A New Vertex Centrality Algorithm For Social Networks Based On Matrix Factorization Technique

The importance of vertices,also known as "centrality",is an important field of social network analysis.In recent years,it has received extensive attention and research from scholars.Identifying important vertices in the network has a significant impact on soci-ety,economy and everyone's lives.Important vertices in the network have more influence than other vertices,and they can influence the structure and function of the network to a greater extent.Studying the vertex centrality algorithm of the network and identifying important vertices in the network are important for many practical application scenarios,such as controlling the outbreak of infectious diseases,predicting the future trend of epi-demics,controlling public opinion,conducting successful advertisements for e-commercial products and so on.Many classical centrality algorithms have been proposed in recent years,such as de-gree centrality,closeness centrality,betweenness centrality,eigenvector centrality and so on.Whether it is a simple algorithm based on calculating the number of neighbor ver-tices or a complex algorithm based on the number and length of shortest paths.They all give indicators to measure the importance of vertices from different perspectives.These classical algorithms consider all edges in the network.But there are situations in which researchers hope to ignore certain dyads in the computation of centrality to avoid biased or misleading results,but simply deleting these dyads will result in wrong conclusions.In 2015,P.Bonacich proposed eigenvector-like centrality for this case.When minimizing the squared error function,eigenvector-like centrality mainly uses the first-order approx-imation of adjacency matrix to exclude the errors caused by the edges which need to be ignored.As far as we know,there is no other algorithm for this problem except the eigenvector-like centrality algorithm.Therefore,this paper proposes a new centrality algorithm for ignoring some edges in the network:degree-like centrality.The main idea of degree-like centrality algorithm is to generalize the first-order matrix approximation xxT in the eigenvector-like centrality to the K-order matrix approximation UUT.Then compute the row sum of the approximate matrix UUT as the final degree-like centrality score.The high order(but low rank)approximation of the adjacency matrix preserves more information about the network.By eliminating the errors caused by the dyads which need to be "ignored" in the objective function of optimizing the high order approximation,we can realize the purpose of ignoring these dyads when calculating the vertex centrality.To calculate the degree-like scores of vertices,we use a new matrix factorization form,namely weighted symmetric non-negative matrix decomposition(WSNMF),and design a hybrid EM-WSNMF algorithm to solve this matrix factorization problem.In this paper,a weighted symmetric nonnegative matrix factorization technique is applied to vertex centrality for the first time.Experimental results on several data sets show the effectiveness of the algorithm.