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