## 1. Introduction

## 2. Problem Definition

**Definition 1.**

**Definition 2.**

## 3. Recent Work

## 4. Related Work

#### 4.1. Existing Similarity Measures

**Local Similarity Measures:**Local similarity measures focus on examining the immediate neighbors of a node in the network. Some well-known measures include the common neighbor (CN) [15], Jaccard coefficient (JC) [3], preferential attachment (PA) [30], Adamic–Adar (AA) [31], resource allocation (RA) [32], etc.**Common Neighbor:**The likelihood of a link being formed between two nodes, v and u, is higher when they share a significant number of common neighbors.$${S}_{v,u}^{CN}=|\mathsf{\Gamma}\left(v\right)\cap \mathsf{\Gamma}\left(u\right)|$$In Equation (1), ${S}_{v,u}^{CN}$ denotes the size of the nodes’ neighborhoods’ intersection; $\mathsf{\Gamma}\left(v\right)$ is the set of neighbors of node v.**Jaccard Coefficient:**The common neighbor is comparable to this metric, which normalizes the score of the common neighbor, as given below.$${S}_{v,u}^{JC}=\frac{\left|\mathsf{\Gamma}\right(v)\cap \mathsf{\Gamma}(u\left)\right|}{\left|\mathsf{\Gamma}\right(v)\cup \mathsf{\Gamma}(u\left)\right|}$$In Equation (2), ${S}_{v,u}^{JC}$ is the size of the intersection of two nodes’ neighborhoods, out of the total neighbors of nodes v and u, where $\mathsf{\Gamma}\left(v\right)$ is the set of neighbors of node v.**Preferential Attachment:**It counts the richness of two nodes instead of shared neighbors between non-adjacent node pairs. The degrees of nodes v and u are multiplied collectively.$${S}_{v,u}^{PA}=\left|d\left(v\right)\right|\ast \left|d\left(u\right)\right|$$$PA$ requires the degree of nodes and does not consider common neighbors. In Equation (3), $d\left(v\right)$ is the degree of node v.**Resource Allocation:**We assume two non-adjacent node pairs, v and u. The amount of resources provided from node v to node u determines how similar the two nodes are when they are transferring resources through their shared nodes.$${S}_{v,u}^{RA}=\sum _{r\in \mathsf{\Gamma}\left(v\right)\cap \mathsf{\Gamma}\left(u\right)}\frac{1}{{d}_{r}}$$**Adamic–Adar:**Adamic–Adar is a variant of resource allocation. In real-world scenarios, for example, individuals with a larger number of friends tend to allocate less time and resources to particular friend compared to those with fewer friends. This is defined as follows:$$S}_{v,u}^{AA}=\sum _{r\in \mathsf{\Gamma}\left(v\right)\cap \mathsf{\Gamma}\left(u\right)}\frac{1}{log|{d}_{r}|$$

#### 4.2. Recent Measures

**Common Neighbor and Centrality-based Parameterized Algorithm (CCPA):**To recommend the creation of new linkages in complex networks, CCPA uses two essential node characteristics—the number of shared neighbors between node pairs, and their centrality measures. In this case, closeness centrality is taken into account as a parameter for missing link prediction. The term “common neighbor” describes the nodes that are shared by two nodes. The term “centrality” refers to the significance of a node inside the network.$${S}_{v,u}^{CCPA}=\alpha \xb7\left(\left|\mathsf{\Gamma}\right(v)\cap \mathsf{\Gamma}(u\left)\right|\right)+(1-\alpha )\xb7\frac{N}{{D}_{v,u}}$$In Equation (6), the user-generated parameter $\alpha \in [0,1]$ regulates the centrality and common neighbor relevance. The set of neighbors of node v is represented by $\mathsf{\Gamma}\left(v\right)$, and ${D}_{v,u}$ is the shortest path length between v and u.**Keyword Network Link Prediction Algorithm (KNLP):**KNLP depends on the nodes’ clustering coefficient, and their centrality measure like eigenvector centrality [33]. The stronger correlation between eigenvector centrality and node degree shows that nodes with the highest eigenvector have more connections. For nodes u and v, KNLP is defined as follows:$${S}_{v,u}^{KNLP}=\frac{C{S}_{v}+C{S}_{u}}{C{C}_{v}+C{C}_{u}+\u03f5}$$In Equation (7), $C{S}_{v}$ and $C{S}_{u}$ are the centrality scores for nodes v and u, $C{C}_{v}$ and $C{C}_{u}$ are clustering coefficient values for nodes v and u, and their values always range between 0 and 1. Here, $\u03f5$ is used to avoid the division by zero error.

