# Spatiotemporal Patterns of Risk Propagation in Complex Financial Networks
###### Abstract  
The methods of complex networks have been extensively used to characterize information flow in complex systems, such as risk propagation in complex financial networks. However, network dynamics are ignored in most cases despite systems with similar topological structures exhibiting profoundly different dynamic behaviors. To observe the spatiotemporal patterns of risk propagation in complex financial networks, we combined a dynamic model with empirical networks. Our analysis revealed that hub nodes play a dominant role in risk propagation across the network and respond rapidly, thus exhibiting a degree-driven effect. The influence of key dynamic parameters, i.e., infection rate and recovery rate, was also investigated. Furthermore, the impacts of two typical characteristics of complex financial systems--the existence of community structures and frequent large fluctuations--on the spatiotemporal patterns of risk propagation were explored. 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 the community and, in turn, control the whole network. In extreme market states, hub nodes become more dominant, indicating better risk control.  
complex financial systems; complex financial networks; econophysics; risk propagation; network dynamics +
Footnote †: journal: Applied  
of the network topology, a typical characteristic of complex financial systems is the existence of community structures [39; 40; 41; 42; 43; 44]. Stocks in the same community share common economic properties and information; thus, the stock prices in a community tend to move simultaneously. Based on the random matrix decomposition (RMD) method, the cross-correlation matrix can be decomposed into three categories of modes: market mode, sector mode, and random mode [43; 44; 45; 46; 47]. In particular, local interactions among the communities are mainly included in sector mode [43; 44]. In terms of dynamics, frequent large fluctuations, such as a financial crisis and a financial bubble, are another distinctive feature of complex financial systems [48; 49; 50]. Additionally, investigating the non-stationary dynamic properties is very important in understanding financial systems comprehensively [51; 52; 53; 54; 55; 56].  
On the one hand, the research on complex financial systems is still focused on a pure structure [21; 22; 23; 24] or a pure model [57; 58; 59; 60; 61; 62; 63]. The new method of complex networks combining structure and dynamics has not been applied to complex financial systems [29; 38]. On the other hand, we still do not know exactly what role each node and link in the network plays in the risk propagation processes and how much time each node takes to respond. In this paper, we are mainly committed to solving these two problems. Due to the similarity between the spread of financial risk and infectious disease, epidemic models are widely applied to financial systems [58; 62; 63]. Therefore, we combined the susceptible-infected-susceptible (SIS) infectious disease model with the empirical networks to explore the spatiotemporal patterns of risk propagation flow in complex financial networks.  
In Section 2, we review the literature. Section 3 presents the data and methods. In Section 4, we conduct the empirical analysis: First, we obtain the risk propagation flow of complex financial networks, and then, we investigate the influence of two key dynamic parameters. Next, we detect the communities in the sector mode and further characterize the risk propagation flow between and within different communities. Lastly, we classify different categories of large fluctuations with the threshold method and explore the risk propagation flow in extreme market states. The conclusions are stated in Section 5.