Detection of hubs in complex networks by the Laplacian matrix
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Online ISSN 2005-2863 Print ISSN 1226-3192
RESEARCH ARTICLE
Detection of hubs in complex networks by the Laplacian matrix Younghee Hong1 · Iksoo Chang2 · Choongrak Kim3 Received: 31 December 2019 / Accepted: 16 September 2020 © Korean Statistical Society 2020
Abstract We propose a definition of hub in complex networks by using the eigenvectors of the Laplacian matrix, and suggest a method of detecting hubs. The proposed definition provides a different concept from the classical measures such as the centrality or degree. Also, a method of determining the number of hubs is suggested using a scree plot. Illustrative examples based on artificial data sets and real data sets are given. Keywords Adjacency matrix · Conditional dependency · Eigenvalue · Eigenvector
1 Introduction Lots of academic interests are in complex networks such as the world-wide web, the internet, biological networks, social networks, and so on. A wonderful scientific fact is that any complex network can be represented by a graph, and any graph can be represented by an adjacency matrix. Further, a degree matrix and/or a Laplacian matrix are derived by an adjacency matrix. Graph theory has been developed by mathematicians for a long time, and mathematicians are interested in properties of the eigenvalues of an adjacency matrix; distribution of eigenvalues (Wigner 1955; Marcenko and Pastur 1967) and distribution of the largest eigenvalue (Tracy and Widom 1996), lower and/or upper bound of eigenvalues, relationship between
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)(No.2019R1A2C1007193 to C. Kim) and the Ministry of Science, Technology and ICT(NRF-2017R1E1A1A03070854 to I.Chang). * Choongrak Kim [email protected] 1
Statistics Research Institute, Daejeon, Korea
2
Daegu Gyeongbuk Institute of Science and Technology, Daegu, Korea
3
Pusan National University, Busan, Korea
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Journal of the Korean Statistical Society
degree and eigenvalues, and so on. There are numerous books on the graph theory, and one of good references is (Mieghem 2010). On the other hand, statisticians have paid attention to the graph theory quite recently, and they are mainly interested in graphical models and estimation of an adjacency matrix using available observations. For a graph to be defined, the adjacency between two vertices should be estimated (or given, sometimes). Therefore, appropriate estimation of adjacency is necessary for a graph to be well defined, and estimation of adjacency is highly related with estimation of covariance matrix or precision matrix (Dempster 1972), i.e., the inverse of covariance matrix. We say, under Gaussian assumption, the adjacency is marginal or conditional if it is based on the covariance or the precision matrix, respectively. Recently, statisticians are interested in the inference on high dimensional data where p (number of variables) is larger than n (sample size) because lots of networks such as biological networks and social netw
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