cdetrio/blockchains-and-channel-capacities.ipynb

## blockchains-and-channel-capacities.ipynb
{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from scipy.stats import entropy\n",
    "import numpy as np\n",
    "import math\n",
    "\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Channel capacities and 51% attacks\n",
    "\n",
    "If we view a blockchain as a communication channel, an attacker's hash rate corresponds to adversarial noise. The level of adversarial noise places an upper bound on the [channel capacity](https://en.wikipedia.org/wiki/Channel_capacity).\n",
    "\n",
    "Our starting point is [LaViola and Miller's model](https://socrates1024.s3.amazonaws.com/consensus.pdf) of bitcoin's PoW blockchain as a statistical test between two exponential distributions, i.e. the KL-divergence between two rate parameters (the arrival rate of blocks from correct processes vs. the arrival rate of blocks from byzantine processes):\n",
    "\n",
    ">  We can thus relate the Byzantine consensus problem to a statistical test. At each instant, every process prefers the value that is most likely, given the available information, to be preferred by the largest number of processes. For the particular puzzle we defined, the only discriminating information is the number of distinct puzzle solutions found. Consider the number of bits of certainty obtained per message, (i.e., k/x). From Equation 1, this is equal to the KL-divergence between two exponential distributions, where δ is the ratio between the respective scale parameters. ([LaViola and Miller, 2012](https://socrates1024.s3.amazonaws.com/consensus.pdf))\n",
    "\n",
    "\n",
    "We'll look at the relation between this KL-divergence, the capacity of a binary symmetric channel, and the capacity of a Poisson channel."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# KL-divergence between two exponential rate parameters (from Miller 2012 paper)\n",
    "# > Let B be a subset of n − f correct processes. ...\n",
    "# > the subset B finds puzzle solutions according to a Poisson arrival process\n",
    "# > with expected interarrival time µ_B = (δ/2)µ, where δ = 2(n−f)/n > 1 (in particular, faster than half the overall rate).\n",
    "\n",
    "\n",
    "# KL-divergence formula: https://en.wikipedia.org/wiki/Exponential_distribution#Kullback.E2.80.93Leibler_divergence\n",
    "# p and q are rate (or scale) parameters\n",
    "# rate parameter for overall rate = 2\n",
    "# rate parameter for half of overall rate = 1\n",
    "\n",
    "def ExponentialKLd(p, q):\n",
    "    nats = (p/q - np.log(p/q) - 1) # KL-divergence in nats\n",
    "    return nats*np.log2(math.e) # convert from nats to bits"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's calculate the bits of information per block when 70% of the processes are correct."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.091651189185343443"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# scale parameter for 70% = 0.7*2 = 1.4\n",
    "# compare 70% of hashrate (1.4) to half of overall hashrate (1)\n",
    "\n",
    "ExponentialKLd(1.4, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "So when 70% of processes are correct, each block carries 0.09165.. bits of information.\n",
    "\n",
    "Since 70% of processes are correct, the noise rate is 30%. As a sanity check, let's compare to the capacity of a [binary symmetric channel](https://en.wikipedia.org/wiki/Binary_symmetric_channel)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def H(x): # entropy of a biased coin flip\n",
    "    return -(1 - x)*np.log2(1 - x) - x*np.log2(x)\n",
    "\n",
    "def BSCcapacity(x): # capacity of a binary symmetric channel\n",
    "    # x = bit error rate\n",
    "    return 1 - H(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.1187091007693073"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BSCcapacity(0.30) # capacity when noise rate is 30%"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "The bits per block (0.09165..) is below the binary symmetric channel capacity (0.1187..), so it passes our sanity check.\n",
    "\n",
    "Let's check when 55% of processes are correct."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0067659803389613724"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 0.55*2 = 1.1\n",
    "ExponentialKLd(1.1, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0072255460121917192"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 55% correct, or 45% noise\n",
    "BSCcapacity(0.45)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Good, with 55% of processes correct, the bits per block is still below the channel capacity. Let's try with 51% correct."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00028474862100832558"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 0.51*2 = 1.02\n",
    "ExponentialKLd(1.02, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00028855824719009604"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BSCcapacity(0.49)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "0.000284.. < 0.000288.. so the sanity check is still good.\n",
    "\n",
    "Observe that as the proportion of correct processes decreases to 0.5, the bits of information per block approaches 0. That's what we should expect, since the theoretical capacity of a binary symmetric channel with a 50% noise rate is 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BSCcapacity(0.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's go in the direction of less noise, and try with 90% correct processes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.30615912615622071"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 0.9*2 = 1.8\n",
