class03_degdist_R_template.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# CS446/546 - Class Session 3 - Degree Distribution\n",
    "\n",
    "In this class session we are going to plot the degree distribution of the undirected human\n",
    "protein-protein interaction network (PPI), without using `igraph`. We'll obtain the interaction data from the Pathway Commons SIF file (in the `shared/` folder) and we'll \n",
    "manually compute the degree of each vertex (protein) in the network. We'll then compute\n",
    "the count `N(k)` of vertices that have a given vertex degree `k`, for all `k` values.\n",
    "Finally, we'll plot the degree distribution and discuss whether it is consistent with the \n",
    "results obtained in the Jeong et al. article for the yeast PPI."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We're going to use `ggplot2` in this notebook"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "library(ggplot2)\n",
    "suppressPackageStartupMessages(library(igraph))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 1:  load in the SIF file as the data frame `sif_data` (with column names `species1`, `interaction_type`, and `species2`), using the function `read.table`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "sif_data <- "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's take a look at the top of the data frame, using the `head` function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A data.frame: 6 × 3</caption>\n",
       "<thead>\n",
       "\t<tr><th scope=col>species1</th><th scope=col>interaction_type</th><th scope=col>species2</th></tr>\n",
       "\t<tr><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><td>A1BG</td><td>controls-expression-of     </td><td>A2M   </td></tr>\n",
       "\t<tr><td>A1BG</td><td>interacts-with             </td><td>ABCC6 </td></tr>\n",
       "\t<tr><td>A1BG</td><td>controls-phosphorylation-of</td><td>AKT1  </td></tr>\n",
       "\t<tr><td>A1BG</td><td>controls-state-change-of   </td><td>AKT1  </td></tr>\n",
       "\t<tr><td>A1BG</td><td>interacts-with             </td><td>ANXA7 </td></tr>\n",
       "\t<tr><td>A1BG</td><td>interacts-with             </td><td>CDKN1A</td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 6 × 3\n",
       "\\begin{tabular}{r|lll}\n",
       " species1 & interaction\\_type & species2\\\\\n",
       " <chr> & <chr> & <chr>\\\\\n",
       "\\hline\n",
       "\t A1BG & controls-expression-of      & A2M   \\\\\n",
       "\t A1BG & interacts-with              & ABCC6 \\\\\n",
       "\t A1BG & controls-phosphorylation-of & AKT1  \\\\\n",
       "\t A1BG & controls-state-change-of    & AKT1  \\\\\n",
       "\t A1BG & interacts-with              & ANXA7 \\\\\n",
       "\t A1BG & interacts-with              & CDKN1A\\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 6 × 3\n",
       "\n",
       "| species1 &lt;chr&gt; | interaction_type &lt;chr&gt; | species2 &lt;chr&gt; |\n",
       "|---|---|---|\n",
       "| A1BG | controls-expression-of      | A2M    |\n",
       "| A1BG | interacts-with              | ABCC6  |\n",
       "| A1BG | controls-phosphorylation-of | AKT1   |\n",
       "| A1BG | controls-state-change-of    | AKT1   |\n",
       "| A1BG | interacts-with              | ANXA7  |\n",
       "| A1BG | interacts-with              | CDKN1A |\n",
       "\n"
      ],
      "text/plain": [
       "  species1 interaction_type            species2\n",
       "1 A1BG     controls-expression-of      A2M     \n",
       "2 A1BG     interacts-with              ABCC6   \n",
       "3 A1BG     controls-phosphorylation-of AKT1    \n",
       "4 A1BG     controls-state-change-of    AKT1    \n",
       "5 A1BG     interacts-with              ANXA7   \n",
       "6 A1BG     interacts-with              CDKN1A  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(sif_data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 2:  restrict the interactions to undirected protein-protein interactions. We will do this by subsetting the data frame to only rows for which the `interaction_type` column contains either `in-complex-with` or `interacts-with`. We'll also subset the data frame to include only the `species1` and `species2` columns.  There are two ways to do it, (i) using standard indexing and (ii) using the `subset` function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is how you would do it using standard indexing:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "interac_ppi <- "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is how you would do it using the `subset` function; this code is arguably more \"readable\":"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "interac_ppi <- "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use the `head` function to show the first few rows of the data frame, and use the `nrow` function to see how many rows it is:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<caption>A data.frame: 6 × 2</caption>\n",
       "<thead>\n",
       "\t<tr><th></th><th scope=col>species1</th><th scope=col>species2</th></tr>\n",
       "\t<tr><th></th><th scope=col>&lt;chr&gt;</th><th scope=col>&lt;chr&gt;</th></tr>\n",
       "</thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>2</th><td>A1BG</td><td>ABCC6 </td></tr>\n",
