  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 ($\alpha < 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 cavity method is used to clarify these results.
