CITS4403 Computational Modelling

Self Organized Criticality

Critical SystemsA system is “critical” if it is in transition between two phases; for example, water at its freezing point is a critical system.Many natural systems exhibit characteristic behaviors of criticality, but if critical points are unstable, they should not be common in nature.

Self organizing Critical SystemsBak, Tang and Wiesenfeld propose a phenomena called self-organized criticality (SOC), where “self-organized” means that from any initial condition, the system tends to move toward a critical state, and stay there, without external control.The demonstrated the phenomena using a very basic hourglass model:

Sandpiles

Bak, Tang and Wiesenfeld allowed the system to reach a stable equilibria and then examined the size of clusters (avalanches), compared to their frequency. This exhibited the property of pink noise, where the power (magnitude) is inversely proportional to the frequency:

Python implementationThe Game of Life or Fire implementation can be modified to reflect the dyanmics of the sandpile model:

When z(x,y) exceeds some critical value.

CriticalityFor each perturbation, they measure D, the total number of cells that are affected by the resulting avalanche. Most of the time, D is small, usually 1. But occasionally a large avalanche affects a substantial fraction of the grid. The distribution of D turns out to be long-tailed, which supports the claim that the system is in a critical state.

These phenomena can be observed in many other dynamic and real world systems. What we are observing is the correlation between power and frequency.

Spectral Density

• If h(t) is a signal that varies in time, it can be described by its power spectral density, P(f), which is a function that maps from a frequency, f, to the amount of power the signal contains at that frequency.

Spectral AnalysisSpectral analysis is the process of taking a signal and computing its spectral density. The first step is to compute the Fourier Transform:

where ω = 2 π f is the angular frequency in radians per second (rather than cycles per second). The power spectral density is related to the Fourier transform by the following relation:

P(f) = |H(2 π f)|2

Discrete Fourier TransformSuppose that we have N values hk with k in the range from 0 to N−1. The DFT is written Hn, where n is an index related to frequency:

Each element of this sequence corresponds to a particular frequency. If the elements of hk are equally spaced in time, with time step d, the frequency that corresponds to Hn is

Fast Fourier Transform

The Fast Fourier Transform (FFT) is an efficient algorithm for computing the DFT.• The first step toward the FFT is to apply a substitution W = e2π/iN

The second step is the Danielson-Lanczos Lemma which states

This Lemma suggests a recursive divide-and-conquer approach for calulcuating the Fourier Transform in O(n lg n).

Fast Fourier Transform

• If h has only a single element, then H=h. Otherwise:• Split h into he and ho.• Compute He and Ho by making two recursive calls.• Use the lemma to combine He and Ho to form H.• If H has 2N elements, He and Ho have only N. In

order to merge them, you have to wrap around, but you can do that because Hn+N

e = Hne.

• This recursive algorithm is the Fast Fourier Transform.

Reductionism vs HolismA reductionist model describes a system by describing its parts and their interactions. When a reductionist model is used as an explanation, it depends on an analogy between the components of the model and the components of the system

Holistic models are more focused on similarities between systems and less interested in analogous parts. A holistic approach to modeling often consists of two steps, not necessarily in this order:• Identify a kind of behavior that appears in a variety of systems.• Find the simplest model that demonstrates that behavior.

Causation and PredictionIf a stock market index drops by a fraction of a percent in a day, there is no need for an explanation. But if it drops 10%, people want to know why. Pundits on television are willing to offer explanations, but the real answer may be that there is no explanation.It has been suggested that failures in large engineered systems, like nuclear power plants, are like avalanches in the sand pile model. Most failures are small, isolated and harmless, but occasionally a coincidence of bad fortune yields a catastrophe.