class06_degdist_R.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# CS446/546 - Class Session 6 - 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": 36,
   "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": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "sif_data <- read.table(\"shared/pathway_commons.sif\",\n",
    "                       sep=\"\\t\",\n",
    "                       header=FALSE,\n",
    "                       stringsAsFactors=FALSE,\n",
    "                       col.names=c(\"species1\",\n",
    "                                   \"interaction_type\",\n",
    "                                   \"species2\"),\n",
    "                       quote=\"\",\n",
    "                       comment.char=\"\")"
   ]
  },
  {
   "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": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>species1</th><th scope=col>interaction_type</th><th scope=col>species2</th></tr></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>interacts-with        </td><td>ACE2                  </td></tr>\n",
       "\t<tr><td>A1BG                  </td><td>interacts-with        </td><td>ADAM10                </td></tr>\n",
       "\t<tr><td>A1BG                  </td><td>interacts-with        </td><td>ADAM17                </td></tr>\n",
       "\t<tr><td>A1BG                  </td><td>interacts-with        </td><td>ADAM9                 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lll}\n",
       " species1 & interaction\\_type & species2\\\\\n",
       "\\hline\n",
       "\t A1BG                   & controls-expression-of & A2M                   \\\\\n",
       "\t A1BG                   & interacts-with         & ABCC6                 \\\\\n",
       "\t A1BG                   & interacts-with         & ACE2                  \\\\\n",
       "\t A1BG                   & interacts-with         & ADAM10                \\\\\n",
       "\t A1BG                   & interacts-with         & ADAM17                \\\\\n",
       "\t A1BG                   & interacts-with         & ADAM9                 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "species1 | interaction_type | species2 | \n",
       "|---|---|---|---|---|---|\n",
       "| A1BG                   | controls-expression-of | A2M                    | \n",
       "| A1BG                   | interacts-with         | ABCC6                  | \n",
       "| A1BG                   | interacts-with         | ACE2                   | \n",
       "| A1BG                   | interacts-with         | ADAM10                 | \n",
       "| A1BG                   | interacts-with         | ADAM17                 | \n",
       "| A1BG                   | interacts-with         | ADAM9                  | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  species1 interaction_type       species2\n",
       "1 A1BG     controls-expression-of A2M     \n",
       "2 A1BG     interacts-with         ABCC6   \n",
       "3 A1BG     interacts-with         ACE2    \n",
       "4 A1BG     interacts-with         ADAM10  \n",
       "5 A1BG     interacts-with         ADAM17  \n",
       "6 A1BG     interacts-with         ADAM9   "
      ]
     },
     "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": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "interac_ppi <- sif_data[sif_data$interaction_type %in% c(\"in-complex-with\",\n",
    "                                                         \"interacts-with\"), c(1,3)]"
   ]
  },
  {
   "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": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "interac_ppi <- subset(sif_data, interaction_type %in% c(\"in-complex-with\", \"interacts-with\"), \n",
    "                      select=c(\"species1\",\"species2\"))"
   ]
  },
  {
   "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": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>species1</th><th scope=col>species2</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>2</th><td>A1BG  </td><td>ABCC6 </td></tr>\n",
       "\t<tr><th scope=row>3</th><td>A1BG  </td><td>ACE2  </td></tr>\n",
       "\t<tr><th scope=row>4</th><td>A1BG  </td><td>ADAM10</td></tr>\n",
       "\t<tr><th scope=row>5</th><td>A1BG  </td><td>ADAM17</td></tr>\n",
       "\t<tr><th scope=row>6</th><td>A1BG  </td><td>ADAM9 </td></tr>\n",