    "ExponentialKLd(1.8, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.53100440641071878"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BSCcapacity(0.1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "The bits per block (0.306..) is way below the BSC capacity (0.531..), and the gap only grows as the correct proportion increases to 100% (noise rate decreases to 0%).\n",
    "\n",
    "Let's check the gap when 99.9% of processes are correct."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.44125306767685413"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 0.999 * 2 = 1.998\n",
    "ExponentialKLd(1.998, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.98859224226253883"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BSCcapacity(0.001)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So with higher noise rates (e.g. 30% to 50%), the information per block approaches the capacity of a binary symmetric channel. But with lower noise rates (e.g. 0% to 20%), the information per block is well below the BSC capacity.\n",
    "\n",
    "Let's look for a channel model that works better than BSC for lower noise rates.\n",
    "\n",
    "Recall that the information per block is the KL-divergence between two exponential distributions. Since blocks arrive according to a Poisson process (i.e. the times between arrivals form an exponential distribution), this suggests trying a Poisson channel as the channel model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Poisson channel capacity formula taken from (1.5) in http://wiki.epfl.ch/edicpublic/documents/Candidacy%20exam/wyner.pdf\n",
    "# > for codes which achieve capacity, the average number of received photons per second is q_star*A*T. \n",
    "\n",
    "# A = peak power constraint of input\n",
    "# ortho*A = average power constraint. when average-power == peak-power, ortho=1\n",
    "# lam0 = \"dark current\" noise. nonnegative constant\n",
    "# total rate of poisson channel = A + lam0\n",
    "# s = lam0/A\n",
    "# q_star = min(ortho, q0(s))\n",
    "# q0(s) = [ ((1+s)^(1+s)) / ((s^s)*e) ] - s\n",
    "\n",
    "\n",
    "def poissonCapacity(A):\n",
    "    lam0 = 1 - A # A is rate of \"correct\" input, lam0 is the noise rate\n",
    "    s = lam0/A\n",
    "    q0 = ( ((1+s)**(1+s)) / ((s**s)*math.e) ) - s\n",
    "    q_star = min(1, q0)\n",
    "    C = A * ( (q_star*(1+s)*math.log(1+s)) + ((1-q_star)*s*math.log(s)) - ((q_star+s)*math.log(q_star+s)) )\n",
    "    return C*np.log2(math.e) # convert from nats to bits\n",
    "\n",
    "\n",
    "# formula for high noise.. s -> infinity\n",
    "# (1.7a) in http://wiki.epfl.ch/edicpublic/documents/Candidacy%20exam/wyner.pdf\n",
    "# couldn't get sensible results from this formula, but might try again in the future\n",
    "#def noisyPoissonC(A, lam0):\n",
    "#    s = lam0/A\n",
    "#    q_star = min(1, 1/2)\n",
    "#    big_o = (1 / s**2)\n",
    "#    C = ( A*q_star*(1-q_star) ) / ( 2*s )  +  big_o\n",
    "#    return C*np.log2(math.e)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's check the gap between the information per block and the poisson channel capacity when 90% of processes are correct."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.31637116481270106"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# A is the correct rate\n",
    "# lam0 is the noise rate\n",
    "\n",
    "# check the capacity when 90% of processes are correct\n",
    "poissonCapacity(0.90)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.30615912615622071"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 0.9*2 = 1.8\n",
    "ExponentialKLd(1.8, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using a Poisson channel rather than a binary symmetric channel, the gap is much closer!\n",
    "\n",
    "Let's go ahead and plot all three: the information per block, the capacity of a binary symmetric channel, and the capacity of a Poisson channel."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def bitsPerBlock(correct_rates):\n",
    "    data_points = [[x, ExponentialKLd(2*x, 1)] for x in correct_rates]\n",
    "    return np.array(data_points)\n",
    "\n",
    "\n",
    "def BSCdataPoints(correct_rates):\n",
    "    data_points = [[x, BSCcapacity(1-x)] for x in correct_rates]\n",
    "    return np.array(data_points)\n",
    "\n",
    "\n",
    "def poissonDataPoints(correct_rates):\n",
    "    data_points = [[x, poissonCapacity(x)] for x in correct_rates]\n",
    "    return np.array(data_points)\n",
    "\n",
    "\n",
    "\n",
    "noise_rates = np.linspace(0.01, 0.49, num=49) # from 0.01 to 0.49 because 0.50 causes warning\n",
    "correct_rates = (1 - noise_rates)[::-1]\n",
    "\n",
    "poisson_data = poissonDataPoints(correct_rates)\n",
    "bsc_data = BSCdataPoints(correct_rates)\n",
    "pow_data = bitsPerBlock(correct_rates)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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wyy3+XoeISA4EY7nbS0Ay8Bk+3G8ESgE7gIvMbGDwys2eghjsv/xyanjHx/vJ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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1047f7e10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(figsize=(8,6))\n",
    "\n",
    "plt.plot(poisson_data[:,0], poisson_data[:,1], label='Poisson channel capacity')\n",
    "plt.plot(bsc_data[:,0], bsc_data[:,1], label='binary symmetric channel capacity')\n",
    "plt.plot(pow_data[:,0], pow_data[:,1], label='bits per PoW block')\n",
    "\n",
    "xticks = [poisson_data[:,0][i] for i in range(len(poisson_data[:,0])) if (i%4==0)]\n",
    "xticks.append(0.99)\n",
    "xticklabels = (1 - np.array(xticks))\n",
    "\n",
    "ax.set_xticks(xticks)\n",
    "ax.set_xticklabels(xticklabels)\n",
    "plt.xlabel('error rate')\n",
    "plt.ylabel('channel capacity (bits per message)')\n",
    "ax.set_yscale('log')\n",
    "legend = ax.legend(loc='best', shadow=True)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}