Modeling the Paths of China’s Systemic Financial Risk Contagion: A Ripple Network Perspective Analysis

Abstract

Considering financial contagion has ripple effects, this paper proposes using contagion ripple-spreading network model to reveal the paths of financial contagion from different contagion source to the whole Chinese financial system, and study financial institutions’ systemic importance. We first study the contagion ripple-spreading process triggered by oil market. Then we select four financial institutions from banks, brokerages, insurance and other institutions as contagion source respectively to study the how contagion will spread once these financial institutions trigger financial contagion. Finally, centrality comprehensive evaluation method is applied with heterogeneous networks generated from different contagion ripple-spreading processes to study the institutions’ systemic importance. The empirical results show that the contagion triggered by oil market first spreads to other financial institutions, and then to banks, brokerages and insurance. The contagion triggered by a financial institution first spreads in financial sectors that the contagion source belongs to, and then to other sectors. Moreover, most brokerages and banks have highest systemic importance. Insurance and the rest of brokerages, banks have middle level of systemic importance. Other financial institutions are the least important institutions.

Appendices

Appendix

Appendix 1: Simulation Steps of Contagion Ripple-Spreading Process

With the specified parameters, given any external shock with energy $E_{0}$, we can perform the contagion ripple-spreading process and explore the financial contagion spreading paths between financial institutions. The mathematical description of this process is as follows.

• Step 1 Initialize the current time instant, i.e., $t = 0$. Initialize the current energy of the contagion source as $e_{center} (t) = E_{0}$. Because each node has no initial energy, i.e., $E_{nodes} (i) = 0$,$i = 1,2,...,N$, therefore its current energy is $e_{nodes} (i,t)$ = $E_{nodes} (i) = 0$. Assume that each node or contagion source has a ripple with a current radius of 0, i.e., $r_{center} (t) = 0$ and $r_{nodes} (i,t) = 0$.

• Step 2 If the stopping criterion is not satisfied, do the following:

  • Step 2.1 Let $t = t + 1$.

• Step 2.2 Update the current radius and energy of the contagion source as follows: $r_{center} (t)$ = $r_{center} (t - 1)$ + $s_{0}$, $e_{center} (t)$ = $f_{decay} (E_{0} ,r_{center} (t))$, where $s_{0}$ is the spreading speed of the ripple from the contagion source, i.e., the change in the radius of a ripple during one time instant, and $f_{decay}$ is a function defining how the energy decays as the ripple spreads out. A typical decay function is

  $$ f_{delay} \left( {E_{0} ,r_{center} (t)} \right) = \frac{{\eta E_{0} }}{{2\pi r_{center} (t)}}, $$

  (A.1)

  where $\eta$ is a coefficient and π is the mathematical constant. Clearly, $\eta$ has an important influence on the decay speed of ripples. In this paper, we set $\eta = 1$.

• Step 2.3 Check which new nodes are reached by the ripples from the contagion source. Suppose $D(0,j)$ is the distance between the contagion source and node j. If $D(0,j)$$\le$$r_{center} (t)$ and $e_{center} (t)$$\ge$$\beta_{j}$, then node j is activated by the contagion source, i.e., a link from the contagion source to node j is established. It generates a responding ripple with $E_{nodes} (j)$ = $\alpha_{j}$$e_{center} (t)$, and $e_{nodes} (j,t)$ = $E_{nodes} (j)$. We use adjacency matrix $A$ to record the link, i.e., $A(0,j)$ = 1.

• Step 2.4 If $e_{nodes} \left( {i,t - 1} \right) > 0$, then update the current radius and energy of the ripple starting from node $i$ in a similar way to the ripple from the contagion source, i.e., $r_{nodes} (i,t)$ = $r_{nodes} (i,t - 1)$ + $s_{i}$, $e_{nodes} (i,t)$ = $f_{decay} (E_{nodes} (i),r_{nodes} (i,t))$.

