# Spatiotemporal Patterns of Risk Propagation in Complex Financial Networks
## 3 Data and Methods
### Method and Basics  
In this section, we combine the dynamic differential equations of the SIS infectious disease model with the empirical networks of financial systems and use Monte Carlo simulation to reveal the spatiotemporal patterns of risk propagation in complex financial networks [29; 38]. First, we constructed an empirical financial network with \(N\) components (nodes), linked via the cross-correlation matrix \(C_{ij}\) of returns [73; 74; 75; 76], which is defined by  
\[C_{ij}=<r_{i}(t^{\prime})r_{j}(t^{\prime})>. \tag{3}\]  
where \(C_{ij}\) measures the equal-time correlation between the i-th stock and the j-th stock. The Planar Maximally Filtered Graph (PMFG) tool is often implemented to filter redundant information in complex networks [77]. Taking the absolute values of the negative matrix elements, we pruned the modified cross-correlation matrix \(|C_{ij}|\) with the PMFG method and generated the PMFG graph \(A_{ij}\)[44]. Epidemic models are widely applied to financial systems [58; 62; 63] due to the similarity between the spread of financial risk and infectious diseases. Thus, we combined the susceptible-infected-susceptible (SIS) infectious disease model with empirical networks to obtain the risk propagation flow of complex financial networks. In the SIS infectious disease model, each node exhibits two distinct states: susceptible (S) and infected (I) [78; 79; 80; 81]. Analogically, a stock is infected if the stock is exposed to risk, and the remaining stocks that have not been exposed to risk yet are susceptible. Two key parameters exist in this model, i.e., the infection rate \(R\) and recovery rate \(B\). A susceptible node may become infected through interaction with an infected individual at an infection rate \(R\). \[I(j)+S(i)\stackrel{{ R}}{{\longrightarrow}}I(j)+I(i). \tag{4}\]  
An infected node recovers at a recovery rate \(B\), becoming susceptible again:  
\[I(i)\stackrel{{ B}}{{\longrightarrow}}S(i) \tag{5}\]  
Here we introduce the dynamic differential equations for the SIS model [38]:  
\[\frac{dx_{i}(t)}{dt}=-Bx_{i}(t)+\Sigma_{j=1}^{N}A_{ij}R(1-x_{i}(t))x_{j}(t). \tag{6}\]  
where \(x_{i}(t)\) is the probability of infection for stock i, i.e., the probability of the stock i being exposed to a risk. \(-Bx_{i}(t)\) represents the recovery process of stock i, proportional to the ratio of infected (exposed) \(x_{i}(t)\). The second term describes the process of infection, which depends on i's neighbor \(j\) being infected (probability \(x_{i}(t)\)) and on i being susceptible (probability \(1-x_{i}(t)\)). The weighted link of the PMFG graph \(A_{ij}\) represents the rate of influence between \(x_{i}\) and \(x_{j}\). Once stock i is exposed to a risk, the risk spreads to stocks adjacent to stock i in the network. Increasingly more stocks will progressively become exposed to the risk. Hence, we introduce a perturbation (risk) \(\Delta x_{j}\) to the \(j\)-th stock and, then, track its propagation across the network. To explore the temporal patterns of risk propagation in complex financial networks, the individual response time is defined as \(\tau_{i}\):  
\[\Delta x_{i}(t=\tau_{i})=\eta\Delta x_{i}(t\rightarrow\infty), \tag{7}\]  
where \(\tau_{i}\) represents the time when stock \(i\) reaches an \(\eta\)-fraction of its final response to the perturbation (risk) from \(i\)'s neighbor \(j\)[29]. For the simplicity of notation, we use \(t\rightarrow\infty\) to represent the time when stock \(i\) has reached its final response to the perturbation (risk) from stock \(j\). Typically, the half-life of \(i\)'s response corresponds to \(\eta=0.5\). Over time, the perturbation (risk) \(\Delta x_{j}\) leads to a cascade of responses; then, the whole network reaches a steady-state.