WS and the Small World Effect.ipynb

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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"width: 100%; overflow: hidden;\">\n",
    "    <div style=\"width: 150px; float: left;\"> <img src=\"https://raw.githubusercontent.com/DataForScience/Networks/master/data/D4Sci_logo_ball.png\" alt=\"Data For Science, Inc\" align=\"left\" border=\"0\" width=160px> </div>\n",
    "    <div style=\"float: left; margin-left: 10px;\"> <h1>Watts-Strogatz and the Small-World Effect</h1>\n",
    "        <p>Bruno Gonçalves<br/>\n",
    "        <a href=\"http://www.data4sci.com/\">www.data4sci.com</a><br/>\n",
    "            @bgoncalves, @data4sci</p></div>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from collections import Counter\n",
    "from pprint import pprint\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt \n",
    "import networkx as nx\n",
    "\n",
    "from tqdm import tqdm\n",
    "\n",
    "import watermark\n",
    "\n",
    "%load_ext watermark\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Watermark the notebook with current versions of all loaded libraries"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python implementation: CPython\n",
      "Python version       : 3.8.5\n",
      "IPython version      : 7.19.0\n",
      "\n",
      "Compiler    : Clang 10.0.0 \n",
      "OS          : Darwin\n",
      "Release     : 20.5.0\n",
      "Machine     : x86_64\n",
      "Processor   : i386\n",
      "CPU cores   : 16\n",
      "Architecture: 64bit\n",
      "\n",
      "Git hash: 09938236a5375b6d0e9cb4a1618d9e4ed73cf99d\n",
      "\n",
      "watermark : 2.1.0\n",
      "json      : 2.0.9\n",
      "networkx  : 2.5\n",
      "pandas    : 1.1.3\n",
      "matplotlib: 3.3.2\n",
      "numpy     : 1.19.2\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%watermark -n -v -m -g -iv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load default figure style"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.style.use('./d4sci.mplstyle')\n",
    "colors = plt.rcParams['axes.prop_cycle'].by_key()['color']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Graph Models"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Regular Ring"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We now proceed to defining a regular ring graph where each node is connected with K/2 neighbors on the left and K/2 neighbours on the right. This will be a stepping stone towards the full fledged Watts-Strogats model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ring_graph(N, K):\n",
    "    nodes = np.arange(N)\n",
    "    edges = []\n",
    "    \n",
    "    K = K//2\n",
    "    \n",
    "    for i in range(N):\n",
    "        for j in range(i+1, i+K+1):\n",
    "            edges.append((i, j%N))\n",
    "        \n",
    "    G = nx.Graph()\n",
    "    G.add_nodes_from(nodes)\n",
    "    G.add_edges_from(edges)\n",
    "    \n",
    "    return G"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To generate a small regular ring we simply do"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 100\n",
    "K = 4\n",
    "RG = ring_graph(N, K)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Naturally the degree distribution is pretty simple"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "Pk = pd.DataFrame(Counter(dict(RG.degree()).values()).items(), columns=['k', 'Pk']).sort_values('k')\n",
    "Pk.Pk/=N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With all nodes having degree 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>k</th>\n",
       "      <th>Pk</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>4</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   k   Pk\n",
       "0  4  1.0"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Pk"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Watts-Strogatz model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first step towards defining a WS model is to implementing the rewiring procedure. This takes an existing graph and randomly rewires it's edges with probability $p$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def rewire(G, p):\n",
    "    node_labels = list(G.nodes)\n",
    "    nodes = dict(zip(node_labels, range(len(node_labels))))\n",
    "    \n",
    "    edges = G.edges()\n",
    "        \n",
    "    for node_i, node_j in edges:\n",
    "        # Make sure we just rewire one end of the edge\n",
    "        if nodes[node_j] > nodes[node_i]: \n",
    "            prob = np.random.random()\n",
    "\n",
    "            if prob < p:\n",
    "                new_node_j = node_labels[np.random.randint(len(node_labels))]\n",
    "                G.add_edge(node_i, new_node_j)\n",
    "                G.remove_edge(node_i, node_j)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So if we apply it to the previous graph, we significantly impact the degree distribution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "RG = ring_graph(10000, 4)\n",
    "rewire(RG, 0.3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "Pk = pd.DataFrame(Counter(dict(RG.degree()).values()).items(), columns=['k', 'Pk']).sort_values('k')\n",
    "Pk.Pk/=10000"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Which is now significantly broader while still peaked at 4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'P(k)')"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
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      "text/plain": [
       "<Figure size 3840x2640 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = Pk.plot(x='k', y='Pk', legend=None)\n",
    "ax.set_xlabel('k')\n",
    "ax.set_ylabel('P(k)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Watts-Strogatz model is then, simply"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "def watts_strogatz_graph(N, K, p):\n",
    "    G = ring_graph(N, K)\n",
    "    rewire(G, p)\n",
    "    \n",
    "    return G"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Small World Effect"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We measure the clustering and average shortest path for various values of p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|██████████| 15/15 [00:15<00:00,  1.05s/it]\n"
     ]
    }
   ],
   "source": [
    "p = 1\n",
    "values = []\n",
    "\n",
    "for i in tqdm(range(1, 16), total=15):\n",
    "    G = watts_strogatz_graph(1_000, 4, p)\n",
    "    c = nx.algorithms.cluster.average_clustering(G)\n",
    "    l = nx.average_shortest_path_length(G)\n",
    "    values.append([c, l, p])\n",
    "    p /= 2."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And plot them"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "WS_stats = pd.DataFrame(values, columns=['clustering', 'diameter', 'p'])\n",
    "WS_stats = WS_stats.div(WS_stats.max())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 3840x2640 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = WS_stats.plot.scatter(x='p', y='clustering', marker='*', \n",
    "                           label=r'$\\frac{C[p]}{C[0]}$', \n",
    "                           color=colors[0], s=300)\n",
    "WS_stats.plot.scatter(x='p', y='diameter', marker='.', \n",
    "                           label=r'$\\frac{L[p]}{L[0]}$', \n",
    "                           color=colors[1], s=300, ax=ax)\n",
    "ax.axvspan(xmin=0.001, xmax=0.05, color=colors[3], alpha=0.3, zorder=-3)\n",
    "ax.set_xscale('log')\n",
    "ax.set_ylabel(None)\n",
    "\n",
    "for tick_label in ax.xaxis.get_majorticklabels():\n",
    "    tick_label.set_y(-.01)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"width: 100%; overflow: hidden;\">\n",
    "     <img src=\"data/D4Sci_logo_full.png\" alt=\"Data For Science, Inc\" align=\"center\" border=\"0\" width=300px> \n",
    "</div>"
   ]
  }
 ],
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