#### 4.3. Centrality Measures

**Local Centrality:**Local centrality involves only immediate neighborhood. Degree centrality (D) [5] and clustering coefficient (CC) [35] are two popular local centralities used in this paper.**Degree Centrality:**The node v’s degree centrality is calculated as the fraction of other nodes adjacent to node v out of the possible total. Nodes characterized by a high degree of centrality are referred to as $Hub$ nodes.$${C}_{D}\left(v\right)=\frac{{d}_{v}}{N-1}$$**Clustering Coefficient:**The clustering coefficient of a specific node is determined by the ratio of closed triangles within the node’s neighborhood, to the total number of triangles present in that neighborhood. It is also known as transitivity.$${C}_{CC}\left(v\right)=\frac{2{K}_{v}}{{d}_{v}({d}_{v}-1)}$$In Equation (9), node v has a degree of ${d}_{v}$, and the number of triangles connected to node v is ${K}_{v}$.**Global Centrality:**Global centrality involves the whole graph. Closeness centrality (C) [34] and betweenness centrality (B) [36] are few popular global centralities used in this paper.**Closeness Centrality:**One method of identifying nodes that can efficiently distribute information throughout a network is through closeness centrality. The closeness centrality of a node, denoted as v, within a graph, is determined by taking the reciprocal of the average shortest path distance from node v to all $N-1$ reachable nodes in the graph.$${C}_{C}\left(v\right)=\frac{N-1}{{\sum}_{u\neg v}{D}_{v,u}}$$In Equation (10), the shortest path length from v to u is denoted by ${D}_{v,u}$. In the network, the node that is nearest to every other node is the one with the highest closeness centrality.**Betweenness Centrality:**A node’s betweenness centrality is a measure of how many shortest paths there are via a particular node.$${C}_{B}\left(v\right)=\sum _{v,u\in V}\frac{{\sigma}_{v,u}\left(r\right)}{{\sigma}_{v,u}}$$In Equation (11), ${\sigma}_{v,u}$ represents the total number of shortest paths between nodes v and u, and ${\sigma}_{v,u}\left(r\right)$ denotes the total number of shortest paths between nodes v and u that pass through node r.

## 5. Proposed Work

#### 5.1. Similarity Based on Average Centrality Measures (SAC)

Algorithm 1: An algorithm for common neighbor-based average centrality |

#### 5.2. Time Complexity of Similarity Based on Average Centrality Measures

## 6. Implementation

#### 6.1. Datasets

#### 6.2. Evaluation Metrics

**AUROC:**AUROC, short for Area Under the Receiver Operating Characteristic (ROC), is a widely used metric for assessing the effectiveness of a prediction model. The ROC curve is a visual representation that illustrates the relationship between the True Positive Rate (TPR) and the (FPR). The TPR (y-axis) vs. FPR (x-axis) is plotted for various threshold values [39]. AUROC gives the area under the ROC curve. AUROC measures the probability of false alarms or incorrect positive predictions. The AUROC score has a range from 0 to 1, where a higher value signifies superior performance. An AUROC of 1 represents a perfect model, while an AUROC of 0.5 indicates a random model.

**AUPR:**AUPR stands for Area Under Precision-Recall (PR) curve, is another metric used to evaluate the performance of a prediction model. AUPR demonstrates superior performance in scenarios where the ROC curve may provide an overly optimistic assessment of a predictor’s performance, especially with imbalanced data [40,41]. The PR curve displays the precision on the y-axis and the recall on the x-axis. Precision quantifies the ratio of correct positive predictions to all positive predictions, while recall calculates the ratio of correct positive predictions to all actual positive instances. AUPR is a single quantity that represents the area under PR curve.

