This paper introduces a generalization of the so-called space-fractional Poisson process by extending the difference operator acting on state space present in the associated difference-differential equations to a much more general form. It turns out
that this generalization can be put in relation to a specific subordination of a homogeneous Poisson process by means of a subordinator for which it is possible to express the characterizing Levy measure explicitly. Moreover, the law of this subordinator solves a one-sided first-order differential equation in which a particular convolution-type integral operator appears, called Prabhakar derivative. In the last section of the paper, a similar model is introduced in which the Prabhakar derivative also acts in time. In this case, too, the probability generating function of the corresponding process and the probability distribution are determined.
Real-world networks may exhibit detachment phenomenon determined by the cancelling of previously existing connections. We discuss a tractable extension of Yule model to account for this feature. Analytical results are derived and discussed both asymp
totically and for a finite number of links. Comparison with the original model is performed in the supercritical case. The first-order asymptotic tail behavior of the two models is similar but differences arise in the second-order term. We explicitly refer to World Wide Web modeling and we show the agreement of the proposed model on very recent data. However, other possible network applications are also mentioned.
