Abstract
In this chapter, we review and compare some methods for the reconstruction of an interbank network from limited information. By exploiting the theory of complex networks and some ideas from statistical physics, we mainly focus on three different methods based on the maximum entropy principle, the relative entropy minimization, and the fitness model. We apply our analysis to the credit network of electronic Market for Interbank Deposit (e-MID) in 2011. In comparing the goodness of fit of the proposed methods, we look at the topological network properties and how reliably each method reproduces the real-world network.
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Notes
- 1.
The weighted adjacency matrix or weights matrix is the generalization of the adjacency matrix in the case of weighted (directed) graphs.
- 2.
Relative entropy is also known as cross-entropy or Kullback-Leibler divergence.
- 3.
The word bosonic is used for the analogy with the Bose gas in Physics since the probability distribution for the considered model turns out to be the Bose-Einstein statistics.
- 4.
\(C_i(G)\) is the value of the observable \(C_i\) associated with graph G of the ensemble. In the micro-canonical ensemble, we choose \(C_i(G)\) equal to the observed quantity \(C_i^*\) for each graph G in the ensemble. In the canonical ensemble, this equality holds only in average.
- 5.
In the grand canonical ensemble, the solution is unique except for a specific shift of the Lagrange multipliers (symmetry of the Hamiltonian of the network models).
- 6.
The standard configuration model is the network model obtained by imposing the degree sequence rather than the strength sequence.
- 7.
Similar results are obtained for the out-degree distribution.
- 8.
We applied the statistical analysis for different aggregation time scales and for different periods in the year 2011 and we obtained always the same results. In this sense, the analyzed properties are stationary for the e-MID market, at least in the year 2011.
- 9.
In the Italian Overnight Money Market, some links are very persistent in time, that is some banks tend to create credit relations with the same counterparts. By looking at the data aggregates in time, total exposure of a bank may be large but all credit transactions occur with the same counterpart. The persistence of credit relations is not capture by fitness model.
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Acknowledgements
This work is supported by the European Community H2020 Program under the scheme INFRAIA-1- 2014–2015: Research Infrastructures, grant agreement no. 654024 SoBigData: Social Mining & Big Data Ecosystem (http://www.sobigdata.eu).
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Mazzarisi, P., Lillo, F. (2017). Methods for Reconstructing Interbank Networks from Limited Information: A Comparison. In: Abergel, F., et al. Econophysics and Sociophysics: Recent Progress and Future Directions. New Economic Windows. Springer, Cham. https://doi.org/10.1007/978-3-319-47705-3_15
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