## 7. Results

#### 7.1. Comparing Proposed Similarity-Based Centralities with Existing Similarity-Based Link Prediction Measures

#### 7.2. Comparing Proposed Measures

#### 7.3. Comparing Proposed Measures with Recent Methods like CCPA and KNLP

#### 7.4. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

LP | Link prediction |

CMs | Centrality measures |

CNs | Common neighbors |

JC | Jaccard coefficient |

AA | Adamic–Adar |

RA | Resource allocation |

PA | Preferential attachment |

D | Degree centrality |

B | Betweenness centrality |

C | Closeness centrality |

CC | Clustering coefficient |

CCPA | Common Neighbor and Centrality-based Parameterized Algorithm |

KNLP | Keyword network link prediction algorithm |

SAC_D | Similarity based on Average Degree |

SAC_B | Similarity based on Average Betweenness |

SAC_C | Similarity based on Average Closeness |

SAC_CC | Similarity based on Average Clustering Coefficient |

AUROC | Area Under the Receiver Operating Characteristic |

AUPR | Area Under Precision-Recall |

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**Figure 2.**AUROC scores for link prediction using common neighbors based on average centrality for top k node pairs, k ranging from 1750 to 35,000, for four datasets.

**Figure 3.**AUPR scores for link prediction using common neighbors based on average centrality for the top 35,000 node pairs across four datasets.

**Figure 4.**AUROC scores for proposed measures of top 35,000 node pairs across four datasets with SAC

_{D}(Similarity based on Average Degree), SAC

_{B}(Similarity based on Average Betweenness), SAC

_{C}(Similarity based on Average Closeness), and SAC

_{CC}(Similarity based on Average Clustering Coefficient).

**Figure 5.**AUPR for proposed measures of top 35,000 node pairs across four datasets with SAC

_{D}(Similarity based on Average Degree), SAC

_{B}(Similarity based on Average Betweenness), SAC

_{C}(Similarity based on Average Closeness), and SAC

_{CC}(Similarity based on Average Clustering Coefficient).

**Table 1.**$SA{C}_{\mathcal{C}}(v,u)$ is the proposed centrality, where $\mathcal{C}$ stands for D (degree), B (betweenness), C (closeness), and CC (clustering coefficient).

S.No. | Centrality $\mathcal{C}$ | Avg $\mathcal{C}$ | ${\mathit{SAC}}_{\mathcal{C}}\mathbf{(}\mathit{v}\mathbf{,}\mathit{u}\mathbf{)}$ |
---|---|---|---|

1 | ${C}_{D}\left(v\right)=\frac{{d}_{v}}{n-1}$ | $AD\left(G\right)=\frac{{\sum}_{v\in V\left(G\right)}{C}_{D}\left(v\right)}{N}$ | $\begin{array}{c}\hfill SA{C}_{D}(v,u)=\mid \{x\mid x\in \hfill \\ \hfill \mathsf{\Gamma}\left(v\right)\cap \mathsf{\Gamma}\left(u\right)\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}D\left(x\right)\ge AD\left(G\right)\}\mid \hfill \end{array}$ |

2 | ${C}_{B}\left(v\right)={\sum}_{p,q\in V}\frac{{\sigma}_{p,q}\left(v\right)}{{\sigma}_{p,q}}$ | $AB\left(G\right)=\frac{{\sum}_{v\in V\left(G\right)}{C}_{B}\left(v\right)}{N}$ | $\begin{array}{c}\hfill SA{C}_{B}(v,u)=\mid \{x\mid x\in \hfill \\ \hfill \mathsf{\Gamma}\left(v\right)\cap \mathsf{\Gamma}\left(u\right)\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}B\left(x\right)\ge AB\left(G\right)\}\mid \hfill \end{array}$ |

3 | ${C}_{C}\left(v\right)=\frac{n-1}{{\sum}_{u\neg v}{d}_{v,u}}$ | $AC\left(G\right)=\frac{{\sum}_{v\in V\left(G\right)}{C}_{C}\left(v\right)}{N}$ | $\begin{array}{c}\hfill SA{C}_{C}(v,u)=\mid \{x\mid x\in \hfill \\ \hfill \mathsf{\Gamma}\left(v\right)\cap \mathsf{\Gamma}\left(u\right)\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}C\left(x\right)\ge AC\left(G\right)\}\mid \hfill \end{array}$ |

4 | ${C}_{CC}\left(v\right)=\frac{2{K}_{v}}{{d}_{v}({d}_{v}-1)}$ | $ACC\left(G\right)=\frac{{\sum}_{v\in V}{C}_{CC}\left(v\right)}{N}$ | $\begin{array}{c}\hfill SA{C}_{CC}(v,u)=\mid \{x\mid x\in \hfill \\ \hfill \mathsf{\Gamma}\left(v\right)\cap \mathsf{\Gamma}\left(u\right)\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}CC\left(x\right)\ge ACC\left(G\right)\}\mid \hfill \end{array}$ |

**Table 2.**SAC

_{D}(Similarity based on Average Degree), SAC

_{B}(Similarity based on Average Betweenness), SAC

_{C}(Similarity based on Average Closeness), SAC

_{CC}(Similarity based on Average Clustering Coefficient), CN (common neighbor), JC (Jaccard coefficient), PA (preferential attachment), RA (resource allocation), AA (Adamic–Adar), CCPA (Common Neighbor and Centrality-based Parameterized Algorithm), and KNLP (keyword network link prediction algorithm) similarity scores for non-adjacent node pairs for a graph are shown in Figure 1.

