639 lines
23 KiB
R
639 lines
23 KiB
R
# FND-MAT-Graphs_and_networks.R
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#
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# Purpose: A Bioinformatics Course:
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# R code accompanying the FND-MAT-Graphs_and_networks unit.
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#
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# Version: 0.1
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#
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# Date: 2017 08 28
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# Author: Boris Steipe (boris.steipe@utoronto.ca)
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#
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# Versions:
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# 0.1 First code copied from 2016 material.
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#
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# TODO:
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#
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#
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# == DO NOT SIMPLY source() THIS FILE! =======================================
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# If there are portions you don't understand, use R's help system, Google for an
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# answer, or ask your instructor. Don't continue if you don't understand what's
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# going on. That's not how it works ...
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# ==============================================================================
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# = 1 ___Section___
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# This tutorial covers basic concepts of graph theory and analysis in R. You
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# should have typed init() to configure some utilities in the background.
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# ==============================================================================
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# PART ONE: REVIEW
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# ==============================================================================
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# I assume you'll have read the Pavlopoulos review of graph theory concepts.
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# Let's explore some of the ideas by starting with a small random graph."
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# To begin let's write a little function that will create random "gene" names;
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# there's no particular purpose to this other than to make our graphs look a
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# little more like what we would find in a publication ...
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makeRandomGenenames <- function(N) {
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nam <- character()
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while (length(nam) < N) {
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a <- paste(c(sample(LETTERS, 1), sample(letters, 2)),
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sep="", collapse="") # three letters
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n <- sample(1:9, 1) # one number
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nam[length(nam) + 1] <- paste(a, n, sep="") # store in vector
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nam <- unique(nam) # delete if this was a duplicate
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}
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return(nam)
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}
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N <- 20
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set.seed(112358)
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Nnames <- makeRandomGenenames(N)
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Nnames
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# One way to represent graphs in a computer is as an "adjacency matrix". In this
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# matrix, each row and each column represents a node, and the cell at the
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# intersection of a row and column contains a value/TRUE if there is an edge,
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# 0/FALSE otherwise. It's easy to see that an undirected graph has a symmetric
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# adjacency matrix (i, j) == (j, i); and we can put values other than {1, 0}
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# into a cell if we want to represent a weighted edge.
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# At first, lets create a random graph: let's say a pair of nodes has
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# probability p <- 0.1 to have an edge, and our graph is symmetric and has no
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# self-edges. We use our Nnames as node labels, but I've written the function so
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# that we could also just ask for any number of un-named nodes, we'll use that later.
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makeRandomGraph <- function(nam, p = 0.1) {
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# nam: either a character vector of unique names, or a single
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# number that will be converted into a vector of integers.
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# p: probability that a random pair of nodes will have an edge.
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#
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# Value: an adjacency matrix
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#
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if (is.numeric(nam) && length(nam) == 1) { # if nam is a single number ...
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nam <- as.character(1:nam)
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}
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N <- length(nam)
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G <- matrix(numeric(N * N), ncol = N) # The adjacency matrix
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rownames(G) <- nam
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colnames(G) <- nam
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for (iRow in 1:(N-1)) { # Note how we make sure iRow != iCol
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for (iCol in (iRow+1):N) {
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if (runif(1) < p) { # runif() creates uniform random numbers
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# between 0 and 1
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G[iRow, iCol] <- 1 # row, col !
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G[iCol, iRow] <- 1 # col, row !
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}
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}
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}
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return(G)
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}
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set.seed(112358)
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G <- makeRandomGraph(Nnames, p = 0.09)
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G
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# Listing the matrix is not very informative - we should plot this graph. We'll
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# go into more details of the igraph package a bit later, for now we just use it
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# to plot:
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if (!require(igraph)) {
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install.packages("igraph")
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library(igraph)
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}
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iG <- graph_from_adjacency_matrix(G)
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iGxy <- layout_with_graphopt(iG, charge=0.001) # calculate layout coordinates
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# The igraph package adds its own function to the collection of plot()
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# functions; R makes the selection which plot function to use based on the class
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# of the object that we request to plot. This plot function has parameters
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# layout - the x,y coordinates of the nodes;
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# vertex.color - which I define to color by node-degree
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# vertex size - which I define to increase with node-degree
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# vertex.label - which I set to use our Nnames vector
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oPar <- par(mar= rep(0,4)) # Turn margins off
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plot(iG,
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layout = iGxy,
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rescale = FALSE,
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xlim = c(min(iGxy[,1]), max(iGxy[,1])) * 1.1,
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ylim = c(min(iGxy[,2]), max(iGxy[,2])) * 1.1,
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vertex.color=heat.colors(max(degree(iG)+1))[degree(iG)+1],
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vertex.size = 800 + (150 * degree(iG)),
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vertex.label = as.character(degree(iG)/2),
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# vertex.label = Nnames,
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edge.arrow.size = 0)
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par(oPar) # reset plot window
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# The simplest descriptor of a graph are the number of nodes, edges, and the
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# degree-distribution. In our example, the number of nodes was given: N; the
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# number of edges can easily be calculated from the adjacency matrix. In our
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# matrix, we have entered 1 for every edge. Thus we simply sum over the matrix:
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sum(G)
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# Is that correct? Is that what you see in the plot?
