Maintenance
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@ -1,11 +1,5 @@
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# tocID <- "FND-MAT-Graphs_and_networks.R"
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#
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# ---------------------------------------------------------------------------- #
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# PATIENCE ... #
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# Do not yet work wih this code. Updates in progress. Thank you. #
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# boris.steipe@utoronto.ca #
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# ---------------------------------------------------------------------------- #
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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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@ -22,7 +16,7 @@
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# 0.1 First code copied from 2016 material.
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#
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#
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# TODO:
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# TODO: Add some explicit tasks and sample solutions.
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#
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#
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# == DO NOT SIMPLY source() THIS FILE! =======================================
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@ -37,28 +31,27 @@
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#TOC> ==========================================================================
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#TOC>
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#TOC> Section Title Line
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#TOC> ------------------------------------------------------------
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#TOC> 1 Review 50
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#TOC> 2 DEGREE DISTRIBUTIONS 204
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#TOC> 2.1 Random graph 210
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#TOC> 2.2 scale-free graph (Barabasi-Albert) 258
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#TOC> 2.3 Random geometric graph 323
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#TOC> 3 A CLOSER LOOK AT THE igraph PACKAGE 445
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#TOC> 3.1 Basics 448
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#TOC> 3.2 Components 520
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#TOC> 4 RANDOM GRAPHS AND GRAPH METRICS 539
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#TOC> 4.1 Diameter 576
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#TOC> 5 GRAPH CLUSTERING 645
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#TOC> --------------------------------------------------------------
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#TOC> 1 REVIEW 50
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#TOC> 2 DEGREE DISTRIBUTIONS 206
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#TOC> 2.1 Random graph 212
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#TOC> 2.2 scale-free graph (Barabasi-Albert) 260
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#TOC> 2.3 Random geometric graph 326
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#TOC> 3 A CLOSER LOOK AT THE igraph:: PACKAGE 448
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#TOC> 3.1 Basics 451
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#TOC> 3.2 Components 523
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#TOC> 4 RANDOM GRAPHS AND GRAPH METRICS 542
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#TOC> 4.1 Diameter 579
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#TOC> 5 GRAPH CLUSTERING 648
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#TOC>
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#TOC> ==========================================================================
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# = 1 Review ==============================================================
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# = 1 REVIEW ==============================================================
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# This tutorial covers basic concepts of graph theory and analysis in R. Make
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# sure you have pulled the latest version of the project from the GitHub
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# repository, and that you have typed init() to load some utility functions and
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# data.
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# repository.
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# Let's explore some of the basic ideas of graph theory by starting with a small
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# random graph.
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@ -90,11 +83,11 @@ set.seed(NULL) # reset the RNG
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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.
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# Let's create an adjacency matrix for random graph: let's say a pair of nodes
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# has probability p <- 0.1 to have an edge, and our graph is symmetric , i.e. it
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# is an undirected graph, and it has neither self-edges, i.e. loops, nor
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# multiple edges between the same nodes, i.e. it is a "simple" graph. We use our
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# the Nnames vector as node labels.
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# Let's create an adjacency matrix for our random graph: let's say a pair
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# of nodes has probability p <- 0.09 to have an edge, and our graph is
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# symmetric , i.e. it is an undirected graph, and it has neither self-edges,
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# i.e. loops, nor multiple edges between the same nodes, i.e. it
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# is a "simple" graph. We use the Nnames vector as node labels.
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makeRandomAM <- function(nam, p = 0.1) {
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# Make a random adjacency matrix for a set of nodes with edge probability p
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@ -106,11 +99,11 @@ makeRandomAM <- function(nam, p = 0.1) {
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#
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N <- length(nam)
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AM <- matrix(numeric(N * N), ncol = N) # The adjacency matrix
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AM <- matrix(numeric(N * N), ncol = N) # Adjacency matrix, pre-filled with 0
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rownames(AM) <- nam
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colnames(AM) <- nam
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for (iRow in 1:(N-1)) { # Note how we make sure iRow != iCol - this prevents
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# loops
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# loops (self-edges)
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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. The expression is TRUE with
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@ -128,8 +121,8 @@ set.seed(NULL) # reset the RNG
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# Listing the matrix is not very informative - we should plot this graph. The
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# standard package for work with graphs in r is "igraph". We'll go into more
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# details of the igraph package a bit later, for now we just use it to plot:
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# standard package for work with graphs in r is "igraph::". We'll go into more
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# details of the igraph:: package a bit later, for now we just use it to plot:
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if (! requireNamespace("igraph", quietly = TRUE)) {
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install.packages("igraph")
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@ -149,7 +142,7 @@ myGxy <- igraph::layout_with_graphopt(myG,
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set.seed(NULL) # reset the RNG
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# The igraph package adds its own function to the collection of plot()
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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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@ -160,14 +153,14 @@ set.seed(NULL) # reset the RNG
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# See ?igraph.plotting for the complete list of parameters
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oPar <- par(mar= rep(0,4)) # Turn margins off
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oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
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plot(myG,
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layout = myGxy,
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rescale = FALSE,
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xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
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ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
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vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1],
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vertex.size = 1600 + (300 * igraph::degree(myG)),
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vertex.size = 1000 + (900 * igraph::degree(myG)),
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vertex.label = sprintf("%s(%i)", names(igraph::V(myG)), igraph::degree(myG)),
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vertex.label.family = "sans",
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vertex.label.cex = 0.7)
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@ -202,10 +195,13 @@ axis(side = 1, at = 0:7)
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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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# A degree distribution is actually a very informative 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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# network formed? Asking about the mechanism, the way the interactions
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# support function and shape the fitness landscape in which the cell evolves,
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# connects our mathematical abstraction with biological insight.
