665 lines
25 KiB
R
665 lines
25 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: 1.0
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#
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# Date: 2017 10 06
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# Author: Boris Steipe (boris.steipe@utoronto.ca)
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#
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# Versions:
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# 1.0 First final version for learning units.
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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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#
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#
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# == DO NOT SIMPLY source() THIS FILE! =======================================
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#
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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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# ==============================================================================
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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 48
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#TOC> 2 DEGREE DISTRIBUTIONS 192
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#TOC> 2.1 Random graph 198
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#TOC> 2.2 scale-free graph (Barabasi-Albert) 242
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#TOC> 2.3 Random geometric graph 304
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#TOC> 3 A CLOSER LOOK AT THE igraph PACKAGE 424
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#TOC> 3.1 Basics 427
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#TOC> 3.2 Components 499
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#TOC> 4 RANDOM GRAPHS AND GRAPH METRICS 518
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#TOC> 4.1 Diameter 553
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#TOC> 5 GRAPH CLUSTERING 621
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#TOC>
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#TOC> ==========================================================================
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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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# 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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# 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 "biological ...
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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 <- paste0(c(sample(LETTERS, 1), sample(letters, 2)),
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collapse="") # one uppercase, two lowercase 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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# 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.
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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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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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# Parameters:
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# nam: a character vector of unique node names.
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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 for a simple, undirected graph
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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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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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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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# probability p. if it is TRUE ...
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AM[iRow, iCol] <- 1 # ... record an edge for the pair (iRow, iCol)
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}
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}
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}
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return(AM)
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}
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set.seed(112358)
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(myRandAM <- makeRandomAM(Nnames, p = 0.09))
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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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if (!require(igraph)) {
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install.packages("igraph")
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library(igraph)
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}
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myG <- graph_from_adjacency_matrix(myRandAM, mode = "undirected")
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set.seed(112358)
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myGxy <- layout_with_graphopt(myG, charge=0.0012) # 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 combine the names of the vertices of the
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# graph - names(V(iG)) - with the node degree - degree(iG).
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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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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(degree(myG)+1))[degree(myG)+1],
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vertex.size = 1600 + (300 * degree(myG)),
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vertex.label = sprintf("%s(%i)", names(V(myG)), degree(myG)),
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vertex.label.family = "sans",
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vertex.label.cex = 0.7)
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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(myRandAM)
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# Is that what you expect?
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# What about the degree distribution? We can get that simply by summing over the
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# rows and summing over the columns and adding the two vectors.
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rowSums(myRandAM) + colSums(myRandAM) # check this against the plot!
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# The function degree() gives the same values
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degree(myG)
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# Let's plot the degree distribution in a histogram:
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degG <- degree(myG)
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brk <- seq(min(degG)-0.5, max(degG)+0.5, by=1) # define histogram breaks
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hist(degG, 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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# = 2 DEGREE DISTRIBUTIONS ================================================
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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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# == 2.1 Random graph ======================================================
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set.seed(31415927)
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my200AM <- makeRandomAM(as.character(1:200), p = 0.015)
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myG200 <- graph_from_adjacency_matrix(my200AM, mode = "undirected")
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myGxy <- layout_with_graphopt(myG200, charge=0.0001) # calculate layout coordinates
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oPar <- par(mar= rep(0,4)) # Turn margins off
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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(degree(myG200)+1))[degree(myG200)+1],
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vertex.size = 150 + (60 * degree(myG200)),
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vertex.label = NA)
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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(myG200)
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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="#A5F5CC",
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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 pronounced peak of this distribution: this is not "scale-free".
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# Here is the 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 = "#A5F5CC",
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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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# ... which shows us that this does NOT correspond to the single-slope linear
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# relationship that we expect for a "scale-free" graph.
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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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# 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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# process that will yield scale-free distributions.
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set.seed(31415927)
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GBA <- sample_pa(200, power = 0.8, directed = FALSE)
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GBAxy <- 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 = GBAxy,
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rescale = FALSE,
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xlim = c(min(GBAxy[,1]) * 0.99, max(GBAxy[,1]) * 1.01),
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ylim = c(min(GBAxy[,2]) * 0.99, max(GBAxy[,2]) * 1.01),
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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 = NA)
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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". Play with the "power" parameter and get a sense for what difference
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# this makes. Also: note that the graph has only a single component - no
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# singletons.
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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="#DCF5B5",
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xlim = c(0,max(dg)+1), 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, max(dg)+1, by=5))
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# Most nodes have a degree of 1, but one node has a degree of 19.
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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 = "#DCF5B5",
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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, directed = FALSE) # 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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# == 2.3 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 entirely 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 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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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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# edges connected with probabilities according to a generalized logistic
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# function.
