# Spatiotemporal Patterns of Risk Propagation in Complex Financial Networks
## 5 Conclusions  
To explore the spatiotemporal patterns of risk propagation in complex financial networks, we combined the SIS infectious disease model with empirical networks. We revealed that hub nodes (high-degree nodes) play important roles in risk propagation across the network and respond rapidly, exhibiting a degree-driven effect. Moreover, we investigated the influence of two key dynamic parameters. The impacts of the two dynamic parameters on the degree-driven effect of the nodes were almost opposite. Figure 4: Spatiotemporal patterns of risk propagation in extreme market states. We measured the risk propagation flow \(F_{i}\) and the local response times \(\tau_{i}\) of all nodes for the S&P500 market and the HS300 market in extreme market states. For comparison, the results of the original network are displayed in triangles. (**a**) \(F_{i}\) vs. the weighted degree \(S_{i}\) for the S&P500 market with different threshold \(\sigma\) for two tails, T. (**b**) \(\tau_{i}\) vs. \(S_{i}\) for the S&P500 market with different threshold \(\sigma\) for two tails, T. (**c**) \(F_{i}\) vs. \(S_{i}\) for the HS300 market with different threshold \(\sigma\) for two tails, T. (**d**) \(\tau_{i}\) vs. \(S_{i}\) for the HS300 market with different threshold \(\sigma\) for two tails, T. (**e**) \(F_{i}\) vs. \(S_{i}\) for the S&P500 market for two tails, T; the negative tail, N; and the positive tail, P. (**f**) \(\tau_{i}\) vs. \(S_{i}\) for the S&P500 market for two tails, T; the negative tail, N; and the positive tail, P. (**g**) \(F_{i}\) vs. \(S_{i}\) for the HS300 market for two tails, T; the negative tail, N; and the positive tail, P. (**h**) \(\tau_{i}\) vs. \(S_{i}\) for the HS300 market for two tails, T; the negative tail, N; and the positive tail, P. For (**a–d**), \(\sigma\in[0.5,2]\), while for (**e–h**), \(\sigma=1.5\). Furthermore, we explored the impacts of two typical characteristics of complex financial systems-- the existence of community structures and frequent large fluctuations--on risk propagation flow. First, we characterized the risk propagation flow between and within different communities of the complex financial networks. About 30% of the total risk propagation flow of each community can be explained by the top 10% nodes. Thus, we can control the risk propagation flow of each community by controlling a few influential nodes in each community and, in turn, control the whole network. Then, we classified different categories of large fluctuations with the threshold method and explored the risk propagation flow in extreme market states. As the threshold increased, the degree-driven effect of the nodes became greater. In other words, hub nodes become more dominant in extreme market states, indicating that better control may be achieved by hub nodes in these cases. The results shed light on the underlying mechanisms of risk propagation flow in complex financial networks, answering the questions about which nodes are more important in the risk propagation process and which nodes have a shorter response time. One limitation of our work is that, although the SIS model is an appropriate approximation to describe the risk propagation process of the financial system for now, it is not very accurate. In the future, we can replace it with a more accurate dynamic equation of the financial system. Moreover, we can design a risk management scheme based on our results and control the influential nodes to slow down or curb the spread of financial risk in the whole network. New methods may also be developed and applied to portfolio optimization based on our results.