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# Yes and no: we entered every edge twice: once for a node [i,j], and again for
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# the node [j, i]. Whether that is correct depends on what exactly we
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# want to do with the matrix. If these were directed edges, we would need to
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# keep track of them separately. Since we didn't intend them to be directed,
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# we'll could divide the number of edges by 2. Why didn't we simply use an
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# upper-triangular matrix? Because then we need to keep track of the ordering of
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# edges if we want to know whether a particular edge exists or not. For example
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# we could sort the nodes alphabetically, and make sure we always query a pair
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# in alphabetical order. Then a triangular matrix would be efficient.
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# What about the degree distribution? We can get that simply by summing over the
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# rows (or the columns):"
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rowSums(G) # check this against the plot!
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# Let's plot the degree distribution in a histogram:
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rs <- rowSums(G)
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brk <- seq(min(rs)-0.5, max(rs)+0.5, by=1) # define breaks for the histogram
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hist(rs, breaks=brk, col="#A5CCF5",
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xlim = c(-1,8), xaxt = "n",
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main = "Node degrees", xlab = "Degree", ylab = "Number") # plot histogram
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axis(side = 1, at = 0:7)
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# Note: I don't _have_ to define breaks, the hist() function usually does so
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# quite well, automatically. But for this purpose I want the columns of the
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# histogram to represent exactly one node-degree difference.
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# A degree distribution is actually quite an important descriptor of graphs,
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# since it is very sensitive to the generating mechanism. For biological
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# networks, that is one of the key questions we are interested in: how was the
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# network formed?
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# ==============================================================================
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# PART TWO: DEGREE DISTRIBUTIONS
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# ==============================================================================
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# Let's simulate a few graphs that are a bit bigger to get a better sense of
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# their degree distributions:
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#
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# === random graph
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set.seed(31415927)
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G200 <- makeRandomGraph(200, p = 0.015)
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iG200 <- graph_from_adjacency_matrix(G200)
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iGxy <- layout_with_graphopt(iG200, charge=0.0001) # calculate layout coordinates
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oPar <- par(mar= rep(0,4)) # Turn margins off
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plot(iG200,
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layout = iGxy,
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rescale = FALSE,
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xlim = c(min(iGxy[,1]), max(iGxy[,1])) * 1.1,
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ylim = c(min(iGxy[,2]), max(iGxy[,2])) * 1.1,
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vertex.color=heat.colors(max(degree(iG200)+1))[degree(iG200)+1],
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vertex.size = 200 + (30 * degree(iG200)),
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vertex.label = "",
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edge.arrow.size = 0)
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par(oPar)
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# This graph has thirteen singletons and one large, connected component. Many
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# biological graphs look approximately like this.
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# Calculate degree distributions
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dg <- degree(iG200)/2 # here, we use the iGraph function degree()
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# not rowsums() from base R.
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brk <- seq(min(dg)-0.5, max(dg)+0.5, by=1)
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hist(dg, breaks=brk, col="#A5CCF5",
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xlim = c(-1,11), xaxt = "n",
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main = "Node degrees", xlab = "Degree", ylab = "Number") # plot histogram
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axis(side = 1, at = 0:10)
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# Note the characteristic peak of this distribution: this is not "scale-free". Here is a log-log plot of frequency vs. degree-rank:
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(freqRank <- table(dg))
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plot(log10(as.numeric(names(freqRank)) + 1),
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log10(as.numeric(freqRank)), type = "b",
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pch = 21, bg = "#A5CCF5",
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xlab = "log(Rank)", ylab = "log(frequency)",
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main = "200 nodes in a random network")
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# === scale-free graph (Barabasi-Albert)
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# What does one of those intriguing "scale-free" distributions look like? The
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# iGraph package has a function to make random graphs according to the
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# Barabasi-Albert model of scale-free graphs. It is: sample_pa(), where pa
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# stands for "preferential attachment", one type of process that will yield
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# scale-free distributions.
