A random graph of moderate density

Abstract

We analyse a randomly growing graph model in which the average degree is asymptotically equal to a constant times the square root of the number of vertices, and the clustering coefficient is rather small. In every step, we choose two vertices uniformly at random, check whether they are connected or not, and we either add a new edge or delete one and add a new vertex of degree two to the graph. This dependence on the status of the connection chosen vertices makes the total number of vertices random after n steps. We prove asymptotic normality for this quantity and also for the degree of a fixed vertex (with normalization $n^{1∕6}$). We also analyse the proportion of vertices with degree greater than a fixed multiple of the average degree, and the maximal degree.