Fat-tailed distribution derived from the first eigenvector of a symmetric random sparse matrix

Many solutions for scientific problems rely on finding the first (largest) eigenvalue and eigenvector of a particular matrix. We explore the distribution of the first eigenvector of a symmetric random sparse matrix. To analyze the properties of the first eigenvalue/vector, we employ a methodology based on the cavity method, a well-established technique in the statistical physics. A symmetric random sparse matrix in this paper can be regarded as an adjacency matrix for a network. We show that if a network is constructed by nodes that have two different types of degrees then the distribution of its eigenvector has fat tails such as the stable distribution (α < 2) under a certain condition; whereas if a network is constructed with nodes that have only one type of degree, the distribution of its first eigenvector becomes the Gaussian approximately. The method consisting of the cavity method and the population dynamical method clarifies these results.