# Spatiotemporal Patterns of Risk Propagation in Complex Financial Networks
## 4 Results
### Quantifying Risk Propagation Flow
As shown in Figure 2a-d, when the recovery rate R decreased, two effects occurred: (i) \(F_{i}\) of the hubs increased, indicating that the nodes tended to be more degree-driven as R decreased; (ii) \(\tau_{i}\) of the hubs also increased as R decreased, weakening the degree-driven \begin{table}
\begin{tabular}{l l l} \hline \hline
**Market** & **Top 10 Nodes** & **Top 10 Links** \\ \hline \multirow{7}{*}{S\&P500} & CAG.N & CZR.O \(\longleftrightarrow\) VFC.N \\  & CPB.N & CZR.O \(\longleftrightarrow\) SPGL.N \\  & DG.N & CNP.N \(\longleftrightarrow\) CZR.O \\  & MNST.O & AME.N \(\longleftrightarrow\) PM.N \\  & AMGNO.O & FITB.O \(\longleftrightarrow\) IQV.N \\  & GIS.N & PM.N \(\longleftrightarrow\) TXT.N \\  & K.N & CNP.N \(\longleftrightarrow\) VFC.N \\  & REGN.O & BOX.N \(\longleftrightarrow\) FITB.O \\  & MSFT.O & CZR.O \(\longleftrightarrow\) DHR.N \\  & ABT.N & BXP.N \(\longleftrightarrow\) IQV.N \\ \hline \multirow{7}{*}{HS300} & 002773.SZ & 002714.SZ \(\longleftrightarrow\) 300498.SZ \\  & 600521.SH & 000876.SZ \(\longleftrightarrow\) 300498.SZ \\  & 002555.SZ & 002311.SZ \(\longleftrightarrow\) 300498.SZ \\  & 600763.SH & 002157.SZ \(\longleftrightarrow\) 300498.SZ \\  & 002624.SZ & 600196.SH \(\longleftrightarrow\) 600521.SH \\  & 600196.SH & 600196.SH \(\longleftrightarrow\) 601607.SH \\  & 300498.SZ & 002773.SZ \(\longleftrightarrow\) 600079.SH \\  & 002008.SZ & 600763.SH \(\longleftrightarrow\) 603939.SH \\  & 600436.SH & 000876.SZ \(\longleftrightarrow\) 002714.SZ \\  & 603939.SH & 002773.SZ \(\longleftrightarrow\) 600763.SH \\ \hline \hline \end{tabular}
\end{table} 
Table 1: This table lists the stock tick names corresponding to the top 10 nodes (links) of risk propagation flow networks, ranked by the size of \(F_{i}(F_{ij})\) for the S&P500 market and HS300 market. For simplicity, here, we take \(B=1\) and \(R=1\). Figure 1: (**a**) Risk propagation flow network of the HS300 market. (**b**) Risk propagation flow network of the S&P500 market. The size of nodes and the width of links represent the \(F_{i}\) of a stock and the \(F_{ij}\) between two stocks, respectively. For simplicity, here, we take \(B=1\) and \(R=1\). effect of nodes. Figure 2e-h demonstrate that \(F_{i}\) and \(\tau_{i}\) of the hubs both increased, meaning that the degree-driven effect also had two effects as the recovery rate B increased. The impacts of these two dynamic parameters on the degree-driven effect of the nodes were almost opposite. As reported in [38], on average, risk propagation flow and the node degree obey the relationship \(F_{i}\sim S_{i}^{\omega}\), and the contribution of \(A_{ij}\) link to \(Fij\) follows \(Fij\sim A_{ij}S_{i}^{\omega-1}S_{j}^{\omega-1}\). For the SIS model of epidemic spread, \(\omega=0\)[38]. In this case, \(F_{i}\) is independent of \(S_{i}\), which represents homogeneous flow. Based on linear response theory, \(\tau_{i}\) is linked with \(S_{i}\) through the relationship \(\tau_{i}\sim S_{i}^{\theta}\)[29]. In other words, \(\tau_{i}\) is determined by the interaction between topology \(S_{i}\) and dynamics \(\theta\). \(\theta\) was determined to be -1 for the SIS model here [29]. However, our empirical results for financial networks were slightly different from the theoretical derivation in References [29; 38], which may arise from the complexity of financial networks, such as an obvious community structure and frequent large fluctuations. In future research, we can determine the values of these two parameters based on data from the real market to describe the risk propagation flow of complex financial networks more accurately.