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set.seed(31415927)
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GBA <- sample_pa(200, power = 0.8)
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iGxy <- layout_with_graphopt(GBA, charge=0.0001) # calculate layout coordinates
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oPar <- par(mar= rep(0,4)) # Turn margins off
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plot(GBA,
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layout = iGxy,
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rescale = FALSE,
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xlim = c(min(iGxy[,1]), max(iGxy[,1])) * 1.1,
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ylim = c(min(iGxy[,2]), max(iGxy[,2])) * 1.1,
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vertex.color=heat.colors(max(degree(GBA)+1))[degree(GBA)+1],
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vertex.size = 200 + (30 * degree(GBA)),
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vertex.label = "",
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edge.arrow.size = 0)
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par(oPar)
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# This is a very obviously different graph! Some biological networks have
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# features that look like that - but in my experience the hub nodes are usually
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# not that distinct. But then again, that really depends on the parameter
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# "power". Feel encouraged to change "power" and get a sense for what difference
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# this makes. Also: note that the graph has only a single component.
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# What's the degree distribution of this graph?
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(dg <- degree(GBA))
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brk <- seq(min(dg)-0.5, max(dg)+0.5, by=1)
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hist(dg, breaks=brk, col="#A5D5CC",
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xlim = c(0,30), xaxt = "n",
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main = "Node degrees 200 nodes PA graph",
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xlab = "Degree", ylab = "Number")
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axis(side = 1, at = seq(0, 30, by=5))
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# Most nodes have a degree of 1, but one node has a degree of 28.
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(freqRank <- table(dg))
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plot(log10(as.numeric(names(freqRank)) + 1),
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log10(as.numeric(freqRank)), type = "b",
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pch = 21, bg = "#A5F5CC",
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xlab = "log(Rank)", ylab = "log(frequency)",
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main = "200 nodes in a preferential-attachment network")
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# Sort-of linear, but many of the higher ranked nodes have a frequency of only
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# one. That behaviour smooths out in larger graphs:
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#
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X <- sample_pa(100000, power = 0.8) # 100,000 nodes
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freqRank <- table(degree(X))
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plot(log10(as.numeric(names(freqRank)) + 1),
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log10(as.numeric(freqRank)), type = "b",
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xlab = "log(Rank)", ylab = "log(frequency)",
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pch = 21, bg = "#A5F5CC",
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main = "100,000 nodes in a random, scale-free network")
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rm(X)
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# === Random geometric graph
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# Finally, let's simulate a random geometric graph and look at the degree
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# distribution. Remember: these graphs have a high probability to have edges
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# between nodes that are "close" together - an entriely biological notion.
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# We'll randomly place our nodes in a box. Then we'll define the
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# probability for two nodes to have an edge to be a function of their distance.
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# Here is a function that makes such graphs. iGraph has sample_grg(), which
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# connects nodes that are closer than a cutoff, the function I give you below is
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# a bit more interesting since it creates edges according to a probability that
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# is determined by a generalized logistic function of the distance. This
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# sigmoidal function gives a smooth cutoff and creates more "natural" graphs.
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# Otherwise, the function is very similar to the random graph function, except
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# that we output the "coordinates" of the nodes together with the adjacency
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# matrix. Lists FTW.
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#
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makeRandomGeometricGraph <- function(nam, B = 25, Q = 0.001, t = 0.6) {
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# nam: either a character vector of unique names, or a single
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# number that will be converted into a vector of integers.
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# B, Q, t: probability that a random pair (i, j) of nodes gets an
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# edge determined by a generalized logistic function
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# p <- 1 - 1/((1 + (Q * (exp(-B * (x-t)))))^(1 / 0.9)))
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#
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# Value: a list with the following components:
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# G$mat : an adjacency matrix
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# G$nam : labels for the nodes
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# G$x : x-coordinates for the nodes
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# G$y : y-coordinates for the nodes
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#
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nu <- 1 # probably not useful to change
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G <- list()
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if (is.numeric(nam) && length(nam) == 1) {
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nam <- as.character(1:nam)
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}
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G$nam <- nam
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N <- length(G$nam)
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G$mat <- matrix(numeric(N * N), ncol = N) # The adjacency matrix
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rownames(G$mat) <- G$nam
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colnames(G$mat) <- G$nam
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G$x <- runif(N)
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G$y <- runif(N)
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for (iRow in 1:(N-1)) { # Same principles as in makeRandomGraph()
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for (iCol in (iRow+1):N) {
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# geometric distance ...
