Abstract
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 lattice Zd, d ≥ 1, is a particular attractive example 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.
Highlights
IntroductionReal-life networks may be understood as sets of particles that are possibly linked with each other
Random graph theory has become very popular to model real-life networks
All particles have infinitely many links which is, not interesting for real-life network applications. For this reason we only consider the non-trivial case min{α, βα} > d. In this latter case the degree distribution is heavy-tailed with tail parameter τ = βα/d > 1, see Theorem 2.2 of [12], which is in line with the stylized facts
Summary
Real-life networks may be understood as sets of particles that are possibly linked with each other. Such networks appear, for example, as virtual social networks, see [1], as financial networks such as the banking system, see [2,3], or the network of interbank transactions, see [4,5]. The connectivity of the network plays a crucial role on the spread of information and the development of default cascades, the latter being crucial for macroeconomic stability, see [6]. It is, of major interest to understand the geometry of such networks. Using empirical data one has observed several stylized facts about large real-life networks, for a detailed outline we refer to [1,7], and Section 1.3 in [8]:
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