Various Measures | Node Pair (v,u) | (1,2) | (2,3) | (2,6) | (4,7) | (4,8) | (5,7) |
---|---|---|---|---|---|---|---|

Proposed Measures | SAC_{D} (v,u) | 1 | 1 | 2 | 2 | 2 | 1 |

SAC_{B} (v,u) | 1 | 1 | 2 | 2 | 1 | 1 | |

SAC_{C} (v,u) | 2 | 1 | 2 | 1 | 1 | 1 | |

SAC_{CC} (v,u) | 0 | 0 | 1 | 0 | 2 | 0 | |

Basic Measures | ${S}_{v,u}^{CN}$ | 2 | 1 | 2 | 2 | 2 | 2 |

${S}_{v,u}^{JC}$ | 0.5 | 0.2 | 0.5 | 0.4 | 0.4 | 0.2 | |

${S}_{v,u}^{AA}$ | 2 | 0.6 | 1.3 | 1.8 | 1.6 | 0.9 | |

${S}_{v,u}^{RA}$ | 0.7 | 0.2 | 0.4 | 0.6 | 0.5 | 0.3 | |

${S}_{v,u}^{PA}$ | 9 | 6 | 9 | 10 | 10 | 8 | |

Recent Measures | ${S}_{v,u}^{CCPA}$ | 2.4 | 1.5 | 2.4 | 2 | 2.4 | 1.5 |

${S}_{v,u}^{KNLP}$ | 0.9 | 0.4 | 0.7 | 2.3 | 0.5 | 1.2 |

Datasets | #Nodes | #Edges | #Max. Degree | #Avg. Degree | #Diameter | #Avg. Clust. Coeff. |
---|---|---|---|---|---|---|

bio-celegans | 453 | 2025 | 237 | 8.94 | 7 | 0.646 |

web-polblogs | 643 | 2280 | 165 | 7.09 | 10 | 0.232 |

CA-Grqc | 5242 | 14,496 | 81 | 5 | 17 | 0.529 |

Facebook-large | 22,470 | 171,002 | 709 | 15.22 | 15 | 0.359 |

**Table 4.**Performance of the proposed measures against existing measures in terms of AUROC for the top k predictions, at various thresholds of k.

Datasets | k | SAC_{D} | SAC_{B} | SAC_{C} | SAC_{CC} | CCPA | KNLP |
---|---|---|---|---|---|---|---|

