Subexponential Interval Graphs Generated by Immigration-death Processes Subexponential Interval Graphs Generated by Immigration-death Processes

Scale-free graphs have recently attracted much attention since so-called scale-free phenomena have really appeared in various physical and social networks, where a graph is said to be scale-free if the distribution of degrees has a power-law tail. In the previous paper, the authors proposed a simple model of random interval graphs generated by immigration-death processes (also known as M/G/∞ queueing processes) and showed that, when the interval lengths follow a power-law distribution, the generated interval graph is scale-free in the steady state. In this paper, we generalize this result to the model where the distribution of interval lengths is subexponential and provide a condition under which the stationary degree distribution is also subexponential. Furthermore, we consider the conditional expectation of the cluster coefficient of a vertex given its degree and derive its limit as the degree goes to infinity under the same condition as that for obtaining the tail asymptotics of the degree distribution.