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# = 2 DEGREE DISTRIBUTIONS ================================================
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@ -225,14 +221,14 @@ myG200 <- igraph::graph_from_adjacency_matrix(my200AM, mode = "undirected")
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myGxy <- igraph::layout_with_graphopt(myG200, charge=0.0001) # calculate layout
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# coordinates
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oPar <- par(mar= rep(0,4)) # Turn margins off, save graphics state
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oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off, save graphics state
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plot(myG200,
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layout = myGxy,
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rescale = FALSE,
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xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
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ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
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vertex.color=heat.colors(max(igraph::degree(myG200)+1))[igraph::degree(myG200)+1],
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vertex.size = 150 + (60 * igraph::degree(myG200)),
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vertex.size = 150 + (70 * igraph::degree(myG200)),
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vertex.label = NA)
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par(oPar) # restore graphics state
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@ -264,9 +260,9 @@ plot(log10(as.numeric(names(freqRank)) + 1),
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# == 2.2 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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# 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". Preferential attachment is one type of
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# stands for "preferential attachment". Preferential attachment is _one_ type of
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# process that will yield scale-free distributions.
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N <- 200
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@ -277,7 +273,7 @@ set.seed(NULL) # reset the RNG
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GBAxy <- igraph::layout_with_graphopt(GBA, charge=0.0001)
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oPar <- par(mar= rep(0,4)) # Turn margins off, save graphics state
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oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off, save graphics state
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plot(GBA,
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layout = GBAxy,
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rescale = FALSE,
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@ -314,9 +310,10 @@ plot(log10(as.numeric(names(freqRank)) + 1),
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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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# one. That behaviour smooths out in larger graphs - let's try 100000 nodes:
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#
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X <- igraph::sample_pa(1e5, power = 0.8, directed = FALSE) # 100,000 nodes
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N <- 10000
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X <- igraph::sample_pa(N, power = 0.8, directed = FALSE) # 100,000 nodes
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freqRank <- table(igraph::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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@ -336,16 +333,16 @@ rm(X)
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# probability for two nodes to have an edge to be a function of their Euclidian
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# distance in the box.
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# Here is a function that makes an adjacency matrix for such graphs. iGraph has
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# a similar function, sample_grg(), which connects nodes that are closer than a
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# cutoff, the function I give you below is a bit more interesting since it
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# creates edges according to a probability that is determined by a generalized
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# logistic function of the distance. This sigmoidal function gives a smooth
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# cutoff and creates more "natural" graphs. Otherwise, the function is very
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# similar to the random graph function, except that we output the "coordinates"
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# of the nodes together with the adjacency matrix which we then use for the
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# layout. list() FTW.
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#
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# Here is a function that makes an adjacency matrix for such graphs. iGraph::
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# has a similar function, sample_grg(), which connects nodes that are closer
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# than a cutoff, the function I give you below is a bit more interesting since
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# it creates edges according to a probability that is determined by a
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# generalized logistic function of the distance. This sigmoidal function gives
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# a smooth cutoff and creates more "natural" graphs. Otherwise, the function is
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# very similar to the random graph function, except that we output the
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# "coordinates" of the nodes together with the adjacency matrix which we then
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# use for the layout. list() FTW.
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makeRandomGeometricAM <- function(nam, B = 25, Q = 0.001, t = 0.6) {
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# Make an adjacency matrix for an undirected random geometric graph from
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@ -400,7 +397,7 @@ makeRandomGeometricAM <- function(nam, B = 25, Q = 0.001, t = 0.6) {
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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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# our nodes: from 0 to sqrt(2) i.e. the diagonal of the unit square 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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@ -415,7 +412,7 @@ set.seed(NULL) # reset the RNG
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myGRG <- igraph::graph_from_adjacency_matrix(rGAM$mat, mode = "undirected")
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oPar <- par(mar= rep(0,4)) # Turn margins off
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oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
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plot(myGRG,
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layout = cbind(rGAM$x, rGAM$y), # use our node coordinates for layout,
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rescale = FALSE,
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@ -448,12 +445,12 @@ plot(log10(as.numeric(names(freqRank)) + 1),
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# = 3 A CLOSER LOOK AT THE igraph PACKAGE =================================
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# = 3 A CLOSER LOOK AT THE igraph:: PACKAGE ===============================
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# == 3.1 Basics ============================================================
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# The basic object of the igraph package is a graph object. Let's explore the
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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(myG)
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@ -464,7 +461,7 @@ summary(myG)
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mode(myG)
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class(myG)
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# This means an igraph graph object is a special list object; it is opaque in
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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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@ -494,12 +491,12 @@ plot(myG) # this is the result of default plot parameters
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# See ?plot.igraph for details."
