What is catastrophe? A catastrophe occurs when a system in a stable state is transformed into another stable state in a time period that is...
What is catastrophe? A catastrophe occurs when a system in a stable state is transformed into another stable state in a time period that is orders of magnitude shorter than the system's lifetime, due to some external or internal cause or causes. The process often takes the form of a collapse, in which a precise, refined equilibrium state ceases to exist.
A catastrophe can be, for example, the sudden collapse of a building, a lightning or an earthquake, or a social revolution, and in the biosphere, the mass extinction of living systems is a rapid transformation, a manifestation of catastrophe as a phenomenon.
Theoretically, the phenomenon of catastrophe is well understood for simpler systems with few variables and mathematically well-describable interactions, known as exact systems. For complex systems with many variables, which may be difficult or impossible to define mathematically, in the so-called inexact systems such as catastrophes in biological or social structures, the developed catastrophe theory does not provide reliable solutions, and is unable to model the collapse.
However, inexact systems are also subject to catastrophe. When an inexact system in equilibrium is suddenly transformed into another state by external or internal influences, it is not a random process but seemingly initiated by and follows direct regularities.
The mathematical tools of the developed catastrophe theory are not suitable for modeling and predicting catastrophes in inexact systems. However, since seemingly rules lead to catastrophes in inexact systems, also, some modeling, which may even lead to the prediction of catastrophe, can certainly be set up. Catastrophe analysis of inexact systems, however, requires new methods that are different from the mathematical tools of traditional catastrophe theory.
Catastrophe as a phenomenon is interesting behavior in the case of complex systems. Inexact systems are necessarily complex systems, so for catastrophe analysis of inexact systems, it is sufficient to consider the behavior of complex systems.
How can a complex system get to a catastrophic state? What can lead to the rapid collapse of a complex system? When examining complex systems, we can distinguish between two types of catastrophes based on the strength of the impact that causes the collapse.
Trivial catastrophe
A trivial catastrophe occurs when the strength of the effect that causes the sudden collapse is greater than the strength of the interaction between the elements of the system in equilibrium. In this case, the presented effect causes an immediate collapse of the system. A trivial catastrophe is when a bridge collapses due to an excessively large load, and a similar example is when a nearby supernova causes mass extinction on Earth. Trivial catastrophes of both exact and inexact systems can be interpreted straightforwardly.
But disasters can happen in other than just trivial ways. A disaster can also occur in a way that the effect causing the collapse is almost imperceptible in the system. In complex systems, even weak effects can lead to catastrophe, as for example a seemingly intact building suddenly collapses without any prior sign or noticeable effect, just as a cause of sudden mass extinction of the biosphere is not always trivial. Disasters of this kind form another possible class of catastrophes in complex systems.
Stealth catastrophe
A stealth catastrophe occurs when the magnitude of the effect causing the collapse is comparable to the strength of the interactions that hold the elements of the complex system together. In a stealth catastrophe, the system seems to collapse spontaneously or only in response to a less significant effect. In the case of a stealth catastrophe, the effect may not necessarily be only one type of effect. A stealth catastrophe can be caused by a variety of weak effects occurring simultaneously or even sequentially.
How can a weak effect cause a complex system in equilibrium to collapse? In this case, the longer persistence of the effect causing the catastrophe is the reason for the sudden collapse of the system.
How does stealth catastrophe work?
Every complex system can be characterized by the elements that constitute the system and the relationships and interactions between the elements. A complex system is a network of elements that build up the system and their interconnections. The network of a complex system can also be modeled as a graph. The graph consists of nodes (the elements of the system) and edges or lines connecting the nodes (the relationships of the system, the interactions between the constituting elements). A complex system is usually a multi-coherent graph, where multiple paths are possible between the nodes in the graph. (A complex system that can be modeled as a single-coherent graph typically acts according to the trivial catastrophe mechanism.)
The stealth catastrophe can be understood through the graph model of complex systems.
