# Spatiotemporal Patterns of Risk Propagation in Complex Financial Networks
## 4 Results
### Risk Propagation Flow in Extreme Market States  
Frequent large fluctuations are another typical characteristic of complex financial network dynamics. A large fluctuation is identified when the volatility is sufficiently large compared with the average one [51].  
First, we filtered out large fluctuations in extreme market states by taking different thresholds. For two tails, we assigned \(r(t^{\prime})=0\) for \(|r(t^{\prime})|<\sigma\), where \(\sigma\) is the threshold. For the negative (positive) tail, we used \(r(t^{\prime})=0\) for \(r(t^{\prime})>-\sigma\left(r(t^{\prime})<\sigma\right)\). Then, we calculated the cross-correlation matrix \(C_{ij}\) of \(r(t^{\prime})\)[73, 74, 75, 76]. Then, we pruned the modified cross-correlation matrix \(|C_{ij}^{\sigma,tail}|\) with the PMFG method and generated the threshold PMFG graph \(A_{ij}^{\sigma,tail}\).  
\begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Market** & **Community Name** & **Top 1 Node** & **Nodes No.** & **Weight** \\ \hline S\&P500 & Medical device and service & BXPN & 59 & 27.75\% \\ Finance & LNC.N & 40 & 22.81\% \\ Food and daily necessities & CAG.N & 37 & 29.42\% \\ Consumer durables and apparel & GPS.N & 14 & 25.18\% \\ Pharmaceuticals and biotechnology & LLY.N & 28 & 23.84\% \\ Energy and materials & CTSH.O & 27 & 21.30\% \\ Retailing, capital goods, and media & DLTR.O & 27 & 25.02\% \\ Software and service & AJG.N & 63 & 30.42\% \\ Tourism and transportation & PENN.O & 44 & 28.43\% \\ Utilities & AEE.N & 75 & 32.52\% \\ Real estate & ITW.N & 48 & 25.74\% \\ Semiconductors & AMAT.O & 21 & 25.51\% \\ \hline HS300 & Securities & 601211.SH & 22 & 26.39\% \\ Food and daily necessities & 600050.SH & 25 & 24.35\% \\ Healthcare & 600763.SH & 28 & 27.90\% \\ Energy and materials & 000876.SZ & 29 & 33.58\% \\ Real estate & 002271.SZ & 14 & 24.43\% \\ Information technology & 600584.SH & 35 & 27.40\% \\ Transportation & 601021.SH & 15 & 37.20\% \\ Capital goods & 601618.SH & 15 & 21.82\% \\ Automobiles & 000625.SZ & 11 & 24.43\% \\ Banks, insurance, and industrials & 601818.SH & 46 & 27.07\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: This table lists the community name, the stock tick name corresponding to the top 1 node, the node/stock number, and the weight of the top 10% stocks’ \(F_{i}\) in each community for the S\&P500 market and the HS300 market.  
Figure 3: (**a**) Risk propagation flow within communities for the S\&P500 market. (**b**) Risk propagation flow within communities for the HS300 market. The size of the nodes and the width of links denote the \(F_{i}\) of a community and the \(F_{ij}\) between two communities, respectively.  
Thus, threshold risk propagation flow networks were constructed from the threshold PMFG graph \(A_{ij}^{\sigma,T}\). As shown in Figure 4a-d, \(F_{i}\) positively scaled with \(S_{i}\), and \(\tau_{i}\) negatively scaled with \(S_{i}\), i.e., the nodes exhibited a degree-driven effect. When the threshold \(\sigma\) increased, two effects were observed: (i) the \(F_{i}\) of the hubs increased, indicating that the nodes tended to be more degree-driven; (ii) the \(\tau_{i}\) of the hubs also increased, weakening the degree-driven effect of the nodes. The former effect was stronger by comparison. Therefore, hub nodes were more dominant in extreme market states, indicating that we may achieve better control using hub nodes in these cases.  
Furthermore, we distinguished between crashes and bubbles and calculated the risk propagation flow with the threshold PMFG graph \(A_{ij}^{\sigma,N}\) and \(A_{ij}^{\sigma,P}\). As shown in Figure 4e-h, \(F_{i}\) and \(\tau_{i}\) of the hubs both increased in the order of two tails, the negative tail, and then, the positive tail. The hub nodes were more dominant, on the one hand, but responded more slowly, on the other hand, during bubbles than during crashes. The dependence of the link \(F_{ij}\) on the product of \(i\) and \(j\)'s weighted degree is listed in Supplementary Material S1.