Reconstruction methods for networks: The case of economic and financial systems

Abstract

The study of social, economic and biological systems is often (when not always) limited by the partial information about the structure of the underlying networks. An example of paramount importance is provided by financial systems: information on the interconnections between financial institutions is privacy-protected, dramatically reducing the possibility of correctly estimating crucial systemic properties such as the resilience to the propagation of shocks. The need to compensate for the scarcity of data, while optimally employing the available information, has led to the birth of a research field known as network reconstruction. Since the latter has benefited from the contribution of researchers working in disciplines as different as mathematics, physics and economics, the results achieved so far are still scattered across heterogeneous publications. Most importantly, a systematic comparison of the network reconstruction methods proposed up to now is currently missing. This review aims at providing a unifying framework to present all these studies, mainly focusing on their application to economic and financial networks. 7The problem of missing information. After an initial activity aimed at determining the structure of real-world networks by measuring standard topological quantities, a more theoretical activity was started, aiming at both defining new quantities and devising proper models to explain observations [45,44,52,53,54]. Given the complexity that can arise even from a simple mathematical model based upon graphs, researchers have recently focused on the development of a topological theory: loosely speaking, topological quantities are employed to define statistical models, rather than reproduced from microscopic dynamical rules [55,56,57,58,59].Unfortunately, when moving to the validation of such models a common problem arises: very often, the data available on the real network are either incomplete or imprecise (or both). This problem is particularly evident in the case of economic and financial networks: in this case, data collection suffers from the problem of partial accounting and the presence of disclosure requirements. In order to illustrate the importance of such an issue, let us think of a bipartite, financial network whose node sets represent investors and the investments they do. Although the knowledge of the whole network structure could help regulators to take immediate countermeasures to stop the propagation of financial distress, this information is seldom available (the knowledge of the whole network of investments would pose immense problems of privacy), thus hindering the possibility of providing a realistic estimate of the extent of the contagion. As confirmed by the analysis of the various papers reported in this review, the incompleteness of network instances seems to be unavoidable [60,61]: since addressing the problem of estimating the resilience of financial networks cannot be addressed without knowing the structural details of national and cross-countries interbank networks, inf...

A faster horse on a safer trail: generalized inference for the efficient reconstruction of weighted networks

“…The same observation holds for CReM B , where the exponential parameter is now given by β ij . We observe then that both models show the same unctional form of the weights conditional distribution as the DECM (see (20)). However the numerical values of the parameters characterizing the distribution are different.…”

“…We have just shown some possible choices for the functional form of p ij . Other possibilities are the ones discussed in [20].…”

“…Our results indicate that the best method is obtained by "dressing" the best-performing method available to date with a certain exponential distribution of link weights. The method is fast, unbiased and accurate and reproduces the empirical networks with highest generalised likelihood.Network reconstruction is an active field of research within the broader field of complex networks [20]. Addressing the network reconstruction problem means facing the double challenge represented by the estimation of topology and link weights.…”

“…Network reconstruction is an active field of research within the broader field of complex networks [20]. Addressing the network reconstruction problem means facing the double challenge represented by the estimation of topology and link weights.…”

“…For instance, in the case of the micro-canonical ensemble, where the degree are constrained sharply and not on average on the ensemble, P (A) is constant on all graphs with the same degree sequence. When P (A) factorizes f ij corresponds to the quantity denoted with p ij in the literature [20], that is the linkage probability the is independent from link to link. Differentiating (12) with respect to β out i and β in i , ∀i, yields the system of 2N coupled equations…”