Recent developments of constructing adjacency matrix in network analysis

Abstract

Abstract In this paper, we review recent developments in network analysis using the graphtheory, and introduce ongoing research area with relevant theoretical results. In spe-ci c, we introduce basic notations in graph, and conditional and marginal approach inconstructing the adjacency matrix. Also, we introduce the Marcenko-Pastur law, theTracy-Widom law, the white Wishart distribution, and the spiked distribution. Finally,we mention the relationship between degrees and eigenvalues for the detection of hubsin a network.Keywords: Adjacency matrix, conditional dependency, graph theory, marginal depen-dency, Tracy-Widom law. 1. Introduction Recently Big data is one of the most hot issues in many branches of science, and complexnetwork problem is the central issue in Big data. Lots of interest arose in complex networkssuch as the world-wide web, the internet, biological networks, social networks, and so on. Awonderful scienti c fact is that any complex network can be represented by a graph, and anygraph can be represented by an adjacency matrix. Further, degree matrix and/or Laplacianmatrix are derived by the adjacency matrix.Graph theory has been developed by mathematicians for a long time, and mathematiciansare interested in properties of eigenvalues of adjacency matrix, distribution of eigenvalues(Wigner, 1955; Marcenko and Pastur, 1967) and distribution of the largest eigenvalue (Tracyand Widom, 1996), lower and/or upper bound of eigenvalues, relationship between degreeand eigenvalues, and so on. There are numerous books on the graph theory, and one of therecent and best references is Mieghem (2010).On the other hand, statisticians paid attention to the graph theory quite recently, and theyare mainly interested in graphical models and estimation of adjacency matrix using availableobservations. Recently, statisticians are interested in the case where p(number of variables)is larger than n(sample size) because lots of recent networks such as biological networks and