The Degree Analysis of an Inhomogeneous Growing Network with Two Types of Vertices

Abstract

We consider an inhomogeneous growing network with two types of vertices. The degree sequences of two different types of vertices are investigated, respectively. We not only prove that the asymptotical degree distribution of type $s$ for this process is power law with exponent $\mathrm{2}+\left(\left(\mathrm{1}+\delta \right)q_s+\beta \left(\mathrm{1}-q_s\right)\right)/\alpha q_s$, but also give the strong law of large numbers for degree sequences of two different types of vertices by using a different method instead of Azuma’s inequality. Then we determine asymptotically the joint probability distribution of degree for pairs of adjacent vertices with the same type and with different types, respectively.