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d <- sqrt((G$x[iRow] - G$x[iCol])^2 +
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(G$y[iRow] - G$y[iCol])^2) # Pythagoras
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# distance dependent probability
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p <- 1 - 1/((1 + (Q * (exp(-B * (d-t)))))^(1 / nu))
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if (runif(1) < p) {
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G$mat[iRow, iCol] <- 1
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G$mat[iCol, iRow] <- 1
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}
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}
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}
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return(G)
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}
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# Getting the parameters of a generalized logistic right takes a bit of
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# experimenting. If you are interested, you can try a few variations. Or you can
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# look up the function at
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# https://en.wikipedia.org/wiki/Generalised_logistic_function
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# This function computes generalized logistics ...
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# genLog <- function(x, B = 25, Q = 0.001, t = 0.5) {
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# # generalized logistic (sigmoid)
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# nu <- 1
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# return(1 - 1/((1 + (Q * (exp(-B * (x-t)))))^(1 / nu)))
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# }
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#
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# ... and this code plots p-values over the distances we could encouter between
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# our nodes: from 0 to sqrt(2) i.e. the diagonal of the unit sqaure in which we
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# will place our nodes.
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# x <- seq(0, sqrt(2), length.out = 50)
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# plot(x, genLog(x), type="l", col="#AA0000", ylim = c(0, 1),
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# xlab = "d", ylab = "p(edge)")
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# 200 node random geomteric graph
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set.seed(112358)
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GRG <- makeRandomGeometricGraph(200, t=0.4)
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iGRG <- graph_from_adjacency_matrix(GRG$mat)
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iGRGxy <- cbind(GRG$x, GRG$y) # use our node coordinates for layout
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oPar <- par(mar= rep(0,4)) # Turn margins off
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plot(iGRG,
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layout = iGRGxy,
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rescale = FALSE,
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xlim = c(min(iGRGxy[,1]), max(iGRGxy[,1])) * 1.1,
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ylim = c(min(iGRGxy[,2]), max(iGRGxy[,2])) * 1.1,
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vertex.color=heat.colors(max(degree(iGRG)+1))[degree(iGRG)+1],
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vertex.size = 0.1 + (0.1 * degree(iGRG)),
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vertex.label = "",
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edge.arrow.size = 0)
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par(oPar)
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# degree distribution:
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(dg <- degree(iGRG)/2)
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brk <- seq(min(dg)-0.5, max(dg)+0.5, by=1)
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hist(dg, breaks=brk, col="#FCD6E2",
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xlim = c(0, 25), xaxt = "n",
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main = "Node degrees: 200 nodes RG graph",
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xlab = "Degree", ylab = "Number")
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axis(side = 1, at = c(0, min(dg):max(dg)))
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# You'll find that this is kind of in-between the random, and the scale-free
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# graph. We do have hubs, but they are not as extreme as in the scale-free case;
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# and we have have no singletons, in contrast to the random graph.
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(freqRank <- table(dg))
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plot(log10(as.numeric(names(freqRank)) + 1),
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log10(as.numeric(freqRank)), type = "b",
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pch = 21, bg = "#FCD6E2",
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xlab = "log(Rank)", ylab = "log(frequency)",
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main = "200 nodes in a random geometric network")
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# ====================================================================
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# PART THREE: A CLOSER LOOK AT THE igraph PACKAGE
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# ====================================================================
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# == BASICS ==========================================================
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# The basic object of the igraph package is a graph object. Let's explore the
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# first graph some more, the one we built with our random gene names:
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summary(iG)
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# This output means: this is an IGRAPH graph, with D = directed edges and N =
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# named nodes, that has 20 nodes and 40 edges. For details, see
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?print.igraph
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mode(iG)
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class(iG)
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# This means an igraph graph object is a special list object; it is opaque in
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# the sense that a user is never expected to modify its components directly, but
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# through a variety of helper functions which the package provides. There are
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# many ways to construct graphs - from adjacency matrices, as we have just done,
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# from edge lists, or by producing random graphs according to a variety of
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# recipes, called _games_ in this package.
