Systemic risk assessment through high order clustering coefficient

Abstract

In this article we propose a novel measure of systemic risk in the context of financial networks. To this aim, we provide a definition of systemic risk which is based on the structure, developed at different levels, of clustered neighbours around the nodes of the network. The proposed measure incorporates the generalized concept of clustering coefficient of order l of a node i introduced in Cerqueti et al. (2018). Its properties are also explored in terms of systemic risk assessment. Empirical experiments on the time-varying global banking network show the effectiveness of the presented systemic risk measure and provide insights on how systemic risk has changed over the last years, also in the light of the recent financial crisis and the subsequent more stringent regulation for globally systemically important banks.

Appendix

We here report the definition of the clustering coefficients in the directed case (see Clemente and Grassi 2018 for details).

The overall clustering coefficient for weighted and directed network is defined as:

$$\begin{aligned} c^{all}_i=\frac{\frac{1}{2}[(\mathbf{W }+\mathbf{W }^T)(\mathbf{A }+\mathbf{A }^T)^2]_{ii}}{{s_{i}\left( d_{i}-1\right) -2s_{i}^{\leftrightarrow }}} \end{aligned}$$

where $s^{\leftrightarrow }_i$ is the strength related to bilateral arcs between the node i and its adjacent nodes, defined as:

$$\begin{aligned} s^{\leftrightarrow }_i=\sum _{j\ne {i}}a_{ij}a_{ji}\frac{(w_{ij}+w_{ji})}{2}. \end{aligned}$$

The numerator of the overall coefficient takes into account all directed triangles that a node i actually forms with its neighbours, weighted with the average weight of the links connecting i to its adjacent nodes. Then, it is divided by all possible (appropriately weighted) directed triangles that i could form. Observe that $2s^\leftrightarrow _i$ represents the number of “false” triangles, being formed by i and by a pair of directed arcs pointing to the same node, e.g., $i \rightarrow j$ and $j \rightarrow i$. Indeed, being the network directed, i can form up to two triangles with each pair of adjacent nodes, including two “false” potential triangles for each bilateral link. To this reason, $2s^\leftrightarrow _i$ does not contribute to the number of possible directed triangles and has to be removed by the denominator. Notice that, if the network is undirected, ${\mathbf {c}}^{all}={\mathbf {c}}$.

Accordingly to the previous definition, in and out weighted local clustering coefficients of node i are defined, respectively, as:

$$\begin{aligned} c^{in}_i= \frac{\frac{1}{2}[\mathbf{W }^{T}(\mathbf{A }+\mathbf{A }^T)\mathbf{A }]_{ii}}{\overleftarrow{s}_{i}\left( \overleftarrow{d}_{i}-1\right) } \end{aligned}$$

and

$$\begin{aligned} c^{out}_i= \frac{\frac{1}{2}[\mathbf{W }(\mathbf{A }+\mathbf{A }^T)\mathbf{A }^{T}]_{ii}}{\overrightarrow{s}_{i}\left( \overrightarrow{d}_{i}-1\right) }. \end{aligned}$$