1 Default contagion in financial networks Chiara Benazzoli and Luca Di Persio Abstract—The preset work aims at giving insights about how the theory behind the study of complex networks can be prof- itably used to analyse the increasing complexity characterizing a wide number of current financial frameworks. In particular we exploit some well known approaches developed within the setting of the graph theory, such as, e.g., the Erd˝ os and R´ eny model, and the Barab´ asi-Albert model, as well as producing an analysis based on the evolving network theory. Numerical simulations are performed to study the spread of financial peak events, as in the case of the default of a single bank belonging to a net of interconnected monetary institutions, showing how the knowledge about the underlying graph theory can be effectively used to withstand a financial default contagion. Keywords—Financial networks, default spread, graph theory, random graphs I. I NTRODUCTION Modern monetary systems are characterized by a high grade of interconnections between players that share finan- cial products of heterogeneous type with high frequency also exploiting fast communication channels. This leads to graph- structures with a rather complicated topologies where each node can be linked to a wide number of other ones by mean of hedges with weights of quite different magnitude scale. The latter implies that isolated peak events, that may occur at a certain node in a given financial network, could spread their effects along the whole structure. As an example, the failure of a key bank may cause a big backlash to the other institutions which are linked to it, hence leading to a cascading failures and global financial crisis. Recently, concrete examples of such a cascading phenomena have been observed during the worldwide financial crisis of 2007- 2008, e.g., concerning the high negative impact caused by the Lehman Brothers bankruptcy. The present work is subdivided into the three following sections: in Section II we show the reasons why complex network theory can be usefully and naturally applied in order to study financial nets of interconnected agents. Moreover, following [1] and [2], some considerations about systemic risk and risk contagion in banking network are discussed; in Section III, the major mathematical results related to the Erd˝ os-R´ eny model, the Barab´ asi-Albert model, and those concerning the evolving network theory, are collected, mainly following [3] and [4]; in Section IV, we generalise the approaches proposed in [5] and [6], where the authors analysed a static interbank network model to describe con- tagion effects, by allowing to new banks to enter in the network, establishing links with the ones already present in the system. The latter dynamic will be shown to produce a reduction in the number of the defaulting banks. C. Benazzoli is with the Department of Mathematics, Univer- sity of Trento, Via Sommarive, 14 - 38123 Povo, Italy (e-mail: chiara.benazzoli@unitn.it). L. Di Persio is with the Department of Computer Science, Univer- sity of Verona, Strada le Grazie, 15 - 37134 Verona, Italy, (e-mail: luca.dipersio@univr.it). II. BANKING NETWORK AND SYSTEMIC RISK The increasing complexity of the banking systems has suggested the use of approaches belonging to the graph theory in general, with particular respect to the possibility of exploiting results coming from the theory of random graphs, see, e.g., [7, Chapt.5], and references therein. The main idea is to represent a banking network by associating to each financial institution composing it, e.g., insurance companies, banks, pension funds, etc., a vertex of a graph whose connecting edges represent the liability or the financial exposure financial player with respect to the others. It follows that latter links can potentially become the tie through which the distress of an individual component of the net propagate to the others ones. Our main goal is to understand how a bank’s failure may damage the whole financial system or, in other words, to analyse how the banking network reacts to the default of one participant. The impact that the failure of a financial institution has with respect to the stability of the whole market of which it is a component is the so called systemic risk. An unambiguous definition of the systemic risk is rather difficult to give, since it depends on multiple factors which vary in time as, e.g., the reflection of the market movements and cycle trends caused by new regulatory constraints. Moreover the systemic risk also varies in space, according to specific regional, national or even international, economic policies. Last but not least systemic risk depends on a wide number of heterogeneous type of financial agents which are interconnected by different types of links it can be transmitted through. However, some key factors are acknowledged as measures of how an institution represents a possible source, or transmission vehicle, of systemic risk. First of all, the size of the entity: the bigger is its the size of the particular financial institution we are interested in, the higher is the impact of its possible default. Secondly, its degree of interconnection substitutability, namely its core business function, e.g., banking, custody, fund management, brokerage, clearing, etc. It is worth to mention that the central bank and a small private bank play a significantly different role in the banking network to which they con- tribute. Every financial collapse begins with the downfall of a single player. The causes of this initial failure are many. In particular they can be of exogenous type as in the case, e.g., of recessions, wars, political crisis, etc., or they can be originated within the financial system itself, as frauds or misapplication of mathematical models. Analogously, there are several ways through which these shocks can be spread over the whole network. They can be divided into three different classes: i propagation due to a direct counterpart exposures: this takes into account all the losses which may occur if banks default in their obligations to other bank in the interbank market; INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATION Volume 10, 2016 ISSN: 1998-0159 112