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# Two basic functions retrieve nodes "Vertices", and "Edges":
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V(iG)
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E(iG)
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# As with many R objects, loading the package provides special functions that
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# can be accessed via the same name as the basic R functions, for example:
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print(iG)
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plot(iG)
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# ... where plot() allows the usual flexibility of fine-tuning the plot. We
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# first layout the node coordinates with the Fruchtermann-Reingold algorithm - a
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# force-directed layout that applies an ettractive potential along edges (which
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# pulls nodes together) and a repulsive potential to nodes (so they don't
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# overlap). Note the use of the degree() function to color and scale nodes and
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# labels by degree and the use of the V() function to retrieve the vertex names.
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# See ?plot.igraph for details."
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iGxy <- layout_with_fr(iG) # calculate layout coordinates
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# Plot with some customizing parameters
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oPar <- par(mar= rep(0,4)) # Turn margins off
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plot(iG,
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layout = iGxy,
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vertex.color=heat.colors(max(degree(iG)+1))[degree(iG)+1],
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vertex.size = 9 + (2 * degree(iG)),
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vertex.label.cex = 0.5 + (0.05 * degree(iG)),
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edge.arrow.size = 0,
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edge.width = 2,
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vertex.label = toupper(V(iG)$name))
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par(oPar)
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# == Components
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# The igraph function components() tells us whether there are components of the
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# graph in which there is no path to other components.
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components(iG)
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# In the _membership_ vector, nodes are annotatd with the index of the component
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# they are part of. Sui7 is the only node of component 2, Cyj1 is in the third
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# component etc. This is perhaps more clear if we sort by component index
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sort(components(iG)$membership)
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# Retrieving e.g. the members of the first component from the list can be done by subsetting:
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|
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components(iG)$membership == 1 # logical ..
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components(iG)$membership[components(iG)$membership == 1]
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names(components(iG)$membership)[components(iG)$membership == 1]
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# == RANDOM GRAPHS AND GRAPH METRICS =================================
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|
|
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# Let's explore some of the more interesting, topological graph measures. We
|
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# start by building a somewhat bigger graph. We aren't quite sure whether
|
|
# biological graphs are small-world, or random-geometric, or
|
|
# preferential-attachment ... but igraph has ways to simulate the basic ones
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# (and we could easily simulate our own). Look at the following help pages:
|
|
|
|
?sample_gnm # see also sample_gnp for the Erdös-Rényi models
|
|
?sample_smallworld # for the Watts & Strogatz model
|
|
?sample_pa # for the Barabasi-Albert model
|
|
|
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# But note that there are many more sample_ functions. Check out the docs!
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|
|
|
# Let's look at betweenness measures for our first graph: here: the nodes again
|
|
# colored by degree. Degree centrality states: nodes of higher degree are
|
|
# considered to be more central. And that's also the way the force-directed
|
|
# layout drawas them, obviously.
|
|
|
|
set.seed(112358)
|
|
iGxy <- layout_with_fr(iG) # calculate layout coordinates
|
|
oPar <- par(mar= rep(0,4)) # Turn margins off
|
|
plot(iG,
|
|
layout = iGxy,
|
|
rescale = FALSE,
|
|
xlim = c(min(iGxy[,1]), max(iGxy[,1])) * 1.1,
|
|
ylim = c(min(iGxy[,2]), max(iGxy[,2])) * 1.1,
|
|
vertex.color=heat.colors(max(degree(iG)+1))[degree(iG)+1],
|
|
vertex.size = 20 + (10 * degree(iG)),
|
|
vertex.label = Nnames,
|
|
edge.arrow.size = 0)
|
|
par(oPar)
|
|
|
|
# == Diameter
|
|
|
|
diameter(iG) # The diameter of a graph is its maximum length shortest path.
