# Measuring the Complexity of Bitcoin

I looked at the distribution of transactions of 1 hour of the block chain (269609 to 269618). I parsed the blockchain into two sets of transactions: originating and receiving addresses. I examined both sets independently and together, attempting to fit the three different trials to Gamma distributions. The receiving addresses proved to have the easiest to interpret set of data, and did not follow a Gamma distribution. Instead, the receiving transactions followed a Log-Normal distribution. Figure 1 shows the results of the data and the resulting fit. The data in figure 1 was parsed into blocks that are 2dB wide (every decade of transactions size is 5 bins, the first bin is $10^{-6}$BTC, 1BTC is the 30th bin)

Figure 1, Receiving address distribution of bitcoin transactions blocks 269609 to 269618 with Log-Normal fit

The Log-Normal distribution is very common especially in biological sciences. It is very simple to estimate using the arithmetic mean and variance.

$p\left(X\right)\sim LN\left(\mu,\sigma\right)$

$\mu=\mathrm{ln}\left[\frac{E\left(X\right)^2}{\sqrt{\mathrm{Var}\left(X\right)}}\right]$

$\sigma^2=\mathrm{ln}\left[1+\frac{\mathrm{Var}\left(X\right)}{E\left(X\right)^2}\right]$

Recall that $\frac{\mathrm{Var}\left(X\right)}{E\left(X\right)^2}\propto\frac{1}{N}$ where $N$ is the number of degrees of freedom of the system.

The entropy of the Log-Normal distribution is:

$S=\frac{1}{2}+\frac{1}{2}\mathrm{ln}\left(2\pi\right)+\frac{1}{2}\mathrm{ln}\left(\sigma^2\right)-\frac{\sigma^2}{2}+\mathrm{ln}\langle X\rangle$       (1)

Where $\langle X\rangle=E\left(X\right)$

We find that the entropy of the system is an increasing function of $N\;\forall\sigma^2\geq 1$.

Differentiating (1), we have

$\mathrm{d}S=\frac{1}{\langle X\rangle}\mathrm{d}\langle X\rangle+\frac{1}{2}\left(\frac{1}{\sigma^2}-1\right)\mathrm{d}\sigma^2$          (2)

This finding provides a simple method to calculate the action of bitcoin,

$T=\langle X\rangle$  (3)

and a metric of the number of independent actors within the community,

$\sigma^2=\mathrm{ln}\left[1+\frac{C}{N}\right]$

where $C$ is a constant of proportionality.

I fit the data in figure 1 by estimating the parameters by:

$\mu=\frac{1}{n}\sum_i^n \mathrm{ln}\left[X_i\right]$

$\sigma^2=\frac{1}{n}\sum_i^n \mathrm{ln}\left[X_i\right]^2-\mu^2$

This approach seemed to provide the best results. The outcome is highly sensitive to how the parameters are estimated. I am still working on developing this concept, as it has implications in other fields. For example, I tested the income distribution of the US as being Log-Normal. Figure 2 shows the result, the maroon curve is the data and the blue curve is the fitted model.

Figure 2 Plot of the Average Wage Index of the United States (maroon) and the Log-Normal fit (blue) for 2012

There are a number of items that need to be worked out, and as such these findings are still incomplete. The next step is to examine the entirety of the block chain to see how well this model holds up. Additionally, changes in the money supply (number of bitcoins in existence) needs to be accounted in the model. However, without examining the data, such model building is impossible.  If anyone has the ability to parse the blockchain any assistance would be appreciated.  Thank you.

# Theft at the Grandest Scale

Theft is a strong word. It requires two things, desire and a lack of consent. One party desires something from another and takes it without their consent. Simple. My current project is developing an understanding of macroeconomics based on a formal aggregation of microeconomics. I did not anticipate the understanding that I found. This last weekend I worked on understanding claims against wealth inequality. I began by looking at the distribution of income derived form the Average Wage Index (AWI) from 1990-2012. Plotted on a Log-Log scale the distribution appeared to follow a canonical distribution Figure 1.

Figure 1 distribution of wealth (# people/\$ v. income) in United States (2012)

# The Chilling Effect of Inflation

There has been some discussion in the news about inflation. Japan is actively pursuing inflationary policy with the Yen. The Federal Reserve is not backing off of its Quantitative Easing, taper-off. Meanwhile, India is is watching their currency collapse. And the ECB just announced a surprise lowering of their interest rates. One of the hallmarks of modern economic theory is the stimulatory nature of increasing consumption now. This is financed through two mechanisms, debt and increasing the monetary base. The root of the dual mandate of the Federal Reserve is that inflation as shown by the Phillips Curve causes higher employment. This is dangerous policy. Here is the math explaining why. Continue reading

# My Comments before Georgia’s Public Service Commission 5 Nov 2013

Thank you for the opportunity to speak before the Commission today.

I spent the last five years studying energy policy, leading to two career changes, and my pursuit of a PhD at Tech in Nuclear Engineering. What I thought was a challenging set of circumstances is much more complex–with significant consequences. Energy is a cornerstone of our economy. It’s use represents 80% of our GDP. If we are to grow our economy, increasing our access to energy carries the utmost importance. It should be our goal to reduce energy’s cost. While we do not have control over fuel costs, we do control how we use them and our policies to regulate their use. Continue reading

# Liberty as a Consequence of Human Action

I came to being a libertarian because of the math. I started with the idea that if we start with microeconomics, game theory, and then formally aggregate a group of individuals we can come to macro economics. I did this by mirroring Gibbs approach to deriving statistical mechanics. Translated into econo-jargon, a society is just a bunch of people added together. Thus statistical economics was born. When I explored the consequences of this framework in environmental ethics, I saw I had no choice other than to accept libertarian principles. There is more that needs to be done, and so here we go. Continue reading

# Russell Brand and a Call for Revolution

Russell Brand made an audacious call for revolution in his New Statesman editorial. I share many of his concerns for the current state of affairs and see that if unchecked we will unfortunately have a revolution. However, I do not see his proposed solution as an effective one. It will lead to a greater tragedy within any society that adopts his well intentioned action. His interview with Jeremy Paxman is informative and entertaining to watch.

# Critique of A Theory of Justice (Part 1)

I am in the process of reading John Rawls’ A Theory of Justice. What strikes me is his assumption of the world being fixed. That individuals do not change with time as they grow and experience more. His distribution of wealth is similarly fixed in time and is bequeathed by random chance. The purpose of his conception of justice is to nullify “the accidents of natural endowment.”

In a statistical economic framework, individuals maximize their entropy-freedom, based upon the constraints that they are given.  When we look at the society from a wider veil of ignorance, one where we as the framers of the society are unable to uniquely identify the individual members, we treat those individuals the same. They have different knowledge, characteristics, and allotment of resources. They act with their full knowledge, the sort that Hayek calls practical knowledge. What is uniform is our ignorance of their specific and uniquely identifiable features.  This is very different from Rawls. He assumed that we ignored our ability. We are here allowing it, not denying it.  This has some fundamental impacts on the outcome. Continue reading