BTC_POWER_LA

Stability of the power law exponent over time using several methods, regression and scale invariance.

I emphasized so much that the power law is not a curve fitting exercise as some pseudo-analysts claim. It is about scaling laws and consistency of behavior of Bitcoin.
What scaling means?
In physics, scaling means that a system looks the same — statistically or structurally — when you zoom in or out by any factor. More precisely, a relationship is scale-invariant if multiplying the independent variable by any constant λ changes the dependent variable by a predictable power of λ, with no preferred scale breaking the symmetry.
Formally: a function f(x) obeys scaling if
f(λx) = λ^β · f(x) for al

Fantastic commentary by Italian comedian asking what is the real difference between the Iranian theocracy and the American one (to us this is a super weird scene). "The world is in a competition about how has the longest god". Use your imagination what the innuendo is here.

警告：下面的图表看起来像一个金字塔，但我们刚刚发布了一篇文章解释比特币不是传销。它只是展示网络如何增长，这类似于受限的指数增长。
幂律采用比指数 (S曲线) 采用更可持续。
有什么区别？
指数增长是这样工作的。想象一个细菌每五分钟翻倍。你从一个细菌开始。五分钟后有两个。十分钟后有四个、八个、十六个，依此类推。指数增长的关键特征是增长率在时间上保持不变。
现在想象这些细菌生活在一个资源有限的密闭容器中。假设细菌填满容器的一半需要三天。填满剩下的一半需要多长时间？
仅仅五分钟。
这就是指数增长的悖论。一切在很长一段时间内看起来都是可控的，然后系统突然耗尽资源。一旦容器满了，细菌就没有食物，菌群崩溃。该系统没有以可持续的方式分配资源。
许多遵循指数增长的过程表现如下：它们增长极快，然后崩溃。
幸运的是，比特币不遵循这种模式。其采用方式更接近幂律。
幂律仍然允许增长取决于已经在系统中的人数，但它包含一个与 1/t 成正比的自然制约因素，其中 t 是系统的年龄。随着系统变老，增长逐渐放缓。
简化形式下，比特币采用者数量的变化可以写成
dN/dt = 3N / t
其中
N 是使用比特币的人数
t 是比特币的年龄 (例如以天为单位)
这个方程意味着采用仍然受益于网络效应——更多用户吸引更多用户——但随着系统成熟，增长率逐渐降低。
让我们用一个简单的例子来说明这一点。
从中本聪作为第一个用户开始

This version of the power law is free online. Link in the comments.

You can build a Hidden Markov Model (HMM) using the slopes of Bitcoin’s growth to identify and predict three regimes: bullish, transition, and bearish. In preliminary tests, this approach can classify the regime with around 90% accuracy.
Below is a link to a Veritasium video that explains how Hidden Markov Models work and why they are powerful tools for prediction problems.
HMMs are widely used in quantitative finance; notably, variations of this technique were reportedly among the methods employed by Jim Simons’ Medallion Fund, one of the most successful trading strategies in history.

Come and visit us at the Physics of Bitcoin show.
Starts in 5 minutes on X, YouTube and Twitch.
Many things to cover tonight including 3D power law and scaling test.