## Thursday, December 22, 2016

### Are power laws good for anything?

It is rather amazing that many complex systems, ranging from proteins to stock markets to cities, exhibit power laws, sometimes over many decades.
A critical review is here, which contains the figure below.

Complexity theory makes much of these power laws.

But, sometimes I wonder what the power laws really tell us, and particularly whether for social and economic issues they are good for anything.
Recently, I learnt of a fascinating case. Admittedly, it does not rely on the exact mathematical details (e.g. the value of the power law exponent!).

The case is described in an article by Dudley Herschbach,
Understanding the outstanding: Zipf's law and positive deviance
and in the book Aid at the Edge of Chaos, by Ben Ramalingam.

Here is the basic idea. Suppose that you have a system of many weakly interacting (random) components. Based on the central limit theorem one would expect that a particular random variable would obey a normal (Gaussian) distribution. This means that large deviations from the mean are extremely unlikely. However, now suppose that the system is "complex" and the components are strongly interacting. Then the probability distribution of the variable may obey a power law. In particular, this means that large deviations from the mean can have a probability that is orders of magnitude larger than they would be if the distribution was "normal".

Now, lets make this concrete. Suppose one goes to a poor country and looks at the weight of young children. One will find that the average weight is significantly smaller than in an affluent country, and most importantly the average less than is healthy for brain and physical development. These low weights arise from a complex range of factors related to poverty: limited money to buy food, lack of diversity of diet, ignorance about healthy diet and nutrition, famines, giving more food to working members of the family, ...
However, if the weights of children obeys a power law, rather than a normal, distribution one might be hopeful that one could find some children who have a healthy weight and investigate what factors contribute to that. This leads to the following.
Positive Deviance (PD) is based on the observation that in every community there are certain individuals or groups (the positive deviants), whose uncommon but successful behaviors or strategies enable them to find better solutions to a problem than their peers. These individuals or groups have access to exactly the same resources and face the same challenges and obstacles as their peers.
The PD approach is a strength-based, problem-solving approach for behavior and social change. The approach enables the community to discover existing solutions to complex problems within the community.
The PD approach thus differs from traditional "needs based" or problem-solving approaches in that it does not focus primarily on identification of needs and the external inputs necessary to meet those needs or solve problems. A unique process invites the community to identify and optimize existing, sustainable solutions from within the community, which speeds up innovation.
The PD approach has been used to address issues as diverse as childhood malnutrition, neo-natal mortality, girl trafficking, school drop-out, female genital cutting (FGC), hospital acquired infections (HAI) and HIV/AIDS.

1. Interesting indeed.
I do question a bit the statement that "These (successful) individuals have access to *exactly* the same resources and face the same challenges and obstacles as their peers."

In my view the availability of (multiple) resources will also have a distribution. Couldn't it be that the successful ones just accidentally have access to a (infinitesimally) more/better timed resources as compared totheir peers that admittedly live in the same society with the same mean level of resources?

Quick example: poor nutrition results in bad health. Possibly not continuously though. So the timing of an infection could accidentally prevent someone of being able to take advantage of a certain resource that happens to pop up at that time - or vice versa, just accidentally "not being sick" at the right time could give people a leg up that leaves them successful.

So I question the word "exact" in my quote.
Obviously this book is about ensembles and not individuals. But (especially when one speaks about humans in need) I think the individuality of the problem matters for the individual path out of the problematic situation.

(But indeed policy-makers may be well-advised to look into the thoughts you communicated.)

2. Off-topic: I remember you being interested (or at least posting about) the (hype in) linear magnetoresistance.
Hence I wanted to point you at this recent paper:
http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.117.256601

3. Krug gives a very well written answer to the question in your title from the field of growth physics: https://arxiv.org/pdf/cond-mat/0403267.pdf

4. I think the most important "goodness" of power laws is the lack of a characteristic scale, what is termed "scale invariance". This is the main distinction with respect to Gaussian or exponential behaviour in general, in which case there IS a characteristic scale for the phenomenon being analysed. Scale invariance is at the core of critical phenomena, which is an amazing fact in many systems which suffer a phase transition, another amazing fact ... all these concepts find fertile grounds in the theory of complex systems, being them physical, economic, social or whatever.

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