# Spatiotemporal Patterns of Risk Propagation in Complex Financial Networks
## 4 Results  
### Quantifying Risk Propagation Flow  
As the representatives of emerging and mature markets, Chinese and American stock markets are two typical complex financial networks. Based on the historical data of the components of the S&P500 and HS300 Indices, we calculated the risk propagation flow through all nodes and links, i.e., \(F_{i}\) and \(F_{ij}\), for each market. For simplicity, here, we take \(B=1\) and \(R=1\). The larger the risk propagation flow of a node (link), the more important the corresponding individual stock (interaction between the two stocks) in the information dissemination of the entire market. Thus, we can identify the influential nodes and links in the propagation of information, which may be applicable to portfolio optimization and risk management. The results for the HS300 market and the S&P500 market are displayed in the Figure 1. There are 483 nodes and 1443 links in the S&P500 risk propagation flow network and 240 nodes and 714 links in the HS300 network. The nodes correspond to individual stocks, and the links represent the risk propagation flow between two stocks. In both markets, a few influential nodes play dominant roles in the risk propagation flow networks. We sorted the nodes and links of different networks according to the size of risk propagation flow and list the top 10 nodes and links in Table 1. For the HS300 market, the top 10 nodes mainly came from the business sectors--healthcare, information technology, and consumer staples--and most of the top 10 links were found within or between the healthcare and consumer staples sectors. For the S&P500 market, the top 10 nodes were mainly distributed in the business sectors--consumer staples, healthcare, consumer discretionary, and information technology--and most of the top 10 links were found within the business sectors--consumer discretionary, finance, utilities, consumer staples, etc. The distribution of the top 10 nodes for the two markets was similar, while the top 10 links in the S&P500 market were much more dispersed than those in the HS300 market because the American market, as a mature market, has better risk diversification. To understand the spatiotemporal patterns of risk propagation in complex financial networks, we first investigated the dependence of \(F_{i}\) on the degree distribution of the network by linking it with the weighted degrees of all nodes \(S_{i}=\sum_{j=1}^{N}Aij\). Next, we focused on the dependence of the individual response time \(\tau_{i}\) on the weighted degree \(S_{i}\). As displayed as triangles in Figure 2a,c, \(F_{i}\) positively scales with \(S_{i}\), and as shown as triangles in Figure 2b,d, \(\tau_{i}\) negatively scales with \(S_{i}\). In brief, hub nodes (high-degree nodes) played a more important role in risk propagation across the network and respond rapidly, indicating a degree-driven effect. We also linked \(F_{ij}\) with the product of \(i\) and \(j\)'s weighted degree, and the results are listed in Supplementary Material S1. The links first exhibited a link-driven effect and, then, an anti-link-driven effect as \(S_{i}S_{j}\) increased. As mentioned above, a susceptible node may become infected through interaction with an infected node at a rate R. An infected node recovers at a rate B, becoming susceptible again. Analogically, a stock is infected if the stock is exposed to a risk, and the remaining stocks that have not yet been exposed to a risk are considered susceptible. We further investigated how the infection rate R and the recovery rate B influence the risk propagation flow of complex financial networks. Varying the dynamic parameters R and B from 0 to 1, we investigated the spatiotemporal patterns of the risk propagation flow for the S&P500 and HS300 markets. As displayed in Figure 2, the spatiotemporal patterns remained qualitatively consistent for different R and B, i.e., the nodes exhibit a degree-driven effect. In contrast, spatiotemporal patterns were quantitatively inconsistent for different \(R\) and \(B\).