       "\t<tr><th scope=row>7</th><td>A1BG  </td><td>AGO1  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       "  & species1 & species2\\\\\n",
       "\\hline\n",
       "\t2 & A1BG   & ABCC6 \\\\\n",
       "\t3 & A1BG   & ACE2  \\\\\n",
       "\t4 & A1BG   & ADAM10\\\\\n",
       "\t5 & A1BG   & ADAM17\\\\\n",
       "\t6 & A1BG   & ADAM9 \\\\\n",
       "\t7 & A1BG   & AGO1  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | species1 | species2 | \n",
       "|---|---|---|---|---|---|\n",
       "| 2 | A1BG   | ABCC6  | \n",
       "| 3 | A1BG   | ACE2   | \n",
       "| 4 | A1BG   | ADAM10 | \n",
       "| 5 | A1BG   | ADAM17 | \n",
       "| 6 | A1BG   | ADAM9  | \n",
       "| 7 | A1BG   | AGO1   | \n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  species1 species2\n",
       "2 A1BG     ABCC6   \n",
       "3 A1BG     ACE2    \n",
       "4 A1BG     ADAM10  \n",
       "5 A1BG     ADAM17  \n",
       "6 A1BG     ADAM9   \n",
       "7 A1BG     AGO1    "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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       "508480"
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       "[1] 508480"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "interac_ppi_ordered <- rbind(subset(interac_ppi, species1 < species2),\n",
    "                             subset(interac_ppi, species1 >= species2)[,c(\"species2\",\"species1\")])"
   ]
  },
  {
   "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": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "[1] 508480"
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "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": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "interac_ppi_unique <- unique(interac_ppi_ordered)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "How many rows do we have now? Let's print it out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "475553"
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       "475553"
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      "text/plain": [
       "[1] 475553"
      ]
     },
     "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": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "vertex_degrees <- table(c(interac_ppi_unique$species1,\n",
    "                             interac_ppi_unique$species2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's have a look at `vertex_degrees`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "  A1BG   A1CF    A2M  A2ML1 A4GALT  A4GNT \n",
       "    59     34    171      4      6     19 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(vertex_degrees)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 6: Calculate the histogram of N(k) vs. k, \"suggesting\" 30 bin \"breaks\" (i.e., bin edges) to R; `hist` will actually pick a different number of bins based on its own histogramming algorithm, so we'll find out exactly how many bin breaks that `hist` picked and then we'll call it `nbreaks_actual`, and we'll work with that. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "nbreaks <- 30\n",
    "hist_res <- hist(vertex_degrees, plot=FALSE, nclass=nbreaks)\n",
    "hist_counts <- hist_res$counts\n",
    "hist_breaks <- hist_res$breaks\n",
    "nbreaks_actual <- length(hist_breaks)\n",
    "kvals <- (hist_breaks[1:(nbreaks_actual-1)] + hist_breaks[-1])/2"
   ]
  },
  {
   "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": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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EgPL0/TP93o00p9/Zset7OBttNsSF2f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\nAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "indend <- min(which(hist_counts == 0))\n",
    "indend <- if (length(indend) == 0) { length(kvals) } else { indend - 1 }\n",
    "plot(kvals[1:indend],\n",
    "     hist_counts[1:indend],\n",
    "     type=\"p\",\n",
    "     main=\"\",\n",
    "     log=\"xy\",\n",
    "     xlab=expression(k),\n",
    "     ylab=expression(N[k]))"
   ]
  },
  {
   "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": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AgQEgQICQKEBAFCggAhQYCQIEBIECAk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\nQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(ggplot(data=data.frame(k=kvals[1:indend], Nk=hist_counts[1:indend]), aes(x=k, y=Nk)) + \n",