       "\t<tr><th scope=row>5</th><td>A1BG</td><td>ANXA7 </td></tr>\n",
       "\t<tr><th scope=row>6</th><td>A1BG</td><td>CDKN1A</td></tr>\n",
       "\t<tr><th scope=row>7</th><td>A1BG</td><td>CRISP3</td></tr>\n",
       "\t<tr><th scope=row>8</th><td>A1BG</td><td>GDPD1 </td></tr>\n",
       "\t<tr><th scope=row>9</th><td>A1BG</td><td>GRB2  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "A data.frame: 6 × 2\n",
       "\\begin{tabular}{r|ll}\n",
       "  & species1 & species2\\\\\n",
       "  & <chr> & <chr>\\\\\n",
       "\\hline\n",
       "\t2 & A1BG & ABCC6 \\\\\n",
       "\t5 & A1BG & ANXA7 \\\\\n",
       "\t6 & A1BG & CDKN1A\\\\\n",
       "\t7 & A1BG & CRISP3\\\\\n",
       "\t8 & A1BG & GDPD1 \\\\\n",
       "\t9 & A1BG & GRB2  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "A data.frame: 6 × 2\n",
       "\n",
       "| <!--/--> | species1 &lt;chr&gt; | species2 &lt;chr&gt; |\n",
       "|---|---|---|\n",
       "| 2 | A1BG | ABCC6  |\n",
       "| 5 | A1BG | ANXA7  |\n",
       "| 6 | A1BG | CDKN1A |\n",
       "| 7 | A1BG | CRISP3 |\n",
       "| 8 | A1BG | GDPD1  |\n",
       "| 9 | A1BG | GRB2   |\n",
       "\n"
      ],
      "text/plain": [
       "  species1 species2\n",
       "2 A1BG     ABCC6   \n",
       "5 A1BG     ANXA7   \n",
       "6 A1BG     CDKN1A  \n",
       "7 A1BG     CRISP3  \n",
       "8 A1BG     GDPD1   \n",
       "9 A1BG     GRB2    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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       "523498"
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       "523498"
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       "523498"
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       "[1] 523498"
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     },
     "metadata": {},
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    }
   ],
   "source": [
    "head(interac_ppi)\n",
    "nrow(interac_ppi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 3: for each interaction, reorder `species1` and `species2` (if necessary) so that\n",
    "`species1 < species2` (in terms of lexicographic order); we are doing this because there are many cases where a pair of proteins \"X\" and \"Y\" appear in the database as both [X,Y] and [Y,X], and we want those edges to be recognized as the same so that they are not separately counted. We'll do this using `subset` twice (once for `species1 < species2` and once for `species1 >= species2`) and then stacking the two data frames that were returned from the two calls to `subset`, using the `rbind` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "interac_ppi_ordered <- "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The number of rows in `interac_ppi_ordered` should be the same as for `interac_ppi`; let's verify that."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "523498"
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       "[1] 523498"
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     },
     "metadata": {},
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    }
   ],
   "source": [
    "nrow(interac_ppi_ordered)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 4: get only the unique interaction pairs of proteins (ignoring the interaction type), using the `unique` function. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "interac_ppi_unique <- "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How many rows do we have now? Let's print it out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "491784"
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       "491784"
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       "491784"
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      "text/plain": [
       "[1] 491784"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "nrow(interac_ppi_unique)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 5: compute the degree of each vertex.  We'll do this by making a single character vector that is the concatenation of the `species1` and `species2` columns of `interac_ppi_unique`, and then we'll use the `table` function to compute the number of times each protein name occurs (you could also do the same thing by using the `sort` and `rle` functions, but `table` is easier to remember)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "vertex_degrees <- "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's have a look at `vertex_degrees`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "  A1BG   A1CF    A2M  A2ML1 A4GALT  A4GNT \n",
       "    19     27    169      6      3     14 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(vertex_degrees)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 6: Calculate the histogram of N(k) vs. k, \"suggesting\" 30 bin \"breaks\" (i.e., bin edges) to R; `hist` will actually pick a different number of bins based on its own histogramming algorithm, so we'll find out exactly how many bin breaks that `hist` picked and then we'll call it `nbreaks_actual`, and we'll work with that. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "nbreaks <- 30\n",
    "hist_res <- \n",
    "hist_counts <- \n",
    "hist_breaks <- \n",
    "nbreaks_actual <- \n",
