Inhomogeneous Long-Range Percolation for Real-Life Network Modeling
Article Inhomogeneous long-range percolation for real-life network modeling Risks Provided in Cooperation with: MDPI -Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Deprez, Philippe; Hazra, Rajat Subhra; Wüthrich, Mario V. (2015) : Inhomogeneous long-range percolation for real-life network modeling, Risks, ISSN Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.
... kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. 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 Z d , 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. Keywords: network modeling; stylized facts of real-life networks; small-world effect; long-range percolation; scale-free percolation; graph distance; phase transition; continuity of percolation probability; inhomogeneous long-range percolation; infinite connected component Risks 2015, 3 2 1. Introduction Random graph theory has become very popular to model real-life networks. 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  , as financial networks such as the banking system, see [2, 3] , or the network of interbank transactions, see [4, 5] . In the latter example, banks are modeled by particles and two banks are linked if one bank transacts a payment to the other one. 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  . It is, therefore, 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 : • Distant particles are typically connected by very few links, i.e., although there are possibly a lot of particles in the network, any two particles are typically connected through only a few other particles. This is called the "small-world effect". For example, there is the observation that most particles in real-life networks are connected by at most six links, see also  . For the Facebook network with 721 million users, where there is a link between two users if they are "friends" on Facebook, the average number of minimal links that connect any two users is around 4.5, while around 99% of all users are connected by at most six links, see  . For the movie actor network, where there is a link between two actors if they appeared in the same film, the average number of minimal links that connect any two actors is also around 4.5, while the number of actors in the network is over two hundred thousand. See  for more examples. • Linked particles tend to have common friends, which is called the "clustering property". For instance, if x is friend of both y and z, then it is likely that y and z are also friends. As an example,  discovers the following in the Facebook network: given a user with 100 friends, about 14% of the possible friendships among his friends exist. • The degree distribution, that is, the distribution of the number of links of a given particle, is heavy-tailed, i.e., its survival probability has a power law decay. It is observed that in real-life networks the (power law) tail parameter τ is often between 1 and 2. For instance, for the movie actor network τ is estimated to be around 1.3. For more explicit examples we refer to [7, 8] . Since it is too complicated to model large networks particle by particle, many different random graph models have been developed and their properties were analyzed. A well studied model in the literature is the homogeneous long-range percolation model on Z d , d ≥ 1. In this model, the particles are the vertices of Z d . For fixed λ, α > 0, any two particles x, y ∈ Z d are linked with probability p xy which behaves as λ|x − y| −α for |x − y| → ∞, i.e., the closer particles are the more likely they are connected. This model has therefore a local clustering property. Moreover, for values of α not too large, the graph distance between x, y ∈ Z d , that is, the minimal number of links that connect x and y, behaves roughly logarithmically as |x − y| tends to infinity, see  . This behavior can be interpreted as a version of the small-world effect in the sense that if two particles are separated by large (Euclidean) distance |x|, they are connected by only roughly log |x| links. But homogeneous long-range percolation does not fulfill the stylized fact of having heavy-tailed degree distributions, which makes this model less attractive for real-life network modeling. Therefore,  introduced the inhomogeneous long-range