Rank-one inhomogeneous random graphs, such as the Norros-Reittu model, and the random connection model with weights are random graphs whose vertices are equipped with i.i.d. marks. For each pair of vertices it is decided independently if they are connected by an edge and the corresponding probability depends on the marks of the vertices. We consider particular choices of the mark distributions such that the resulting random graphs exhibit a scale-free behaviour as it is observed in many real-world complex networks. The Norros-Reittu model has a finite number of vertices, while the vertex set of the random connection model is a stationary Poisson process. We study large degrees of vertices and - in some regimes - the sizes of large components, where we consider only the vertices in an observation window for the random connection model. For increasing number of vertices in the Norros-Reittu model and an increasing observation window for the random connection model, respectively, we show that the corresponding point processes converge to inhomogeneous Poisson processes after suitable rescalings. In particular, we obtain the limiting distributions of the largest degree and the size of the largest component. Moreover, the consistency of the Hill estimator for the inverse of the tail exponent of the typical degree distribution is established. The proofs rely on comparing the point processes of degrees and component sizes with the point process of the marks.

This talk is based on joint works with Chinmoy Bhattacharjee and Matthias Lienau.