General approach
Extreme events occur in natural, technical and social environments. Being an “event”, a given Xevent is something that happens within a limited space and time. Its occurrence can arise by chance or necessity or through a combination of both; through natural or human-made causes. The interpretation of “extreme” cannot be defined so easily. It encompasses a collection of attributes, such as rare, exceptional, catastrophic, surprising etc. An insurer would translate “rare” as “low-probability”, and “catastrophic” as “of great consequence”, the later emphasising the event’s potential for impact and change. The above shows that in a general understanding, a hurricane is an Xevent only if it causes loss of life and material damage; on the other hand it is considered as an ordinary event if it hits uninhabited areas. Since, however, from the viewpoint represented by the sciences (including physics) the impact aspect is not the most important (and huge deviations or burst-like behavior are what really matter in respect to Xevents), the popular understanding of Xevents represented by the humanities (including sociology) makes the interdisciplinary research on Xevents is so difficult.
The above indicates that the issue of Xevents is multifaceted, intricate, and subject to various interpretations. For science it is a very uncomfortable situation. From a physical perspective, the aim is to free Xevents from their subjectivity, so that a more objective definition can be obtained. Defining a universal quantity (mathematical, physical, or whatever) characterizing different Xevents, and having similar meaning like order parameters in equilibrium phase transitions would be a milestone in the research. This is not yet available. All one can do at the moment is to characterize Xevents through their statistical and dynamical properties, and to try to look for universalities or common mechanisms underlying Xevents.
Statistical physics deals with systems comprising a very large number of interacting subunits, for which predicting the exact behaviour of the individual subunit is impossible. Hence, one is limited to making statistical predictions regarding the collective behaviour of the subunits. Phase transitions, which are typically observed in such systems, can be easily related to the somewhat vague intuitive concept of Xevents. For example, at critical points in continuous phase transitions large fluctuations of observables exist. These imply large cooperative phenomena in the systems and may, under certain circumstances, imply consequences that are extreme in the popular / intuitive sense of the word. For this reason critical events are often treated as extreme events (i.e. due to vague definition of Xevents the treatment is acceptable), and vice versa – extreme evens are often confused with critical events (i.e. Xevents are phase transitions in only a very few cases). Let us rephrase it once again: even thought the apparent similarity between extreme events and critical events strong differentiation between the both is in order. Namely, phase transitions (and accompanying them critical phenomena) may only occur for particular values of system parameters, and are known to belong to several universality classes, whereas extreme events may arise “by chance”, still being not understood in the sense of development and occurrence.
Xevents in complex networks
Basic approach to Xevents in complex networks
In the new-emerging science of complex networks (and in particular scale-free networks) the most general perception of extremity / exceptionality is related to existence and function of “hubs” i.e. nodes with an extremely high number of connections to other nodes. It was shown that even if such nodes are quite rare (degree distribution follows power law in most of real-world networks) they usually determine the overall network structure and most of dynamical processes that happen on top of the network. In particular, a number of scale-free network models has been proposed where exceptional / extreme events play a crucial role. One has also proved that hubs strongly influence critical properties in complex networks (like spreading of epidemics and robustness under random attacks).
Here you can download a program (Windows version) demonstrating an impact of an extreme event on the structure of the complex network.
(Selected papers on basic approach to Xevents in complex networks)
Network growth is often explained through mechanisms that rely on node prestige measures, such as degree, fitness or a kind of ranking. As a matter of fact, although the models differ in details, they are governed by the same general principle: “the rarer / more exceptional you are the more attractive you become”.
The authors have studied statistical properties of the highest degree, or most popular, nodes in growing networks. Focusing on the simplest scale-free evolving networks (BA model) they have tried to answer the following questions: How does the identity of the leader / hub – i.e. the individual / node who possesses the extreme value of a particular attribute – changes over time? What is the rate at which lead changes occur? And finally, what is the probability that a leader retains the lead as a function of time?
Cascade failures in complex networks
Although cascade effects may appear in different kinds of networks (including food webs and social networks), a short literature survey on the subject shows that the issue is the most relevant in application to infrastructure networks. For the reason, in the following we concentrate on such networks.
Thus, it is commonly known that in most infrastructure networks, the loads carried by each node on the network are dynamically redistri
buted. A general scenario for cascade failures in such networks is as follows: If a network node is lost, due to accident or attack, the load that node carries is rapidly distributed to the other nodes on the network. If a high-load node is removed from the
network, the loads it carries are redistributed to other nodes on the network. This increased flow causes less capable nodes to exceed their
capacity. To protect these nodes from damage, many networks will automatically force the
overloaded node to fail-over (shut down). In other networks, the increased congestion will cause the overloaded node to become inefficient (bog down). Regardless, the result is a series of shut-downs or slow-downs that "cascade" through the network as the excess load is pushed to the next available node. The end result is total network failure.
It is important to stress that cascading failures only occur in heterogeneous networks where there are a few nodes that have the capacity for high-loads and many with the capacity only for low-loads. Homogeneous networks, where all the nodes handle an equal load do not
suffer cascading failure. Unfortunately, all infrastructure networks, and also overwhelming majority of other real-world networks are heterogeneous by design, creating possibilities for cascading effects.
The authors show that natural and man-made disasters can be described by interconnected causality chains – i.e. cascading failures spreading within interaction networks reflecting how one factor / sector of a system affects others. They argue that such causality networks allow one to estimate the development of disasters with time, to anticipate different side effects, and also give hints about when to take certain actions. They also discuss several examples of disasters / Xevents, and present their causality networks.
Endogenous and exogenous origins of crises in complex networks
(Selected papers on endogenous and exogenous origins of crises in complex networks )
The above indicates that the issue of Xevents is multifaceted, intricate, and subject to various interpretations. For science it is a very uncomfortable situation. From a physical perspective, the aim is to free Xevents from their subjectivity, so that a more objective definition can be obtained. Defining a universal quantity (mathematical, physical, or whatever) characterizing different Xevents, and having similar meaning like order parameters in equilibrium phase transitions would be a milestone in the research. This is not yet available. All one can do at the moment is to characterize Xevents through their statistical and dynamical properties, and to try to look for universalities or common mechanisms underlying Xevents.
ure and most of dynamical processes that happen on top of the network. In particular, a number of scale-free network models has been proposed where exceptional / extreme events play a crucial role. One has also proved that hubs strongly influence critical properties in complex networks (like spreading of epidemics and robustness under random attacks).