    "kvals <- "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 7: Plot `N(k)` vs. `k`, on log-log scale.  Restrict the horizontal axis to only values up to (but not including) the first zero value of `N(k)`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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M88WfcXyvu9u9dsNRPi7yk3W2xN7l2opImtruGjH3bLvc3d9Klp9vrYWqff9Qng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\nSUVORK5CYII=",
      "text/plain": [
       "Plot with title “”"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      },
      "text/plain": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's also plot the data using `ggplot2` instead of R base graphics.  We'll use the functions `ggplot`, `geom_point`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
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rl57EkcIcEvZYS0+JRsR+/V/qqGgc54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\nTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      },
      "text/plain": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 8:  Do a linear fit to the log10(N(k)) vs. log10(k) data.  What slope do you get?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = log10Nk ~ log10k, data = degdist_df)\n",
       "\n",
       "Residuals:\n",
       "     Min       1Q   Median       3Q      Max \n",
       "-0.67228 -0.13992  0.04304  0.21232  0.40943 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)   8.5338     0.3994   21.37  < 2e-16 ***\n",
       "log10k       -2.5114     0.1313  -19.13 4.93e-16 ***\n",
       "---\n",
       "Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1\n",
       "\n",
       "Residual standard error: 0.2885 on 24 degrees of freedom\n",
       "Multiple R-squared:  0.9384,\tAdjusted R-squared:  0.9359 \n",
       "F-statistic: 365.8 on 1 and 24 DF,  p-value: 4.927e-16\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Slope is -2.5 with SE 0.13, i.e., gamma = 2.5 with a 95% CI of about +/- 0.26.\n",
    "\n",
    "Now let's compute the slope for the degree distribution Fig. 1b in the Jeong et al. article, for the yeast PPI.  The change in ordinate over the linear range is about -6.5 in units of natural logarithm. The change in abscissa over the linear range is approximately log(45)-log(2), so we can compute the Jeong et al. slope thus:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "-2.09"
      ],
      "text/latex": [
       "-2.09"
      ],
      "text/markdown": [
       "-2.09"
      ],
      "text/plain": [
       "[1] -2.09"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The slope (-2.09) from Jeong et al.'s article, which is from yeast, is pretty close to the slope (-2.5) for the human PPI!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we'll do the same thing in just a few lines of igraph code (note that igraph uses a more accurate maximum-likelihood algorithm for estimating the exponent)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$continuous</dt>\n",
       "\t\t<dd>FALSE</dd>\n",
       "\t<dt>$alpha</dt>\n",
       "\t\t<dd>2.98079994066646</dd>\n",
       "\t<dt>$xmin</dt>\n",
       "\t\t<dd>227</dd>\n",
       "\t<dt>$logLik</dt>\n",
       "\t\t<dd>-5488.2367769949</dd>\n",
       "\t<dt>$KS.stat</dt>\n",
       "\t\t<dd>0.0333633895726593</dd>\n",
       "\t<dt>$KS.p</dt>\n",
       "\t\t<dd>0.281806273469199</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$continuous] FALSE\n",
       "\\item[\\$alpha] 2.98079994066646\n",
       "\\item[\\$xmin] 227\n",
       "\\item[\\$logLik] -5488.2367769949\n",
       "\\item[\\$KS.stat] 0.0333633895726593\n",
       "\\item[\\$KS.p] 0.281806273469199\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$continuous\n",
       ":   FALSE\n",
       "$alpha\n",
       ":   2.98079994066646\n",
       "$xmin\n",
       ":   227\n",
       "$logLik\n",
       ":   -5488.2367769949\n",
       "$KS.stat\n",
       ":   0.0333633895726593\n",
       "$KS.p\n",
       ":   0.281806273469199\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$continuous\n",
       "[1] FALSE\n",
       "\n",
       "$alpha\n",
       "[1] 2.9808\n",
       "\n",
       "$xmin\n",
       "[1] 227\n",
       "\n",
       "$logLik\n",
       "[1] -5488.237\n",
       "\n",
       "$KS.stat\n",
       "[1] 0.03336339\n",
       "\n",
       "$KS.p\n",
       "[1] 0.2818063\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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QxJRYoPfz3ZaU5SOSOBKFJ+ycnqUyVustC7yoDHdN2ypq1lTaloIalI2nZv+iO0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\n8ymu7Y02IUv6mmfeYMvb5Xqe9+tr3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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {
      "image/png": {
       "height": 420,
       "width": 420
      },
      "text/plain": {
       "height": 420,
       "width": 420
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "R",
   "language": "R",
   "name": "ir"
  },
  "language_info": {
   "codemirror_mode": "r",
   "file_extension": ".r",
   "mimetype": "text/x-r-source",
   "name": "R",
   "pygments_lexer": "r",
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 },
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 "nbformat_minor": 1
}