Reconstruction of Financial Networks

Many dynamical processes of great importance, such as the spreading of a disease in a population, the diffusion of information or fake news in social media, or the contagion process on a financial network, are very sensitive to the topology of the network of interactions on which they occur. However in many situations the structure of the network is (at least partially) unknown, due to privacy concerns or limitations in data collection. The need to deal with such missing information has led to the birth of a research field known as Network Reconstruction

A paramount example is that of interbank networks: banks publicly disclose their aggregate exposures in their balance sheets, but individual exposures (who is lending how much to whom) remain confidential. Hence the problem is to reconstruct the whole configuration of network links starting from the knowledge of the strength (the aggregate weight of incident links) of each node. This is a typical situation that can be handled by the Statistical Mechanics approach: the probability distribution which best represents the current state of knowledge on the system is the one with the largest uncertainty that also satisfies the constraints corresponding to the available information. 

Unfortunately, constraining the strengths is not enough to satisfactory reconstruct the network. This is because the entropy maximisation redistributes the strength of each node over all possible links, generating unrealistic fully-connected networks. Constraining also the degree (number of connections) of each node could solve the problem, but unfortunately this information is not available for interbank networks. 

In this paper we proposed a solution based on the so-called “fitness ansatz”, which simply assumes that the larger the exposures of a bank, the more connections it has. This approach allows to inform a configuration model of the network and thus to obtain an estimate of node degrees. Link weights can then be assigned using a simple heuristic technique known as "gravity model". As shown in this paper by central banks and other "horse races" in the literature, our “Cimi” method outperforms competing probabilistic recipes in providing an accurate reconstruction of interbank networks. 

We further extended the Cimi method to bipartite networks of security holdings (investment portfolios), resulting in an enhanced version of the traditional capital-asset pricing model (CAPM). We also made it more rigorous by coupling the preliminary estimation of node degrees from the fitness ansatz with a maximum-entropy inference that constrain both empirical strengths and estimated degrees. Note that for a maximum entropy method, the agreement between the reconstructed network and empirical data not only proves its effectiveness, but also implies that the network to be reconstructed is close to the equilibrium configuration of the maximum entropy ensemble defined by the imposed constraints

Flow diagram for the reconstruction of an interbank network from aggregate balance sheet exposures. The fitness ansatz allows to estimate degrees and link probabilities from the configuration model. This information can then be used to infer link weights with heuristic gravity models or maximum entropy approaches, so to obtain an ensemble of reconstructed interbank networks.
Comparison of different reconstruction methods (labeled by the first authors of the papers where they have been proposed) in reconstructing a real instance of the eMID interbank network (on top left). For the adjacency matrices, black denotes the presence of a link while the tone of yellow its weight.