In the case of a stealth catastrophe, the system is subject to effects over a longer period of time, whose strength can be compared to the strength of the interactions between the elements of the system. While such an effect is exerted on the system, the strength of the effect may at times be sufficiently strong enough to break a connection between the components that constitute the system. The functioning of a multi-coherent system as a whole is not affected by the breaking of a connection, because the connection between the two components of the system is not broken, the connection continues to exist by another, less direct, but established route. However, the scale of the graph, i.e. the complexity of the system is definitely reduced.
If the effect persists, or if it is terminated, but another effect of comparable magnitude to the strength of the connection that holds the elements of the system together appears, another connection between the elements of the system may be destroyed.
Dynamic complex systems can have regenerative functions, capable of establishing new connections in a general way, or re-establishing broken connections as a dedicated process. In complex biological systems, new connections are formed in a general way by the process of evolution, while, for example, the error-correcting mechanism of DNA can repair errors in the structure of DNA in a dedicated way. In the case of society, the formation of new customs and habits creates new relationships in a general way, while the mechanism of legislation and law enforcement for example is dedicated to the specific repair of defective relationships.
However, the effectiveness of the regenerative function is inhibited by the existing condition that the system itself remains functional despite the loss of the connection or connections, and thus there is no necessity for the regenerative function to operate.
Complex systems can lose connections under this type of weak impacts in a continuous but unnoticeable way. In the process, the complexity of the system is continuously reduced. When modeled as a graph, the graph characterizing the system remains connected, but its scale of connectedness, its complexity decreases.
The system does not necessarily lose its functionality, even if it breaks down into several parts in the process, assuming that the complexity of the subsystems are still sufficiently preserved, because the coherent nature of the system can be re-established through the regenerative function through many different possible links. In this case, there are several possibilities for the regenerative function to make the system connected, and therefore the coherence of the graph is highly probable reinstated, and therefore can be quickly restored.
Because of the persistent decoupling effect, although the original complex graph can remain connected for a long time, or regenerate quickly into a connected graph, the complexity of the system, the degree of connectedness of the graph, will necessarily decrease steadily. A weak but long-lasting decoupling effect has no determining effect on the functionality of the system, on the coherence of the graph for a prolonged time, but it will continuously reduce the complexity of the system, the degree of connectedness of the graph.
How long can this process continue? This reduction in complexity does not fundamentally change the functionality of the system, but it does increase the degree of criticality of the system.
The measure of the criticality of a complex system is the metric that shows that if a connection in the graph representing the system is removed which makes the graph incoherent, how many other, new, possible connections are available to create in reality which would restore the coherence of the graph. As this number decreases, the criticality of the system increases.
Criticality can also be expressed in terms of how many other connections in the graph that might exist in practice have to be created to make the graph coherent again, when the loss of a connection in the graph removes the coherence of the graph. The higher this number, the more critical the system is.
In the case of a stealth catastrophe, the system must reach a certain criticality threshold, specific to the system, the destructive effect, and the regenerative function, in order to collapse.
Under the process of stealth catastrophe, the system remains continuously operational until it approaches the criticality limit. Reaching this state, however, even a small impact can cause the system to collapse.
Dynamic systems can approach the criticality limit without significant changes in functionality, and in this state, they can operate properly for a longer period of time. However, this threshold is a hyper-critical state, because any disturbance, even a minor one, can cause the collapse of the system, leading to a permanent breakdown of the graph modeling the system.
It is a characteristic of dynamical complex systems (such as biological or social structures) that the existence of their constituting elements also requires the existence of interactions that create relationships. In the hyper-critical state of a complex system, because the breaking connections break the structure down into non-connected parts, the constituent elements may cease to exist due to isolation or insufficient relations. The loss of the building blocks of the system also makes it impossible to re-establish new connections, causing an avalanche-like collapse of the system, creating a catastrophe.
Catastrophe analysis of inexact complex systems is possible, and catastrophe can be predicted by modeling the system as a graph, identifying the elements that constitute the system and their interrelations, and analyzing criticality.
At present, the complexity of the biosphere is in noticeable decline (at least partially due to the consequence of human activities), although the biosphere is still functioning. The collapse of the biosphere - which could have a decisive impact on human life and even on humanity - can perhaps be predicted by conducting a catastrophe analysis of inexact systems, and perhaps the catastrophe can be prevented by drastic measures recognizing the level of criticality.
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