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labels" + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "e8335881-543c-4409-ad9e-251f5530dfca", + "metadata": {}, + "outputs": [], + "source": [ + "import networkx as nx\n", + "import matplotlib.pyplot as plt" + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "23a9e6f6-af16-4421-a46a-9f8f2dfb78f5", + "metadata": {}, + "outputs": [], + "source": [ + "G = nx.DiGraph()\n", + "G.add_node(0)\n", + "G.add_node(1)\n", + "G.add_edge(0,1, weight=1.0)\n", + "G.add_edge(1,0, weight=1.0)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "a4c9e387-cdd0-4b64-909c-ab75a71a85de", + "metadata": {}, + "outputs": [], + "source": [ + "pos = {\n", + " 0 : [0,0],\n", + " 1 : [1,0]\n", + "}\n", + "\n", + "edge_labels = dict([((n1, n2), d['weight'])\n", + " for n1, n2, d in G.edges(data=True)])" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "id": "336c4249-48bc-41b9-9e88-77b12bdd60a8", + "metadata": {}, + "outputs": [ + { + "data": { + 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"336c4249-48bc-41b9-9e88-77b12bdd60a8", "metadata": {}, "outputs": [ @@ -76,13 +76,13 @@ " edgecolors='black',\n", " node_size=5000,\n", " linewidths=1,\n", - " connectionstyle='arc3,rad=0.2'\n", + " connectionstyle='arc3,rad=0.2' \n", " )\n", "nx.draw_networkx_edge_labels(G, \n", " pos, \n", " edge_labels=edge_labels, \n", " connectionstyle='arc3,rad=0.2')\n", - "plt.savefig('example4.png', dpi=200)" + "plt.savefig('example4.png', dpi=200, bbox_inches=\"tight\")" ] }, { diff --git a/lectures/markov_chains_II.md b/lectures/markov_chains_II.md index 67def2d7..3be115b3 100644 --- a/lectures/markov_chains_II.md +++ b/lectures/markov_chains_II.md @@ -295,6 +295,7 @@ The transition graph shows that this model is irreducible. ```{image} /_static/lecture_specific/markov_chains_II/example4.png :name: mc_example4 :align: center +:scale: 80% ``` Notice that there is a periodic cycle --- the state cycles between the two states in a regular way.