• Complex Networks
• Network Ensembles
• Statistical Mechanics
• Systemic Risk

Systemic risk concerns the stability of systems composed by different parts, specifically the prediction and prevention of systemic events. A systemic event is typically defined as a phenomenon that emerges from the complex interactions of the constituents and compromises the normal functioning of the system. In particular, in finance the elementary constituents are financial institutions, such as banks, and systemic risk pertains the description and prevention of collapses that involve large portions of the financial system. After the recent troubled years
for the global economy, in which two severe crises (the 2007 crisis of financial markets and the 2010 sovereign debt crisis) have put the whole economic system in dramatic distress, vulnerability of banks to systemic events is now the main focus of a growing number of investigations of the academic community, crosswise different disciplines. When a bank undergoes some sort of malfunctioning or distress, and its troubled situation negatively affects other institutions, we say that distress propagates, and that the interactions
among the two institutions serves as channel of contagion. Networks are one of the main tools in the modelling and description systemic risk in financial systems, since they allow for a straightforward description of different channels of contagion. The indirect channel of contagion can be described as a bipartite network, i.e. a network where the set of nodes is sharply divided into two subsets, one containing only banks, the other only assets. Banks can only be connected with assets but not with other banks. Moreover, a link exist if the bank holds the asset in its portfolio, i.e. invests in that particular assets. In order to quantify losses from indirect contagion a full knowledge of how the banks divide their investments is needed. As quantifiers of systemic risk we consider systemicness and vulnerability of a bank as, respectively, the total percentage loss induced on the system by the distress of the bank and the total percentage loss experienced by the bank when the whole system is in distress. An important part of our thesis is dedicated to the empirical analysis of a dataset that we originally developed, collecting data publicly available from the databases of the Federal Reserve, that describes the quarterly networks of US commercial banks’ exposures in the period 2001-2015. Specifically we compute, for each quarter, systemicness and vulnerability of each bank and the aggregate vulnerability of the system. From our network description of the system, it emerges that there is a clear relation between the vulnerability of the system to external shocks and the way in which the largest banks present manage their investments,as we show in Figure 1.
The central topic of the thesis is the problem of reconstructing systemic risk measures from partial information on the network, e.g. knowing only the size of the banks present and the capitalization of the assets available.Our main purpose is to develop efficient methods to
estimate systemic risk from partial information, without the full knowledge of the bipartitenetwork.
A prolific analogy between statistical mechanics and the statistical inference of networks, has been proposed and exploited by physicists.Network ensembles can be defined, akin to grand canonical ensembles in statistical mechanics, and they can be used to efficiently
reconstruct networks from the sole partial information available. Ensembles are endowed witha probability mass function on a set of graphs that depends on a set of parameters, that in turn need to be estimated numerically from the partial information available. Banks statistics,
that are related to the network structure, are estimated through their expected values on the statistical ensemble. We implemented the numerical procedures needed to practically employ ensembles of bipartite networks to systemic risk reconstruction, and applied them to the reconstruction of systemicness and indirect vulnerability.
In order to test the reconstruction capability of our ensembles, we assume that the balance sheet compositions of the banks are not known, estimate the measures of systemic risk, trough network ensembles, and finally compare the real statistics computed for our dataset, with the
values inferred from partial information. This ex-post comparison allows us to asses whether our methods are appropriate when the network structure is actually unknown.
While network ensembles based on Bose-Einstein statistic are now a standard tool in network science, we found them to be unfit for the reconstruction of systemic risk. In fact better suited ensembles, for the purpose of systemic risk reconstruction, are defined trough a bold extension to networks of the correct Boltzmann counting of states. We refer to the resulting ensembles as Maxwell-Boltzmann (MB) ensembles.
Inspired by the analogy with statistical mechanics, we propose an extension of the diffused Max-Ent approach to the definition of network ensembles, and we refer to it as Minx-Ent.
Via Minx-Entr we define a family of ensembles that includes as limiting cases BE and MB ensembles, and propose it as a new tool for the reconstruction of networks statistics.
At the end of this thesis we propose a, previously unexplored, application of grand-canonical ensembles, to a subject not directly related to systemic risk, i.e. “filtering of complex networks”. Often complex networks, that describe real systems, are extremely dense. A large
network with a high edge density may be hard to interpret, or visualize by traditional tools of network analysis. In addition, a large portion of the links might not be informative, or subject to measurement errors. Hence it is sometimes useful to extract sub-networks, that
contain only a portion of the original nodes and links. The reduction of “noisy” networks, in order to maintain only relevant information, is known as information filtering in complex networks. We propose an original technique for the filtering of complex networks based on
grand-canonical ensembles. A Portion of this thesis has been the object of the paper Di Gangi, Lillo, and Pirino (“Assessing systemic risk due to fire sales spillover through maximum entropy network recon-struction”), currently undergoing reviewing process.