• Step 2.5 Check which new nodes are reached by the ripples of other nodes. If $D(i,j)$$\le$$r_{nodes} (i,t)$, and $e_{nodes} (i,t)$$\ge \beta_{j}$, then node $j$ is activated by node $i$, and generates a responding ripple with $E_{nodes} (j) = \alpha_{j} e_{nodes} (i,t)$, and $e_{nodes} (j,t)$ = $E_{nodes} (j)$. Then a connection between node $i$ and node $j$ is established, i.e., update the adjacent matrix with $A(i,j)$ = 1.

Appendix 2: Networks Centrality Measures

For a give graph $G: = (V,E)$ with $\left| V \right|$ number of vertices, let $A = (a_{ij} )$ to be the adjacency matrix, i.e., $a_{ij} = 1$ if there is a directed edge from vertex $i$ to $j$, and $a_{ij} = 0$ otherwise.

• (1) Degree centrality

The normalized degree centrality for a node indexed by i can be formalized as follows:

$$ DC_{i} = \frac{1}{N - 1}\left( {\sum\limits_{j = 1}^{N} {a_{ij} } + \sum\limits_{j = 1}^{N} {a_{ji} } } \right), $$

(B.1)

where n − 1 is the maximum out-degree or in-degree, and the normalization is to remove the effect of network size.

• (2) Closeness centrality

The normalized closeness centrality for node $i$ is calculated as

$$ CC_{i} = \frac{N - 1}{{\sum\nolimits_{j = 1}^{N} {d_{ij} } }}, $$

(B.2)

where the distance d_ij between two nodes i and j is defined as the length of a shortest directed path from i to j consisting of nodes, provided at least one such path exists. Since $N - 1$ is the minimum sum of distances, the normalized measure has removed the impact of graph size.

• (3) Harmonic centrality

Harmonic centrality is formulized as

$$ HC_{i} = \frac{1}{N - 1}\sum\limits_{j = 1,j \ne i}^{N} {\frac{1}{{d_{ij} }}} , $$

(B.3)

where $d_{ij}$ is the length of a shortest directed path from $i$ to $j$ consisting of nodes, and $1/d_{ij} = 0$ if there is no path from node $i$ to node $j$.

• (4) Eigenvector centrality

The relative centrality score of vertex $i$ can be defined as:

$$ x_{i} = \frac{1}{\lambda }\sum\limits_{j \in M(i)} {x_{j} } = \frac{1}{\lambda }\sum\limits_{j \in G} {a_{ij} x_{j} } , $$

(B.4)

where $M(i)$ is a set of the neighbors of vertex $i$ and $\lambda$ is a constant. With a small rearrangement this can be rewritten in vector notation as eigenvector equation

$$ A{\mathbf{x}} = \lambda {\mathbf{x}}. $$

(B.5)

In general, there will be many different eigenvalues $\lambda$ for which a non-zero eigenvector solution exists. Since the entries in the adjacency matrix are non-negative, there is a unique largest eigenvalue, which is real and positive, by the Perro-Frobenius theorem. This greatest eigenvalue results in the desired centrality measure. The $i$th component of the related eigenvector then gives the relative centrality score of vertex $i$ in the network.

• (5) Betweenness centrality

The normalized betweenness centrality of node $i$ is formalized as follows:

$$ BC_{i} = \frac{1}{(n - 1)(n - 2)}\sum\limits_{j,k} {{{g_{jk} (i)} \mathord{\left/ {\vphantom {{g_{jk} (i)} {g_{jk} }}} \right. \kern-\nulldelimiterspace} {g_{jk} }}} , $$

(B.6)

where i ≠ j, j ≠ k is the number of shortest directed path linking node j and node k, g_jk(i) is the number of shortest directed path linking node j and node k that contain node i.

• (6) PageRank centrality

PageRank satisfies the following equation:

$$ x_{i} = \alpha \sum\limits_{j} {a_{ji} \frac{{x_{j} }}{L(j)} + \frac{1 - \alpha }{N}} , $$

(B.7)

where $x_{i}$ is centrality score of vertex $i$, and $L(j) = \sum\limits_{i} {a_{ji} }$ is the number of neighbors of vertex $j$ (or number of outbound links in a directed graph).