Stochastic Differential Equations: Theory and Applications by Arnold L.

By Arnold L.

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Extra resources for Stochastic Differential Equations: Theory and Applications

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We note, however, that in everyday usage phenomena with complicated and intricate features having both the characteristics of randomness and order are called complex. 2 Empirical laws 33 measure of complexity, or what distinguishes a simple from a complex web. Any list of characteristics of complex webs is therefore idiosyncratic. So, with all these caveats, here is our list of properties that a complex web possesses. (i) Diversity and non-extensivity. A complex web typically contains many elements or nodes and the complexity expands in a nonlinear way with the number of elements; it is not monotonic.

Perhaps an example will serve to clarify this point. The black swan was set in an economic context and requires a certain amount of background to understand the influence of extrema, but there is another, more direct phenomenon with which we are familiar; a blackout. Hines [15] defines a blackout as being any power-network event that results in an involuntary interruption of power to customers and lasts longer than 5 minutes. When the power grid fails its influence is geographically dispersed, potentially affecting literally millions of people and crippling entire regions of the country.

The distributions for these complex webs were all inverse power laws, which from a maximum-entropy argument showed that the relevant property was the scale-free nature of the dynamic variable. The replacement of the normal distribution with the hyperbolic distribution implies that it is the extremes of the data, rather than the central tendency, which dominate complex webs. Consequently, our focus was shifted from the average value and the standard deviation to the variability of the process being investigated, a variability whereby the standard deviation diverges.

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