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The physics of financial networks



As the total value of the global financial market outgrew the value of the real economy, financial institutions created a global web of interactions that embodies systemic risks. Understanding these networks requires new theoretical approaches and new tools for quantitative analysis. Statistical physics contributed significantly to this challenge by developing new metrics and models for the study of financial network structure, dynamics, and stability and instability. In this Review, we introduce network representations originating from different financial relationships, including direct interactions such as loans, similarities such as co-ownership and higher-order relations such as contracts involving several parties (for example, credit default swaps) or multilayer connections (possibly extending to the real economy). We then review models of financial contagion capturing the diffusion and impact of shocks across each of these systems. We also discuss different notions of ‘equilibrium’ in economics and statistical physics, and how they lead to maximum entropy ensembles of graphs, providing tools for financial network inference and the identification of early-warning signals of system-wide instabilities.

Key points

  • Modelling the financial system as a network is crucial to capture the complex interactions between financial institutions.

  • Such a network is naturally a time-dependent multiplex, because relationships between financial institutions are of many different kinds and keep changing.

  • Models of financial contagion enable understanding of how shocks propagate from one financial institution to another.

  • Missing information on financial networks can be ‘reconstructed’ using maximum entropy approaches borrowed from statistical mechanics.

  • Techniques based on financial networks have been adopted by practitioners and policy institutions.

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Fig. 1: Correlation-based networks from multivariate time series.
Fig. 2: Interbank networks and their dynamics.
Fig. 3: Construction of statistical ensembles of financial networks and application to network reconstruction and pattern detection.


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G.C. acknowledges support from the EU project HumanE-AI-Net, no. 952026. D.G. acknowledges support from the Dutch Econophysics Foundation (Stichting Econophysics, Leiden, the Netherlands) and the Netherlands Organization for Scientific Research (NWO/OCW). F.S. and T.S. acknowledge support from the EU project SoBigData++, no. 871042.

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Correspondence to Guido Caldarelli.

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Bipartite networks

Networks in which nodes are of two different types, say A and B (for instance, companies and directors), and links exist only between nodes of different type (for instance, a director being connected to a company if they sit on the board of that company).

Multiplex networks

Collections of networks (also called layers) with the same set of nodes, but with different links. In this context, multiplex networks are used to represent different kinds of linkages between financial institutions.


Items on the balance sheet of an institution that have a positive economic value because they generate present or future income.


A network structure characterized by a large clustering coefficient and a small average shortest path length.

Community structure

A network characterization in which nodes can be grouped into sets such that each set of nodes is densely connected internally.


On a network, a sequence of consecutive edges connecting a sequence of distinct nodes. The shortest path between two nodes is the path of minimal length connecting them.


The tendency of nodes to be linked to other nodes with dissimilar degrees. Conversely, assortativity is the tendency for nodes to be linked to other nodes with similar degrees.

One-mode projections

A one-mode projection of a bipartite network contains only nodes of one type (say A) and any two such nodes are connected to each other with an intensity proportional to the number of their common neighbours of the other type in the original bipartite network (for instance, two directors are connected by a link indicating the number of common boards on which they sit).

Filtered matrix

Matrix, for instance representing correlations, that has been statistically validated (or otherwise processed to eliminate the effects of noise) so that, ideally, only statistically significant information is retained.


Refers to the case in which the liquid assets (such as cash) of one institution are larger than its short-term liabilities (such as loans to be paid back overnight).


Items on the balance sheet of an institution that have a negative economic value because they represent debt to be repaid, potentially at different times (or maturities) in the future.


In accounting, equity is defined by the balance sheet identity as the difference between assets and liabilities. Therefore, it represents the net worth of the institution.


Solvency refers to the case in which the assets of one institution are larger than its liabilities and, therefore, its equity is positive.


(VaR). Risk measure defined as a (typically) large quantile of the probability distribution of losses. For example, when the quantile is 0.99 and the distribution of losses is over a time horizon of 1 year, it is interpreted as the loss that occurs once every 100 years.

Expected shortfall

(ES). Risk measure defined as the mean loss exceeding a (typically) large quantile of the probability distribution of losses. It is always larger than the value-at-risk at the same quantile.


Quantities representing the structural properties either to be enforced in the network reconstruction process or to be discounted in the network validation process.

Shannon entropy

Functional quantifying the amount of uncertainty associated with a probability distribution (see Box 2). Its maximum is attained for a uniform distribution.


The fraction of possible connections that are actually realized in a network.

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Bardoscia, M., Barucca, P., Battiston, S. et al. The physics of financial networks. Nat Rev Phys (2021).

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