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    "internals": {
     "slide_type": "subslide"
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    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Networks"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n"
     ]
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='retina'\n",
    "# import libraries\n",
    "import numpy as np\n",
    "import matplotlib as mp\n",
    "import pandas as pd\n",
    "import matplotlib.pyplot as plt\n",
    "import laUtilities as ut\n",
    "import slideUtilities as sl\n",
    "import demoUtilities as dm\n",
    "import pandas as pd\n",
    "from importlib import reload\n",
    "from datetime import datetime\n",
    "from IPython.display import Image\n",
    "from IPython.display import display_html\n",
    "from IPython.display import display\n",
    "from IPython.display import Math\n",
    "from IPython.display import Latex\n",
    "from IPython.display import HTML\n",
    "print('')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
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       " .container.slides .celltoolbar, .container.slides .hide-in-slideshow {\n",
       "    display: None ! important;\n",
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       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%html\n",
    "<style>\n",
    " .container.slides .celltoolbar, .container.slides .hide-in-slideshow {\n",
    "    display: None ! important;\n",
    "}\n",
    "</style>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "internals": {},
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "%Set up useful MathJax (Latex) macros.\n",
    "%See http://docs.mathjax.org/en/latest/tex.html#defining-tex-macros\n",
    "%These are for use in the slideshow\n",
    "$\\newcommand{\\mat}[1]{\\left[\\begin{array}#1\\end{array}\\right]}$\n",
    "$\\newcommand{\\vx}{{\\mathbf x}}$\n",
    "$\\newcommand{\\hx}{\\hat{\\mathbf x}}$\n",
    "$\\newcommand{\\vbt}{{\\mathbf\\beta}}$\n",
    "$\\newcommand{\\vy}{{\\mathbf y}}$\n",
    "$\\newcommand{\\vz}{{\\mathbf z}}$\n",
    "$\\newcommand{\\R}{{\\mathbb{R}}}$\n",
    "$\\newcommand{\\vu}{{\\mathbf u}}$\n",
    "$\\newcommand{\\vv}{{\\mathbf v}}$\n",
    "$\\newcommand{\\vw}{{\\mathbf w}}$\n",
    "$\\newcommand{\\col}{{\\operatorname{Col}}}$\n",
    "$\\newcommand{\\nul}{{\\operatorname{Nul}}}$\n",
    "$\\newcommand{\\vb}{{\\mathbf b}}$\n",
    "$\\newcommand{\\va}{{\\mathbf a}}$\n",
    "$\\newcommand{\\ve}{{\\mathbf e}}$\n",
    "$\\newcommand{\\setb}{{\\mathcal{B}}}$\n",
    "$\\newcommand{\\rank}{{\\operatorname{rank}}}$\n",
    "$\\newcommand{\\vp}{{\\mathbf p}}$"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "internals": {
     "slide_helper": "subslide_end"
    },
    "slide_helper": "slide_end",
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "source": [
    "\\newcommand{\\mat}[1]{\\left[\\begin{array}#1\\end{array}\\right]}\n",
    "\\newcommand{\\vx}{{\\mathbf x}}\n",
    "\\newcommand{\\hx}{\\hat{\\mathbf x}}\n",
    "\\newcommand{\\vbt}{{\\mathbf\\beta}}\n",
    "\\newcommand{\\vy}{{\\mathbf y}}\n",
    "\\newcommand{\\vz}{{\\mathbf z}}\n",
    "\\newcommand{\\vb}{{\\mathbf b}}\n",
    "\\newcommand{\\vu}{{\\mathbf u}}\n",
    "\\newcommand{\\vv}{{\\mathbf v}}\n",
    "\\newcommand{\\vw}{{\\mathbf w}}\n",
    "\\newcommand{\\va}{{\\mathbf a}}\n",
    "\\newcommand{\\ve}{{\\mathbf e}}\n",
    "\\newcommand{\\vp}{{\\mathbf p}}\n",
    "\\newcommand{\\R}{{\\mathbb{R}}}\n",
    "\\newcommand{\\col}{{\\operatorname{Col}}}\n",
    "\\newcommand{\\nul}{{\\operatorname{Nul}}}\n",
    "\\newcommand{\\rank}{{\\operatorname{rank}}}\n",
    "\\newcommand{\\setb}{{\\mathcal{B}}}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "Network analysis - potentials, edge-incidence, spanning trees \n",
    "\n",
    "Based on Strang Ch 2.5 and Spielman/Srivastava\n",
    "\n",
    "And, properties of the Laplacian (if time) - graph embedding"
   ]
  }
 ],
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.7"
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  "rise": {
   "scroll": true,
   "theme": "beige",
   "transition": "fade"
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}