Bootstrap percolation in directed and inhomogeneous random graphs

Abstract

Bootstrap percolation is a process that is used to model the spread of an infection on a given graph. In the model considered here each vertex is equipped with an individual threshold. As soon as the number of infected neighbors exceeds that threshold, the vertex gets infected as well and remains so forever. We perform a thorough analysis of bootstrap percolation on a novel model of directed and inhomogeneous random graphs, where the distribution of the edges is specified by assigning two distinct weights to each vertex, describing the tendency of it to receive edges from or to send edges to other vertices. Under the assumption that the limiting degree distribution of the graph is integrable we determine the typical fraction of infected vertices. Our model allows us to study a variety of settings, in particular the prominent case in which the degree distribution has an unbounded variance. Among other results, we quantify the notion of "systemic risk", that is, to what extent local adverse shocks can propagate to large parts of the graph through a cascade, and discover novel features that make graphs prone/resilient to initially small infections.

Managing Default Contagion in Inhomogeneous Financial Networks

“…Our starting point is paper [19], where the first three authors developed a directed random graph model that makes it possible to study random graphs whose underlying degree distribution is not necessarily required to possess a second moment, while simultaneously preserving the simplicity of the network. Whereas financial networks are clearly weighted in the sense that exposures between banks have a monetary value, [19] is only capable of investigating the following simplified contagion mechanism: each bank is assigned an integer-valued threshold that represents the number of debtors in the network that need to default in order for the bank to default as well. It is not clear how such threshold values could be determined for an observed network since the size of loans issued to defaulted debtors also plays an important role in the contagion mechanism for financial networks.…”

“…In this article, we construct a model for financial networks that combines the simple, directed and inhomogeneous structure of [19] with weighted edges (monetary exposures). Our model directly contains capitals of the banks and exposures between the banks as parameters and is therefore easily calibrated to observed network structures.…”

“…A Random Graph Model for Financial Networks As already described briefly, we define a model that combines many properties observed for real financial networks such as simplicity and directedness of the links between institutions, weighted connections and a strong degree of inhomogeneity. As a basic building block we use the model from [19], and we enhance it with capitals on the vertices and weights on the edges which represent exposures. This allows to use some results from [19] in the proofs for our enhanced model.…”

“…As a basic building block we use the model from [19], and we enhance it with capitals on the vertices and weights on the edges which represent exposures. This allows to use some results from [19] in the proofs for our enhanced model. We are able to make asymptotic statements for large networks about the impact that fundamental defaults have on the financial system and the wider economy due to their propagation through the network via a cascade mechanism.…”

“…Whereas these examples construct undirected networks, however, financial networks clearly are directed. In [19], the first three authors of this article have hence developed a directed version of the random graph in [16] which will also be the base for our model in this article. Also see [27] for a directed fitness model in the context of financial networks.…”