|
|
|
|
# let's plot this path: here are the nodes ...
|
|
get_diameter(iG)
|
|
|
|
# ... and we can get the x, y coordinates from iGxy by subsetting with the node
|
|
# names. The we draw the diameter-path with a transparent, thick pink line:
|
|
lines(iGxy[get_diameter(iG),], lwd=10, col="#ff63a788")
|
|
|
|
# == Centralization scores
|
|
|
|
?centralize
|
|
# replot our graph, and color by log_betweenness:
|
|
|
|
bC <- centr_betw(iG) # calculate betweenness centrality
|
|
nodeBetw <- bC$res
|
|
nodeBetw <- round(log(nodeBetw +1)) + 1
|
|
|
|
oPar <- par(mar= rep(0,4)) # Turn margins off
|
|
plot(iG,
|
|
layout = iGxy,
|
|
rescale = FALSE,
|
|
xlim = c(min(iGxy[,1]), max(iGxy[,1])) * 1.1,
|
|
ylim = c(min(iGxy[,2]), max(iGxy[,2])) * 1.1,
|
|
vertex.color=heat.colors(max(nodeBetw))[nodeBetw],
|
|
vertex.size = 20 + (10 * degree(iG)),
|
|
vertex.label = Nnames,
|
|
edge.arrow.size = 0)
|
|
par(oPar)
|
|
|
|
# Note that the betweenness - the number of shortest paths that pass through a
|
|
# node, is in general higher for high-degree nodes - but not always: Eqr2 has
|
|
# higher betweenness than Itv7: this measure really depends on the detailed
|
|
# local topology of the graph."
|
|
|
|
# Can you use centr_eigen() and centr_degree() to calculate the respective
|
|
# values? That's something I would expect you to be able to do.
|
|
#
|
|
# Lets plot betweenness centrality for our random geometric graph:
|
|
|
|
bCiGRG <- centr_betw(iGRG) # calculate betweenness centrality
|
|
|
|
nodeBetw <- bCiGRG$res
|
|
nodeBetw <- round((log(nodeBetw +1))^2.5) + 1
|
|
|
|
# colours and size proportional to betweenness
|
|
|
|
oPar <- par(mar= rep(0,4)) # Turn margins off
|
|
plot(iGRG,
|
|
layout = iGRGxy,
|
|
rescale = FALSE,
|
|
xlim = c(min(iGRGxy[,1]), max(iGRGxy[,1])),
|
|
ylim = c(min(iGRGxy[,2]), max(iGRGxy[,2])),
|
|
vertex.color=heat.colors(max(nodeBetw))[nodeBetw],
|
|
vertex.size = 0.1 + (0.03 * nodeBetw),
|
|
vertex.label = "",
|
|
edge.arrow.size = 0)
|
|
par(oPar)
|
|
|
|
diameter(iGRG)
|
|
lines(iGRGxy[get_diameter(iGRG),], lwd=10, col="#ff335533")
|
|
|
|
|
|
|
|
# == CLUSTERING ======================================================
|
|
|
|
# Clustering finds "communities" in graphs - and depending what the edges
|
|
# represent, these could be complexes, pathways, biological systems or similar.
|
|
# There are many graph-clustering algorithms. One approach with many attractive
|
|
# properties is the Map Equation, developed by Martin Rosvall. See:
|
|
# http://www.ncbi.nlm.nih.gov/pubmed/18216267 and htttp://www.mapequation.org
|
|
|
|
|
|
iGRGclusters <- cluster_infomap(iGRG)
|
|
modularity(iGRGclusters) # ... measures how separated the different membership
|
|
# types are from each other
|
|
membership(iGRGclusters) # which nodes are in what cluster?
|
|
table(membership(iGRGclusters)) # how large are the clusters?
|
|
|
|
# The largest cluster has 48 members, the second largest has 25, etc.
|
|
|
|
|
|
# Lets plot our graph again, coloring the nodes of the first five communities by
|
|
# their cluster membership:
|
|
|
|
# first, make a vector with as many grey colors as we have communities ...
|
|
commColors <- rep("#f1eef6", max(membership(iGRGclusters)))
|
|
# ... then overwrite the first five with "real colors" - something like rust,
|
|
# lilac, pink, and mauve or so.
|
|
commColors[1:5] <- c("#980043", "#dd1c77", "#df65b0", "#c994c7", "#d4b9da")
|
|
|
|
|
|
oPar <- par(mar= rep(0,4)) # Turn margins off
|
|
plot(iGRG,
|
|
layout = iGRGxy,
|
|
rescale = FALSE,
|
|
xlim = c(min(iGRGxy[,1]), max(iGRGxy[,1])),
|
|
ylim = c(min(iGRGxy[,2]), max(iGRGxy[,2])),
|
|
vertex.color=commColors[membership(iGRGclusters)],
|
|
vertex.size = 0.1 + (0.1 * degree(iGRG)),
|
|
vertex.label = "",
|
|
edge.arrow.size = 0)
|
|
|
|
par(oPar)
|
|
|
|
|
|
# = 1 Tasks
|
|
|
|
|
|
|
|
|
|
# [END]
|