The study of random graphs has become very popular for real-life network modeling, such as social networks or financial networks. Inhomogeneous long-range percolation (or scale-free percolation) on the latticeexample of a random graph model because it fulfills several stylized facts of real-life networks. For this model, various geometric properties, such as the percolation behavior, the degree distribution and graph distances, have been analyzed. In the present paper, we complement the picture of graph distances and we prove continuity of the percolation probability in the phase transition point. We also provide an illustration of the model connected to financial networks.

The study of real-life network modeling has become very popular in recent years. An attractive model is the scale-free percolation model on the lattice Z d , d ≥ 1, because it fulfills several stylized facts observed in large real-life networks. We adopt this model to continuum space which leads to a heterogeneous random-connection model on R d : particles are generated by a homogeneous marked Poisson point process on R d , and the probability of an edge between two particles is determined by their marks and their distance. In this model we study several properties such as the degree distributions, percolation properties and graph distances.Since it is too complicated to model large real-life networks particle by particle, many theoretical random graph models have been developed and their geometrical properties studied. One of these models is the homogeneous long-range percolation model on Z d , d ≥ 1, first introduced in [25] for d = 1. The set of particles is the lattice Z d and any two particles x, y ∈ Z d are independently linked with probability p xy which behaves as λ|x − y| −α for |x − y| → ∞, with fixed constants λ, α > 0. Since close particles are likely linked, this model has a local clustering property. Moreover, depending on α, the graph distance of two connected particles is roughly of logarithmic order as their separation tends to infinity, see [6]. This is a version of the smallworld effect. However, this model does not fulfill the stylized fact of having heavy-tailed degree distributions. Therefore, [11] extended the homogeneous long-range percolation model to a scalefree percolation model on Z d (also known as inhomogeneous long-range percolation model). In their model they consider a collection (W x ) x∈Z d of i.i.d. positive weights that are heavy-tailed with tail parameter β > 0, and they assign to each particle x ∈ Z d the random weight W x . Given these weights, any two particles x, y ∈ Z d are independently linked with probability p xy which is approximately λW x W y |x − y| −α for large |x − y| and given constants λ, α > 0. Note that p xy is increasing in the weights W x and W y , and decreasing in the distance between x and y. This means that the weights make particles more or less attractive, i.e. particles with large weights play the role of hubs in this network. This extension of the homogeneous model is very natural since the existence of hubs is often observed in real-life networks. Again, this model has a local clustering property. Depending on α and β, [11] showed that the degree distribution is heavy-tailed, i.e. this model fulfills the stylized fact of having heavy-tailed degree distributions. Moreover, they showed that whenever the degree distribution has finite mean but infinite variance, the graph distance of two particles behaves doubly logarithmically as their separation tends to infinity. This is again a version of the small-world effect, sometimes called the ultra-small-world effect.In this article we adopt the scale-free percolation model on Z d to the continuum space R d...

Various stochastic models have been proposed to estimate mortality rates. In this paper we illustrate how machine learning techniques allow us to analyze the quality of such mortality models. In addition, we present how these techniques can be used for differentiating the different causes of death in mortality modeling.

Various stochastic models have been proposed to estimate mortality rates. In this paper we illustrate how machine learning techniques allow us to analyze the quality of such mortality models. In addition, we present how these techniques can be used for differentiating the different causes of death in mortality modeling.

The theory of random graphs goes back to the late 1950s when Paul Erdős and Alfréd Rényi introduced the Erdős-Rényi random graph. Since then many models have been developed, and the study of random graph models has become popular for real-life network modelling such as social networks and financial networks. The aim of this overview is to review relevant random graph models for real-life network modelling. Therefore, we analyse their properties in terms of stylised facts of real-life networks.