Dynamics and statistics of multi-scale systems
From COSNet
Complex systems often evolve and interact on scales spanning several order of magnitudes in space and in time. It is therefore not surprising that scaling law analysis is one of the most useful ways to characterize and to understand these systems. Indeed, features on one spatial or temporal scale are replicated similarly on other scales, and when such behaviour is found it is an indication of a common guiding mechanism extending across scales. The fact that similar scaling behaviours are found in several different systems points to the existence of universal guiding mechanisms.
In recent years, multifractal scaling analysis has proven useful in uncovering scaling properties in a diverse range of temporal and spatial data sets. A fractal is a scale-invariant object for which the box counting dimension is less than the topological dimension. The box counting dimension is inadequate for quantifying the complexity of a non-homogeneous object because there are large fluctuations in the local scaling behaviour over different regions of the object. Multifractal analysis provides a measurement of these fluctuations and provides a set of measurements for the local scaling properties across different subsets that share similar scaling behaviours.
Scaling laws are also a common feature in complex systems that evolve self-organized structures. Self organization driven by power law scaling is called Self-Organized-Criticality. As an example, if sand is slowly dropped onto a surface then, after a while, a self-organized sandpile with a stable and well defined slope emerges. The enabling mechanism for this behaviour is a power law distribution in the size of the avalanches that led to the sandpile. Self-Organized Criticality (SOC) is an important area of research with applications ranging from financial data analysis to earthquake prediction.
Gaussian distributions are well known in characterizing randomness, but in many complex inhomogeneous systems other kind of distributions may be more appropriate. For instance the probability distribution describing the price fluctuations in stock markets has a larger probability for extreme events than the one predicted by the Gaussian distribution ('fat tails').
Moreover, the stochastic behaviour of complex systems is often characterized by power-law-tailed distributions, indicating that the fluctuations have no typical length-scale ('scale-free behaviour'). Non-Gaussian stochastic processes with fractal or multifractal scaling properties are of fundamental importance in applications ranging from diffusion in biological media to turbulence in fluids to fluctuations in financial options.
For instance, the density distribution of diffusing species in a homogeneous environment is Gaussian. The diffusion in this case is also characterized by a quadratic power law scaling between the mean square displacement of diffusing particles and time. On the other hand, in non-homogeneous environments containing traps and binding sites the mean square displacement of diffusing particles still scales as a power law with time but the scaling exponent is less than two. This is an example of anomalous diffusion. The density distribution of diffusing species in this sort of environment is non-Gaussian.
