# Spatiotemporal Patterns of Risk Propagation in Complex Financial Networks
## 4 Results
### Risk Propagation Flow of Communities  
A distinctive feature of complex financial networks is the existence of community structures. A community is a group of nodes linked with high-density internal edges, which often corresponds to different business sectors in financial markets. Therefore, when calculating risk propagation flow, we need to consider the influence of community structure. We first detected the communities in the sector mode and, then, characterized the risk propagation flow between and within different communities of the complex financial networks. Figure 2: Spatiotemporal patterns of risk propagation with different dynamic parameters. We measured the risk propagation flow \(F_{i}\) and the local response times \(\tau_{i}\) of all nodes/links with different dynamic parameters for the S&P500 market and the HS300 market. **(a)**\(F_{i}\) vs. the weighted degree \(S_{i}\) for the S&P500 market with different infection rates \(R\). **(b)**\(\tau_{i}\) vs. \(S_{i}\) for the S&P500 market with different infection rates \(R\). **(c)**\(F_{i}\) vs. \(S_{i}\) for the HS300 market with different infection rates \(R\). **(d)**\(\tau_{i}\) vs. \(S_{i}\) for the HS300 market with different infection rates \(R\). **(e)**\(F_{i}\) vs. \(S_{i}\) for the S&P500 market with different recovery rates \(B\). **(f)**\(\tau_{i}\) vs. \(S_{i}\) for the S&P500 market with different recovery rates \(B\). **(g)**\(F_{i}\) vs. \(S_{i}\) for the HS300 market with different recovery rates \(B\). **(h)**\(\tau_{i}\) vs. \(S_{i}\) for the HS300 market with different recovery rates \(B\). The Wishart matrix is \(C_{ij}\) of N non-correlated time series with length L [83; 84]. In the constraints of \(N>>1\), \(L>>1\), and \(Q=L/N\geq 1\), the probability distribution of the eigenvalue \(\lambda\) is given by  
\[P_{rm}(\lambda)=\frac{Q}{2\pi}\frac{\sqrt{(\lambda_{max}^{ran}-\lambda)( \lambda_{min}^{ran}-\lambda)}}{\lambda}. \tag{11}\]  
The upper and lower bounds of \(\lambda\) are  
\[\lambda_{max(min)}^{ran}=[1\pm 1/\sqrt{Q}]^{2}, \tag{12}\]  
Based on the RMD method, we decompose \(C_{ij}\) into N-dimensional orthogonal eigenmodes:  
\[C_{ij}=\Sigma_{\alpha=1}^{N}\lambda_{\alpha}C_{ij}^{\alpha},\quad C_{ij}^{ \alpha}=u_{i}^{\alpha}u_{j}^{\alpha}, \tag{13}\]  
where \(\lambda_{\alpha}\) is the \(\alpha\)-th largest eigenvalue, \(u_{i}^{\alpha}\) is the \(i\)-th component in the eigenvector of \(\lambda_{\alpha}\), and \(C_{ij}^{\alpha}\) represents the cross-correlation in the \(\alpha\)-th eigenmode [45; 46; 47]. According to the size of the eigenvalue \(\lambda_{\alpha}\), the eigenmodes of real financial markets can be divided into three categories: market mode, sector mode, and random mode [43; 44]. For the market mode, \(C_{ij}^{mar}=\Sigma_{\alpha=1}^{mode}C_{ij}^{\alpha}\lambda_{\alpha}\), which represents the global price movement of the entire market. The sector mode corresponds to the local price motion of business sectors, \(C_{ij}^{sec}=\Sigma_{\alpha=1}^{n-1}C_{ij}^{\alpha}\lambda_{\alpha}\). Here, \(\lambda_{\alpha}>\lambda_{max}^{ran}\), where \(\lambda_{max}^{ran}\) is the upper bound of the Wishart matrix [83; 84]. \(n\) is the number of large eigenvalues exceeding \(\lambda_{max}^{ran}\). For the random mode, we take \(C_{ij}^{func}=\Sigma_{\alpha=1}^{N-1}C_{ij}^{\alpha}\lambda_{\alpha}\). The random mode contains the non-stationary random background of complex financial systems. Taking the absolute values of the negative matrix elements, we then generated the PMFG graph \(A_{sector,ij}\) from \(|C_{sector,ij}|\)[44; 77; 85]. Next, the community structure was extracted in the Gephi software [86]. After removing the non-stationary random background, we can see the community structure more clearly, and the sector mode further removed the influence of the macro changes of the entire market on the network, allowing us to see the local interactions of communities. Lastly, we characterized the risk propagation flow between and within different communities of the complex financial networks.