t-flora/Network metrics and percolation.ipynb

## Network metrics and percolation.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Network metrics and percolation\n",
    "## Percolation\n",
    "\n",
    "\n",
    "Following the mathematical analysis in the textbook, $n$ is defined as the number of nodes in a network, and $q$ is defined as the probability that a random node from the network is *not* part of the largest connected component (LCC) of the network. The derivation in the textbook shows that $q$ is a solution to the equation\n",
    "\n",
    "$$q = e^{\\langle k \\rangle (q−1)}$$\n",
    "\n",
    "\n",
    "where $\\langle k \\rangle$ is the average degree of the network.\n",
    "\n",
    "\n",
    "**Question**: Given the information above, what is the theoretical estimate for the number of nodes in the LCC, expressed in terms of the known variables, $n$, $q$, and $\\langle k \\rangle$?\n",
    "\n",
    "\n",
    "**Task**: Plot how the size of the LCC depends on the average degree $\\langle k \\rangle$ by using the theoretical result in (1). This equation does not have a nice analytical solution, so we use a numerical root finder in `SciPy` to determine the value of $q$ that solves the equation for a given $\\langle k \\rangle$. A root finder computes a numerical solution to an equation of the form $f(x) = 0$, so we need to rewrite (1) as\n",
    "\n",
    "$$q-e^{\\langle k \\rangle (q-1)} = 0$$\n",
    "\n",
    "We give the expression on the left-hand side to the root finding function. Use the code below to compute $q$ for different values of $\\langle k \\rangle$ in the range [1, 10]. Note that $\\langle k \\rangle$ will not necessarily be an integer since it is an average degree.\n",
    "\n",
    "\n",
    "Use the value of $q$ to determine the theoretical estimate for the size of the LCC in a network with average degree $\\langle k \\rangle$. Plot your results of how the size of the LCC depends on $\\langle k \\rangle$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Answer to question**:\n",
    "\n",
    "The number of nodes in the LCC should be given by:\n",
    "$$n_{LCC} = n(1-q) = n(1-e^{\\langle k \\rangle(q-1)})$$\n",
    "\n",
    "Considering a frequentist approach to calculating the probability $q$. If $1-q$ is the probability of a node being in the LCC, then the number of nodes in the LCC should be the equivalent fraction of the total nodes in the network."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def calculate_q(k):\n",
    "    '''\n",
    "    Use a numerical root finder to determine q from the equation\n",
    "    q = exp(k*(q-1)).\n",
    "    '''\n",
    "    from scipy.optimize import root\n",
    "    return root(lambda q: q - np.exp(k * (q - 1)), 0).x[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_LCC = lambda n, k, q: n*(1-np.exp(k*(q-1)))\n",
    "ks = np.linspace(1, 10, 500)\n",
    "\n",
    "qs = [calculate_q(k) for k in ks]\n",
    "\n",
    "ns = [n_LCC(1000, k, q) for k, q in zip(ks, qs)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(ks, ns, color = 'black')\n",
    "plt.xlabel('<k>')\n",
    "plt.ylabel('n_LCC')\n",
    "plt.title('LCC size by average degree of network')\n",
    "plt.show()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}
	{
	"cells": [
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"# Network metrics and percolation\n",
	"## Percolation\n",
	"\n",
	"\n",
	"Following the mathematical analysis in the textbook, $n$ is defined as the number of nodes in a network, and $q$ is defined as the probability that a random node from the network is not part of the largest connected component (LCC) of the network. The derivation in the textbook shows that $q$ is a solution to the equation\n",
	"\n",
	"$$q = e^{\\langle k \\rangle (q−1)}$$\n",
	"\n",
	"\n",
	"where $\\langle k \\rangle$ is the average degree of the network.\n",
	"\n",
	"\n",
	"Question: Given the information above, what is the theoretical estimate for the number of nodes in the LCC, expressed in terms of the known variables, $n$, $q$, and $\\langle k \\rangle$?\n",
	"\n",
	"\n",
	"Task: Plot how the size of the LCC depends on the average degree $\\langle k \\rangle$ by using the theoretical result in (1). This equation does not have a nice analytical solution, so we use a numerical root finder in `SciPy` to determine the value of $q$ that solves the equation for a given $\\langle k \\rangle$. A root finder computes a numerical solution to an equation of the form $f(x) = 0$, so we need to rewrite (1) as\n",
	"\n",
	"$$q-e^{\\langle k \\rangle (q-1)} = 0$$\n",
	"\n",
	"We give the expression on the left-hand side to the root finding function. Use the code below to compute $q$ for different values of $\\langle k \\rangle$ in the range [1, 10]. Note that $\\langle k \\rangle$ will not necessarily be an integer since it is an average degree.\n",
	"\n",
	"\n",
	"Use the value of $q$ to determine the theoretical estimate for the size of the LCC in a network with average degree $\\langle k \\rangle$. Plot your results of how the size of the LCC depends on $\\langle k \\rangle$."
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"Answer to question:\n",
	"\n",
	"The number of nodes in the LCC should be given by:\n",
	"$$n_{LCC} = n(1-q) = n(1-e^{\\langle k \\rangle(q-1)})$$\n",
	"\n",
	"Considering a frequentist approach to calculating the probability $q$. If $1-q$ is the probability of a node being in the LCC, then the number of nodes in the LCC should be the equivalent fraction of the total nodes in the network."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 1,
	"metadata": {},
	"outputs": [],
	"source": [
	"import numpy as np\n",
	"import matplotlib.pyplot as plt\n",
	"\n",
	"def calculate_q(k):\n",
	" '''\n",
	" Use a numerical root finder to determine q from the equation\n",
	" q = exp(k*(q-1)).\n",
	" '''\n",
	" from scipy.optimize import root\n",
	" return root(lambda q: q - np.exp(k * (q - 1)), 0).x[0]"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 2,
	"metadata": {},
	"outputs": [],
	"source": [
	"n_LCC = lambda n, k, q: n(1-np.exp(k(q-1)))\n",
	"ks = np.linspace(1, 10, 500)\n",
	"\n",
	"qs = [calculate_q(k) for k in ks]\n",
	"\n",
	"ns = [n_LCC(1000, k, q) for k, q in zip(ks, qs)]"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 3,
	"metadata": {},
	"outputs": [
	{
	"data": {
	"image/png": 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\n",
	"text/plain": [
	"<Figure size 432x288 with 1 Axes>"
	]
	},
	"metadata": {
	"needs_background": "light"
	},
	"output_type": "display_data"
	}
	],
	"source": [
	"plt.plot(ks, ns, color = 'black')\n",
	"plt.xlabel('<k>')\n",
	"plt.ylabel('n_LCC')\n",
	"plt.title('LCC size by average degree of network')\n",
	"plt.show()"
	]
	}
	],
	"metadata": {
	"kernelspec": {
	"display_name": "Python 3",
	"language": "python",
	"name": "python3"
	},
	"language_info": {
	"codemirror_mode": {
	"name": "ipython",
	"version": 3
	},
	"file_extension": ".py",
	"mimetype": "text/x-python",
	"name": "python",
	"nbconvert_exporter": "python",
	"pygments_lexer": "ipython3",
	"version": "3.7.8"
	}
	},
	"nbformat": 4,
	"nbformat_minor": 4
	}