Joint vertex degrees in the inhomogeneous random graph model \(\mathcal g(n, \{p_{ij}\})\) (Q2879911)

This paper provides a detailed study of the asymptotic distribution of the degree sequence of the an inhomogeneous random graph. We consider a set of \(n\) vertices and each edge between vertices \(i\) and \(j\) appears independently with probability that depends on \(i\) and \(j\). The main result of the paper has to do with the asymptotic distribution of the degree sequence of such a random graph with \(n\) vertices, that is, the vector whose \(i\)th element is the number of vertices that have degree \(i\) in the graph, for \(i=0,\ldots, n-1\). The paper also considers the truncated degree sequence, that is, the number of vertices that have degree \(i\) where \(i\) is larger than some constant \(M\). In this case, the results are expressed in terms of point processes on the product space that combines the vertex set of the graph with degrees. The main ingredient of the proof is the construction of what is called a size-biased coupling between the random graph and the random graph with the degree of a certain vertex having been fixed to a certain value. Thereafter, the Chen-Stein method is applied to derive these results.