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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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oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
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plot(myG,
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layout = igraph::layout_with_fr(myG),
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vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1],
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vertex.size = 9 + (2 * igraph::degree(myG)),
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vertex.label.cex = 0.5 + (0.05 * igraph::degree(myG)),
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vertex.size = 12 + (5 * igraph::degree(myG)),
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vertex.label.cex = 0.4 + (0.15 * igraph::degree(myG)),
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edge.width = 2,
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vertex.label = igraph::V(myG)$name,
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vertex.label.family = "sans",
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@ -507,25 +504,25 @@ plot(myG,
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par(oPar)
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# ... or with a different layout:
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oPar <- par(mar= rep(0,4)) # Turn margins off
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oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
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plot(myG,
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layout = igraph::layout_in_circle(myG),
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vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1],
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vertex.size = 9 + (2 * igraph::degree(myG)),
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vertex.label.cex = 0.5 + (0.05 * igraph::degree(myG)),
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vertex.size = 12 + (5 * igraph::degree(myG)),
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vertex.label.cex = 0.4 + (0.15 * igraph::degree(myG)),
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edge.width = 2,
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vertex.label = igraph::V(myG)$name,
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vertex.label.family = "sans",
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vertex.label.cex = 0.9)
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par(oPar)
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# igraph has a large number of graph-layout functions: see
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# igraph:: has a large number of graph-layout functions: see
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# ?layout_ and try them all.
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# == 3.2 Components ========================================================
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# The igraph function components() tells us whether there are components of the
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# The function igrap::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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igraph::components(myG)
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@ -548,7 +545,7 @@ names(c1)
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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
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# biological graphs are small-world, or random-geometric, or
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# preferential-attachment ... but igraph has ways to simulate the basic ones
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# 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:
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?igraph::sample_gnm # see also sample_gnp for the Erdös-Rényi models
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@ -573,10 +570,10 @@ plot(myG,
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xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
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ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
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vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1],
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vertex.size = 20 + (10 * igraph::degree(myG)),
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vertex.size = 40 + (15 * igraph::degree(myG)),
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vertex.label.cex = 0.4 + (0.15 * igraph::degree(myG)),
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vertex.label = igraph::V(myG)$name,
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vertex.label.family = "sans",
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vertex.label.cex = 0.8)
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vertex.label.family = "sans")
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par(oPar) # restore graphics state
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# == 4.1 Diameter ==========================================================
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@ -600,17 +597,17 @@ bC <- igraph::centr_betw(myG) # calculate betweenness centrality
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nodeBetw <- bC$res
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nodeBetw <- round(log(nodeBetw +1)) + 1
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oPar <- par(mar= rep(0,4)) # Turn margins off
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oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
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plot(myG,
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layout = myGxy,
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rescale = FALSE,
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xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
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ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
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vertex.color=heat.colors(max(nodeBetw))[nodeBetw],
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vertex.size = 20 + (10 * igraph::degree(myG)),
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vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1],
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vertex.size = 40 + (15 * igraph::degree(myG)),
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vertex.label.cex = 0.4 + (0.15 * igraph::degree(myG)),
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vertex.label = igraph::V(myG)$name,
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vertex.label.family = "sans",
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vertex.label.cex = 0.7)
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vertex.label.family = "sans")
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par(oPar)
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# Note that the betweenness - the number of shortest paths that pass through a
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@ -629,7 +626,7 @@ nodeBetw <- bCmyGRG$res
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nodeBetw <- round((log(nodeBetw +1))^2.5) + 1
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# colours and size proportional to betweenness
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oPar <- par(mar= rep(0,4)) # Turn margins off
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oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
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plot(myGRG,
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layout = cbind(rGAM$x, rGAM$y), # use our node coordinates for layout,
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rescale = FALSE,
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@ -677,18 +674,18 @@ commColors <- rep("#f1eef6", max(igraph::membership(myGRGclusters)))
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commColors[1:5] <- c("#980043", "#dd1c77", "#df65b0", "#c994c7", "#d4b9da")
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oPar <- par(mar= rep(0,4)) # Turn margins off
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oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
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plot(myGRG,
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layout = cbind(rGAM$x, rGAM$y),
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rescale = FALSE,
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xlim = c(min(rGAM$x) * 0.9, max(rGAM$x) * 1.1),
|
||||
ylim = c(min(rGAM$y) * 0.9, max(rGAM$y) * 1.1),
|
||||
vertex.color=commColors[igraph::membership(myGRGclusters)],
|
||||
vertex.size = 0.1 + (0.1 * igraph::degree(myGRG)),
|
||||
vertex.size = 0.3 + (0.4 * igraph::degree(myGRG)),
|
||||
vertex.label = NA)
|
||||
par(oPar)
|
||||
|
||||
|
||||
# That's all.
|
||||
|
||||
|
||||
# [END]
|
||||
|
||||
Loading…
Reference in New Issue
Block a user