Visibility graph approach to extreme event series

An extreme event may lead to serious disaster to a complex system. In an extreme event series there exist generally non-trivial patterns covering different time scales. Investigations on extreme events are currently based upon statistics, where the patterns are merged into averages. In this paper from extreme event series we constructed extreme value series and extreme interval series. And the visibility graph is then adopted to display the patterns formed by the increases/decreases of extreme value or interval faster/slower than the linear ones. For the fractional Brownian motions, the properties for the constructed networks are the persistence, threshold, and event-type-independent, e.g., the degree distributions decay exponentially with almost identical speeds, the nodes cluster into modular structures with large and similar modularity degrees, and each specific network has a perfect hierarchical structure. For the volatilities of four stock markets (NSDQ, SZI, FTSE100, and HSI), the properties for the former three's networks are threshold- and market-independent. Comparing with the factional Brownian motions, their degree distributions decay exponentially but with slower speeds, their modularity behaviors are significant but with smaller modularity degrees. The fourth market behaves similar qualitatively but different quantitatively with the three markets. Interestingly, all the transition frequency networks share an identical backbone composed of nine edges and the linked graphlets. The universal behaviors give us a framework to describe extreme events from the viewpoint of network.