CA-Grqc | 1750 | 0.909 | 0.746 | 0.875 | 0.919 | 0.859 | 0.549 |

8750 | 0.91 | 0.784 | 0.818 | 0.918 | 0.859 | 0.344 | |

17,500 | 0.911 | 0.847 | 0.878 | 0.918 | 0.842 | 0.392 | |

26,250 | 0.907 | 0.828 | 0.895 | 0.923 | 0.851 | 0.482 | |

35,000 | 0.913 | 0.766 | 0.862 | 0.926 | 0.853 | 0.444 | |

Facebook-large | 1750 | 0.532 | 0.606 | 0.533 | 0.622 | 0.626 | 0.317 |

8750 | 0.597 | 0.625 | 0.617 | 0.679 | 0.571 | 0.304 | |

17,500 | 0.625 | 0.623 | 0.638 | 0.683 | 0.607 | 0.251 | |

26,250 | 0.648 | 0.627 | 0.648 | 0.695 | 0.59 | 0.257 | |

35,000 | 0.658 | 0.628 | 0.668 | 0.697 | 0.591 | 0.392 | |

web-polblogs | 1750 | 0.856 | 0.718 | 0.743 | 0.679 | 0.721 | 0.456 |

8750 | 0.877 | 0.772 | 0.78 | 0.747 | 0.771 | 0.384 | |

17,500 | 0.883 | 0.729 | 0.749 | 0.709 | 0.785 | 0.407 | |

26,250 | 0.893 | 0.739 | 0.762 | 0.713 | 0.788 | 0.385 | |

35,000 | 0.884 | 0.761 | 0.801 | 0.7 | 0.771 | 0.376 | |

bio-celegans | 1750 | 0.863 | 0.627 | 0.674 | 0.905 | 0.803 | 0.79 |

8750 | 0.92 | 0.636 | 0.672 | 0.913 | 0.822 | 0.785 | |

17,500 | 0.896 | 0.639 | 0.723 | 0.9 | 0.803 | 0.824 | |

26,250 | 0.9 | 0.697 | 0.734 | 0.921 | 0.816 | 0.802 | |

35,000 | 0.898 | 0.666 | 0.749 | 0.915 | 0.832 | 0.788 |

**Table 5.**Performance of the proposed measures against existing measures in terms of AUPR for the top k predictions, at various thresholds of k.

Datasets | k | SAC_{D} | SAC_{B} | SAC_{C} | SAC_{CC} | CCPA | KNLP |
---|---|---|---|---|---|---|---|

CA-Grqc | 1750 | 0.908 | 0.6756 | 0.9019 | 0.9002 | 0.5403 | 0.0002 |

8750 | 0.7507 | 0.4843 | 0.6783 | 0.7915 | 0.5368 | 0.0002 | |

17,500 | 0.7043 | 0.4498 | 0.6517 | 0.7123 | 0.5104 | 0.008 | |

26,250 | 0.6569 | 0.4327 | 0.6487 | 0.7074 | 0.5341 | 0.0023 | |

35,000 | 0.6244 | 0.3336 | 0.5791 | 0.7249 | 0.5345 | 0.0001 | |

Facebook-large | 1750 | 0.5815 | 0.4803 | 0.5973 | 0.8353 | 0.2132 | 0.0001 |

8750 | 0.488 | 0.3619 | 0.5098 | 0.6943 | 0.2092 | 0.0002 | |

17,500 | 0.433 | 0.299 | 0.4432 | 0.6023 | 0.2151 | 0.0002 | |

26,250 | 0.4115 | 0.285 | 0.4081 | 0.5603 | 0.2431 | 0.0001 | |

35,000 | 0.3758 | 0.2482 | 0.3953 | 0.5222 | 0.2268 | 0.0314 | |

web-polblogs | 1750 | 0.2998 | 0.2419 | 0.2428 | 0.2362 | 0.094 | 0.003 |

8750 | 0.1822 | 0.1802 | 0.1948 | 0.2126 | 0.0676 | 0.0015 | |

17,500 | 0.1568 | 0.1338 | 0.1273 | 0.2009 | 0.0687 | 0.0024 | |

26,250 | 0.1769 | 0.1447 | 0.1171 | 0.1642 | 0.0733 | 0.0021 | |

35,000 | 0.1403 | 0.1559 | 0.1418 | 0.1446 | 0.0911 | 0.0016 | |

bio-celegans | 1750 | 0.2232 | 0.1433 | 0.2572 | 0.453 | 0.0753 | 0.0211 |

8750 | 0.1744 | 0.1177 | 0.1483 | 0.3867 | 0.095 | 0.0273 | |

17,500 | 0.1396 | 0.091 | 0.1345 | 0.4003 | 0.0755 | 0.0356 | |

26,250 | 0.108 | 0.0786 | 0.1461 | 0.3736 | 0.0921 | 0.0297 | |

35,000 | 0.0887 | 0.0806 | 0.1265 | 0.4505 | 0.081 | 0.0265 |

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## Share and Cite

**MDPI and ACS Style**

Nandini, Y.V.; Lakshmi, T.J.; Enduri, M.K.; Sharma, H.
Link Prediction in Complex Networks Using Average Centrality-Based Similarity Score. *Entropy* **2024**, *26*, 433.
https://doi.org/10.3390/e26060433

**AMA Style**

Nandini YV, Lakshmi TJ, Enduri MK, Sharma H.
Link Prediction in Complex Networks Using Average Centrality-Based Similarity Score. *Entropy*. 2024; 26(6):433.
https://doi.org/10.3390/e26060433

**Chicago/Turabian Style**

Nandini, Y. V., T. Jaya Lakshmi, Murali Krishna Enduri, and Hemlata Sharma.
2024. "Link Prediction in Complex Networks Using Average Centrality-Based Similarity Score" *Entropy* 26, no. 6: 433.
https://doi.org/10.3390/e26060433