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# Parameters:
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# nam: a character vector of unique names
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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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# AM$mat : an adjacency matrix
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# AM$nam : labels for the nodes
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# AM$x : x-coordinates for the nodes
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# AM$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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AM <- list()
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AM$nam <- nam
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N <- length(AM$nam)
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AM$mat <- matrix(numeric(N * N), ncol = N) # The adjacency matrix
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rownames(AM$mat) <- AM$nam
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colnames(AM$mat) <- AM$nam
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AM$x <- runif(N) # Randomly place nodes into the unit square
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AM$y <- runif(N)
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for (iRow in 1:(N-1)) { # Same principles as in makeRandomGraph()
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for (iCol in (iRow+1):N) {
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# geometric distance ...
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d <- sqrt((AM$x[iRow] - AM$x[iCol])^2 +
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(AM$y[iRow] - AM$y[iCol])^2) # Pythagoras
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# distance dependent probability
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p <- 1 - 1/((1 + (Q * (exp(-B * (d-t)))))^(1 / nu))
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if (runif(1) < p) {
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AM$mat[iRow, iCol] <- 1
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}
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}
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}
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return(AM)
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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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rGAM <- makeRandomGeometricAM(as.character(1:200), t=0.4)
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myGRG <- graph_from_adjacency_matrix(rGAM$mat, mode = "undirected")
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oPar <- par(mar= rep(0,4)) # Turn margins off
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plot(myGRG,
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layout = cbind(rGAM$x, rGAM$y), # use our node coordinates for layout,
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rescale = FALSE,
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xlim = c(min(rGAM$x) * 0.9, max(rGAM$x) * 1.1),
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ylim = c(min(rGAM$y) * 0.9, max(rGAM$y) * 1.1),
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vertex.color=heat.colors(max(degree(myGRG)+1))[degree(myGRG)+1],
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vertex.size = 0.1 + (0.2 * degree(myGRG)),
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vertex.label = NA)
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par(oPar)
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# degree distribution:
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(dg <- degree(myGRG))
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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 = "#FCC6D2",
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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 = "#FCC6D2",
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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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# = 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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# first graph some more, the one we built with our random gene names:
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summary(myG)
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# This output means: this is an IGRAPH graph, with U = UN-directed edges
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# and N = named nodes, that has 20 nodes and 20 edges. For details, see
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?print.igraph
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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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# 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(myG)
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E(myG)
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# additional properties can be retrieved from the Vertices ...
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V(myG)$name
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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(myG)
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plot(myG) # this is the result of default plot parameters
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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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# Plot with some customizing parameters
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oPar <- par(mar= rep(0,4)) # Turn margins off
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plot(myG,
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layout = layout_with_fr(myG),
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vertex.color=heat.colors(max(degree(myG)+1))[degree(myG)+1],
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vertex.size = 9 + (2 * degree(myG)),
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vertex.label.cex = 0.5 + (0.05 * degree(myG)),
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edge.width = 2,
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vertex.label = V(myG)$name,
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vertex.label.family = "sans",
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vertex.label.cex = 0.9)
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par(oPar)
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# ... or with a different layout:
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oPar <- par(mar= rep(0,4)) # Turn margins off
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plot(myG,
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layout = layout_in_circle(myG),
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vertex.color=heat.colors(max(degree(myG)+1))[degree(myG)+1],
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vertex.size = 9 + (2 * degree(myG)),
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vertex.label.cex = 0.5 + (0.05 * degree(myG)),
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edge.width = 2,
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vertex.label = V(myG)$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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|
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|
# igraph has a large number of graph-layout functions: see
|
|
# ?layout_ and try them all.
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|
|
|
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|
# == 3.2 Components ========================================================
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|
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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.
|
|
components(myG)
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|
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|
# In the _membership_ vector, nodes are annotated 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(myG)$membership, decreasing = TRUE)
|
|
|
|
# Retrieving e.g. the members of the first component from the list can be done by subsetting:
|
|
|
|
(sel <- components(myG)$membership == 1) # boolean vector ..
|
|
(c1 <- components(myG)$membership[sel])
|
|
names(c1)
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|
|
|
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|
# = 4 RANDOM GRAPHS AND GRAPH METRICS =====================================
|
|
|
|
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|
# Let's explore some of the more interesting, topological graph measures. We
|
|
# start by building a somewhat bigger graph. We aren't quite sure whether
|
|
# biological graphs are small-world, or random-geometric, or
|
|
# preferential-attachment ... but igraph has ways to simulate the basic ones
|
|
# (and we could easily simulate our own). Look at the following help pages:
|
|
|
|
?sample_gnm # see also sample_gnp for the Erdös-Rényi models
|
|
?sample_smallworld # for the Watts & Strogatz model
|
|
?sample_pa # for the Barabasi-Albert model
|
|
|
|
# But note that there are many more sample_ functions. Check out the docs!
|
|
|
|
# Let's look at betweenness measures for our first graph. Here: the nodes again
|
|
# colored by degree. Degree centrality states: nodes of higher degree are
|
|
# considered to be more central. And that's also the way the force-directed
|
|
# layout drawas them, obviously.
|
|
|
|
set.seed(112358)
|
|
myGxy <- layout_with_fr(myG) # calculate layout coordinates
|
|
oPar <- par(mar= rep(0,4)) # Turn margins off
|
|
plot(myG,
|
|
layout = myGxy,
|
|
rescale = FALSE,
|
|
xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
|
|
ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
|
|
vertex.color=heat.colors(max(degree(myG)+1))[degree(myG)+1],
|
|
vertex.size = 20 + (10 * degree(myG)),
|
|
vertex.label = V(myG)$name,
|
|
vertex.label.family = "sans",
|
|
vertex.label.cex = 0.8)
|
|
par(oPar)
|
|
|
|
# == 4.1 Diameter ==========================================================
|
|
|
|
diameter(myG) # The diameter of a graph is its maximum length shortest path.