    "      geom_point() + \n",
    "      theme_classic(base_size=24) +\n",
    "      scale_x_log10() +\n",
    "      scale_y_log10() +\n",
    "      ylab(expression(N[k])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Step 8:  Do a linear fit to the log10(N(k)) vs. log10(k) data.  Restrict your analysis to the k bin values for which the log(Nk) vs. log(k) relationship is linear-looking, which is the first four points on the plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Call:\n",
       "lm(formula = log10Nk ~ log10k, data = degdist_df)\n",
       "\n",
       "Residuals:\n",
       "       1        2        3        4 \n",
       "-0.01571  0.02675  0.02349 -0.03453 \n",
       "\n",
       "Coefficients:\n",
       "            Estimate Std. Error t value Pr(>|t|)    \n",
       "(Intercept)  7.72896    0.12811   60.33 0.000275 ***\n",
       "log10k      -2.08465    0.05752  -36.24 0.000760 ***\n",
       "---\n",
       "Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1\n",
       "\n",
       "Residual standard error: 0.03679 on 2 degrees of freedom\n",
       "Multiple R-squared:  0.9985,\tAdjusted R-squared:  0.9977 \n",
       "F-statistic:  1314 on 1 and 2 DF,  p-value: 0.0007604\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# put the data from the linear range, into a data frame \n",
    "degdist_df <- data.frame(log10Nk=log10(hist_counts[1:4]),\n",
    "                         log10k=log10(kvals[1:4]))\n",
    "\n",
    "# linear regression\n",
    "summary(lm(log10Nk ~ log10k, \n",
    "           data=degdist_df))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Slope is -2.08 with SE 0.06, i.e., gamma = 2.08 with a 95% CI of about +/- 0.12.\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": 52,
   "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": [
    "round(-6.5/(log(45)-log(2)), digits=2)"
   ]
  },
  {
   "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 (-2.08) for the human PPI!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we'll do the same thing in just a few lines of igraph code"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl>\n",
       "\t<dt>$continuous</dt>\n",
       "\t\t<dd>FALSE</dd>\n",
       "\t<dt>$alpha</dt>\n",
       "\t\t<dd>3.48175806452225</dd>\n",
       "\t<dt>$xmin</dt>\n",
       "\t\t<dd>245</dd>\n",
       "\t<dt>$logLik</dt>\n",
       "\t\t<dd>-4321.08945321598</dd>\n",
       "\t<dt>$KS.stat</dt>\n",
       "\t\t<dd>0.0498304830663898</dd>\n",
       "\t<dt>$KS.p</dt>\n",
       "\t\t<dd>0.0557169127937001</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description}\n",
       "\\item[\\$continuous] FALSE\n",
       "\\item[\\$alpha] 3.48175806452225\n",
       "\\item[\\$xmin] 245\n",
       "\\item[\\$logLik] -4321.08945321598\n",
       "\\item[\\$KS.stat] 0.0498304830663898\n",
       "\\item[\\$KS.p] 0.0557169127937001\n",
       "\\end{description}\n"
      ],
      "text/markdown": [
       "$continuous\n",
       ":   FALSE\n",
       "$alpha\n",
       ":   3.48175806452225\n",
       "$xmin\n",
       ":   245\n",
       "$logLik\n",
       ":   -4321.08945321598\n",
       "$KS.stat\n",
       ":   0.0498304830663898\n",
       "$KS.p\n",
       ":   0.0557169127937001\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "$continuous\n",
       "[1] FALSE\n",
       "\n",
       "$alpha\n",
       "[1] 3.481758\n",
       "\n",
       "$xmin\n",
       "[1] 245\n",
       "\n",
       "$logLik\n",
       "[1] -4321.089\n",
       "\n",
       "$KS.stat\n",
       "[1] 0.04983048\n",
       "\n",
       "$KS.p\n",
       "[1] 0.05571691\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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gQPXTIm2u1Q5icJEjom0KPLMD82+MJTfFa2cBD2G+nvv4eaMuamOEWRLEUTW5dr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\n+H/V/E2Nc/cW1wAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "g <- graph_from_data_frame(interac_ppi_unique, directed=FALSE)\n",
    "degree_distrib <- degree_distribution(g)\n",
    "suppressWarnings(plot(degree_distrib, log=\"xy\"))\n",
    "fit_power_law(degree(g))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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",
   "version": "3.4.1"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}