Saturday, May 22, 2010

Back of the envelop Mathematical Model Number 1: Power Laws

After a comment from Stuart about the scale invariance of power laws (See here) I pondered the subject a bit and decided I would do a post on it.

The ubiquity of the power law probably ranks it with the Gaussian bell curve. The latter arises whenever there is a random walk, a very general and common phenomenon. Another general curve is the Boltzmann distribution, an example of which can be seen in the way atmospheric density changes with altitude. But power law distributions differ markedly from the Gaussian and Boltzmann distributions in one important respect. Unlike the latter two, power law distributions don’t return well behaved means and variances. (See here)

The Boltzmann and Gaussian distributions contain negative exponentials and these create asymptotic cut offs which ensure that the integrals used to calculate averages and variances are finite. This cut off behaviour is essential given that both probability and energy are limited by conservation laws and finite resources. How then can we make sense of a power law distribution, like say the size of meteors, which if taken too literally would suggest that there are bodies out there of infinite mass?

Some back of the envelop theorising may help to explain this.

Some power laws seem to have their logical roots in the conventional concept of space constructed by taking the Cartesian product of the coordinates of this space. If an object in a Cartesian space has a size defined by some linear parameter x then that object will have a surface area or volume that will be some power of x. That is, the surface area is proportional to x to the power of p where p is a real number.

Taking my cue from things as diverse as interplanetary bodies and internet nodes, I envisage such objects being capable of attracting further material thereby growing in size. If this is the case then I suggest that the object’s surface area (or volume) is the parameter that determines its growth rate because it is via this surface area that the object interfaces with the “outside world”. If the object grows by the assimilation of material through the membrane of this surface then we might expect the growth of this object to be proportional to this surface area. That is, the rate of growth of the object, G, is given by:
... where k is a constant.

So, an object of size x is effectively ‘moving’ along the x axis with a ‘velocity’ equal to G. If the density of the distribution of objects on the x axis at point x is D(x), then the flow F(x) at point x will be given by:
Now, let us assume that the objects are coming into existence with constant rate at the lower end of the x axis. This means that when equilibrium is eventually reached the flow along the x axis will be a constant independent of x Therefore:Hence:

Given the assumptions I have built into this calculation we see a natural power law distribution in x that ultimately traces back to Cartesian dimensionality.

The above simple model really provides a starting point from which more sophisticated models can be contemplated and built. I actually feel rather unsure about the assumption that objects are envisaged to reside in a conventional Cartesian space that allows their surface area/volume to be calculated using a simple power law. In a manifold where nodes are connected randomly, rather than connected in sequence as in Cartesian spaces, the volumes/surfaces areas are an exponential function of the number of steps between the furthest nodes comprising the object. Notice also the assumption that objects can grow indefinitely – that is, it is assumed that there are no limits on the material available driving the growth of the objects. This, of course, may not be valid – or it may be valid only for a limited time. If the latter is the case then only during the period when material is available will the power law hold as a reasonable approximation.

Returning to my original query about how power laws, which don’t return convergent means and variances, can exist in a cosmos of limited resources, then it seems the answer is this: Power laws only work in open systems, that is in systems where there is an input from without. As long as that input lasts the system will move toward an equilibrium that displays a power law distribution. However, this power law distribution will only be approximate; in a cosmos of limited resources the input can never be maintained long enough for an absolute equilibrium to be arrived at. Thus we expect power laws, like geological lakes, to be a temporary phenomenon, eventually causing a maxing out of resources.

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