# Influence of clustering coefficient on network embedding in link prediction

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## Abstract

Multiple network embedding algorithms have been proposed to perform the prediction of missing or future links in complex networks. However, we lack the understanding of how network topology affects their performance, or which algorithms are more likely to perform better given the topological properties of the network. In this paper, we investigate how the clustering coefficient of a network, i.e., the probability that the neighbours of a node are also connected, affects network embedding algorithms’ performance in link prediction, in terms of the AUC (area under the ROC curve). We evaluate classic embedding algorithms, i.e., Matrix Factorisation, Laplacian Eigenmaps and node2vec, in both synthetic networks and (rewired) real-world networks with variable clustering coefficient. Specifically, a rewiring algorithm is applied to each real-world network to change the clustering coefficient while keeping key network properties. We find that a higher clustering coefficient tends to lead to a higher AUC in link prediction, except for Matrix Factorisation, which is not sensitive to the change of clustering coefficient. To understand such influence of the clustering coefficient, we (1) explore the relation between the link rating (probability that a node pair is the missing link) derived from the aforementioned algorithms and the number of common neighbours of the node pair, and (2) evaluate these embedding algorithms’ ability to reconstruct the original training (sub)network. All the network embedding algorithms that we tested tend to assign higher likelihood of connection to node pairs that share an intermediate or high number of common neighbours, independently of the clustering coefficient of the training network. Then, the predicted networks will have more triangles, thus a higher clustering coefficient. As the clustering coefficient increases, all the algorithms but Matrix Factorisation could also better reconstruct the training network. These two observations may partially explain why increasing the clustering coefficient improves the prediction performance.