|
|
|
|
# let's plot this path: here are the nodes ...
|
|
get_diameter(myG)
|
|
|
|
# ... 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(myGxy[get_diameter(myG),], lwd=10, col="#ff63a788")
|
|
|
|
# == Centralization scores
|
|
|
|
?centralize
|
|
# replot our graph, and color by log_betweenness:
|
|
|
|
bC <- centr_betw(myG) # calculate betweenness centrality
|
|
nodeBetw <- bC$res
|
|
nodeBetw <- round(log(nodeBetw +1)) + 1
|
|
|
|
oPar <- par(mar= rep(0,4)) # Turn margins off
|
|
plot(myG,
|
|
layout = myGxy,
|
|
rescale = FALSE,
|
|
xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
|
|
ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
|
|
vertex.color=heat.colors(max(nodeBetw))[nodeBetw],
|
|
vertex.size = 20 + (10 * degree(myG)),
|
|
vertex.label = V(myG)$name,
|
|
vertex.label.family = "sans",
|
|
vertex.label.cex = 0.7)
|
|
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:
|
|
|
|
bCmyGRG <- centr_betw(myGRG) # calculate betweenness centrality
|
|
|
|
nodeBetw <- bCmyGRG$res
|
|
nodeBetw <- round((log(nodeBetw +1))^2.5) + 1
|
|
|
|
# colours and size proportional to betweenness
|
|
oPar <- par(mar= rep(0,4)) # Turn margins off
|
|
plot(myGRG,
|
|
layout = cbind(rGAM$x, rGAM$y), # use our node coordinates for layout,
|
|
rescale = FALSE,
|
|
xlim = c(min(rGAM$x) * 0.9, max(rGAM$x) * 1.1),
|
|
ylim = c(min(rGAM$y) * 0.9, max(rGAM$y) * 1.1),
|
|
vertex.color=heat.colors(max(nodeBetw))[nodeBetw],
|
|
vertex.size = 0.1 + (0.03 * nodeBetw),
|
|
vertex.label = NA)
|
|
par(oPar)
|
|
|
|
diameter(myGRG)
|
|
lines(rGAM$x[get_diameter(myGRG)],
|
|
rGAM$y[get_diameter(myGRG)],
|
|
lwd = 10,
|
|
col = "#ff335533")
|
|
|
|
|
|
|
|
# = 5 GRAPH CLUSTERING ====================================================
|
|
|
|
|
|
# Clustering finds "communities" in graphs - and depending what the edges
|
|
# represent, these could be complexes, pathways, biological systems or similar.
|
|
# There are many graph-clustering algorithms. One approach with many attractive
|
|
# properties is the Map Equation, developed by Martin Rosvall. See:
|
|
# http://www.ncbi.nlm.nih.gov/pubmed/18216267 and htttp://www.mapequation.org
|
|
|
|
|
|
myGRGclusters <- cluster_infomap(myGRG)
|
|
modularity(myGRGclusters) # ... measures how separated the different membership
|
|
# types are from each other
|
|
membership(myGRGclusters) # which nodes are in what cluster?
|
|
table(membership(myGRGclusters)) # how large are the clusters?
|
|
|
|
# The largest cluster has 48 members, the second largest has 25, etc.
|
|
|
|
|
|
# Lets plot our graph again, coloring the nodes of the first five communities by
|
|
# their cluster membership:
|
|
|
|
# first, make a vector with as many grey colors as we have communities ...
|
|
commColors <- rep("#f1eef6", max(membership(myGRGclusters)))
|
|
# ... then overwrite the first five with "real colors" - something like rust,
|
|
# lilac, pink, and mauve or so.
|
|
commColors[1:5] <- c("#980043", "#dd1c77", "#df65b0", "#c994c7", "#d4b9da")
|
|
|
|
|
|
oPar <- par(mar= rep(0,4)) # Turn margins off
|
|
plot(myGRG,
|
|
layout = cbind(rGAM$x, rGAM$y),
|
|
rescale = FALSE,
|
|
xlim = c(min(rGAM$x) * 0.9, max(rGAM$x) * 1.1),
|
|
ylim = c(min(rGAM$y) * 0.9, max(rGAM$y) * 1.1),
|
|
vertex.color=commColors[membership(myGRGclusters)],
|
|
vertex.size = 0.1 + (0.1 * degree(myGRG)),
|
|
vertex.label = NA)
|
|
par(oPar)
|
|
|
|
|
|
|
|
|
|
# [END]
|