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https://github.com/pezkuwichain/consensus.git
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Randao analysis Jupyter
This commit is contained in:
AlistairStewart
@@ -0,0 +1,350 @@
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"def cdfmaxgeo(p,t,m):\n",
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" return (1-(1-p)^(m+1))^t\n",
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"def pdfmaxgeo(p,t,m):\n",
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" return cdfmaxgeo(p,t,m)-cdfmaxgeo(p,t,m-1)\n",
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" "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [],
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"source": [
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"def nextstateprob(x,y):\n",
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" if x ==63:\n",
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" return 1-cdfmaxgeo(2/3+0.0,2^x,62)\n",
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" return pdfmaxgeo(2/3+0.0,2^x,y)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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"tm=matrix([[nextstateprob(x,y) for x in range(64)] for y in range(64)])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 80,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"0.999999999952366\n"
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]
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},
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{
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"data": {
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"text/plain": [
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"(0.476563095838547, 0.290008277069172, 0.139917968194352, 0.0586770548078090, 0.0224471788438172, 0.00810421240432098, 0.00282576354047732, 0.000965620671507453, 0.000326252792938701, 0.000109544499335094, 0.0000366568819313766, 0.0000122441918654699, 4.08585723181877e-6, 1.36273835423470e-6, 4.54384351862505e-7, 1.51485734113910e-7, 5.04995072826173e-8, 1.68339168055658e-8, 5.61143674448838e-9, 1.87050189995125e-9, 6.23504803604958e-10, 2.07835570232040e-10, 6.92786571866131e-11, 2.30930253742276e-11, 7.69755676003532e-12, 2.56585251964193e-12, 8.55525570183441e-13, 2.84933826330784e-13, 9.52193095509939e-14, 3.14984028202337e-14, 1.04994676112577e-14, 3.62050607336612e-15, 1.08615182206194e-15, 3.62050607359248e-16, 3.62050607361851e-16, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000)"
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]
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},
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"execution_count": 80,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"pm=tm\n",
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"for _ in range(20):\n",
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" pm = pm^2\n",
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"stat=pm*vector({1:1,63:0})\n",
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"print(sum(stat))\n",
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"stat"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 81,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"1.51485734113910e-7"
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]
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},
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"execution_count": 81,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"stat[15]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 47,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(0.666666666666667, 0.222222222222222, 0.0740740740740741, 0.0246913580246914, 0.00823045267489719, 0.00274348422496573, 0.000914494741655170, 0.000304831580551723, 0.000101610526850648, 0.0000338701756168458, 0.0000112900585389486, 3.76335284635321e-6, 1.25445094878440e-6, 4.18150316261467e-7, 1.39383438679808e-7, 4.64611462636100e-8, 1.54870487545367e-8, 5.16234954783812e-9, 1.72078318261271e-9, 5.73594394204235e-10, 1.91198168408846e-10, 6.37326857955145e-11, 2.12442285985048e-11, 7.08144654026910e-12, 2.36044517265555e-12, 7.86815057551848e-13, 2.62345700718924e-13, 8.73745520379998e-14, 2.91988655476416e-14, 9.65894031423886e-15, 3.21964677141295e-15, 1.11022302462516e-15, 3.33066907387547e-16, 1.11022302462516e-16, 1.11022302462516e-16, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000)"
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]
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},
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"execution_count": 47,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"unbiased=tm*vector({0:1,63:0})\n",
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"unbiased"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 48,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"4.64611462636100e-8"
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]
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},
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"execution_count": 48,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"pdfmaxgeo(2/3+0.0,1,15)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 49,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"4.64611462636100e-8"
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]
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},
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"execution_count": 49,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"unbiased[15]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 52,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"a single transition from ubiased gives expected tail length 0.500000000000001 and expected options 1.99999968601354\n",
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"the steady state distribution has expected tail length 0.904072888428179 and expected options 3.26106196081950\n"
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]
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}
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],
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"source": [
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"def expectedoptions(dist):\n",
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" return sum([2^i*x for (i,x) in zip(range(64),dist)])\n",
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"def expectedtail(dist):\n",
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" return sum([i*x for (i,x) in zip(range(64),dist)])\n",
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"print(\"a single transition from ubiased gives expected tail length \",expectedtail(unbiased),\" and expected options \",expectedoptions(unbiased))\n",
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"print(\"the steady state distribution has expected tail length \", expectedtail(stat), \" and expected options \",expectedoptions(stat))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 22,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"1.21765601217656e-8"
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]
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},
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"execution_count": 22,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"epochsinacentury=(5*60*24*365*1000/32)\n",
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"1/epochsinacentury+0.0"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 63,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"tail length 15 occurs every 8.03809031457012 years in expectation under the stationary distribution\n",
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"tail length 15 occurs every 26.2080493078638 years in expectation under the single transition from unbiased distribution\n",
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"tail length 16 occurs every 24.1122354988936 years in expectation under the stationary distribution\n",
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"tail length 16 occurs every 78.6241479235913 years in expectation under the single transition from unbiased distribution\n",
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"tail length 17 occurs every 72.3334935202939 years in expectation under the stationary distribution\n",
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"tail length 17 occurs every 235.872445461677 years in expectation under the single transition from unbiased distribution\n"
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]
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}
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],
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"source": [
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"print(\"tail length 15 occurs every\",100/(stat[15]*epochsinacentury),\"years in expectation under the stationary distribution\")\n",
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"print(\"tail length 15 occurs every\",100/(unbiased[15]*epochsinacentury),\"years in expectation under the single transition from unbiased distribution\")\n",
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"print(\"tail length 16 occurs every\",100/(stat[16]*epochsinacentury),\"years in expectation under the stationary distribution\")\n",
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"print(\"tail length 16 occurs every\",100/(unbiased[16]*epochsinacentury),\"years in expectation under the single transition from unbiased distribution\")\n",
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"print(\"tail length 17 occurs every\",100/(stat[17]*epochsinacentury),\"years in expectation under the stationary distribution\")\n",
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"print(\"tail length 17 occurs every\",100/(unbiased[17]*epochsinacentury),\"years in expectation under the single transition from unbiased distribution\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 85,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"172.837461872421"
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]
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},
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"execution_count": 85,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"expectedoptions(tm*tm*vector({16:1,63:0}))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 86,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"6.41125360736917"
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]
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},
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"execution_count": 86,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"expectedtail(tm*tm*vector({16:1,63:0}))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 65,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"1901.01954391692"
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]
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},
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"execution_count": 65,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"expectedoptions(tm*vector({16:1,63:0}))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 87,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"36.3762992608834"
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]
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},
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"execution_count": 87,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"expectedoptions(tm*tm*tm*vector({16:1,63:0}))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 46,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"True"
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]
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},
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"execution_count": 46,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"tm.transpose()[0]==tm*vector({0:1,63:0})"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 9.2",
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"language": "sage",
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"name": "sagemath"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.5"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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