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FND-MAT-Graphs_and_networks.R
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@ -1,11 +1,5 @@
# tocID <- "FND-MAT-Graphs_and_networks.R" # tocID <- "FND-MAT-Graphs_and_networks.R"
# #
# ---------------------------------------------------------------------------- #
# PATIENCE ... #
# Do not yet work wih this code. Updates in progress. Thank you. #
# boris.steipe@utoronto.ca #
# ---------------------------------------------------------------------------- #
#
# Purpose: A Bioinformatics Course: # Purpose: A Bioinformatics Course:
# R code accompanying the FND-MAT-Graphs_and_networks unit. # R code accompanying the FND-MAT-Graphs_and_networks unit.
# #
@ -22,7 +16,7 @@
# 0.1 First code copied from 2016 material. # 0.1 First code copied from 2016 material.
# #
# #
# TODO: # TODO: Add some explicit tasks and sample solutions.
# #
# #
# == DO NOT SIMPLY source() THIS FILE! ======================================= # == DO NOT SIMPLY source() THIS FILE! =======================================
@ -37,28 +31,27 @@
#TOC> ========================================================================== #TOC> ==========================================================================
#TOC> #TOC>
#TOC> Section Title Line #TOC> Section Title Line
#TOC> ------------------------------------------------------------ #TOC> --------------------------------------------------------------
#TOC> 1 Review 50 #TOC> 1 REVIEW 50
#TOC> 2 DEGREE DISTRIBUTIONS 204 #TOC> 2 DEGREE DISTRIBUTIONS 206
#TOC> 2.1 Random graph 210 #TOC> 2.1 Random graph 212
#TOC> 2.2 scale-free graph (Barabasi-Albert) 258 #TOC> 2.2 scale-free graph (Barabasi-Albert) 260
#TOC> 2.3 Random geometric graph 323 #TOC> 2.3 Random geometric graph 326
#TOC> 3 A CLOSER LOOK AT THE igraph PACKAGE 445 #TOC> 3 A CLOSER LOOK AT THE igraph:: PACKAGE 448
#TOC> 3.1 Basics 448 #TOC> 3.1 Basics 451
#TOC> 3.2 Components 520 #TOC> 3.2 Components 523
#TOC> 4 RANDOM GRAPHS AND GRAPH METRICS 539 #TOC> 4 RANDOM GRAPHS AND GRAPH METRICS 542
#TOC> 4.1 Diameter 576 #TOC> 4.1 Diameter 579
#TOC> 5 GRAPH CLUSTERING 645 #TOC> 5 GRAPH CLUSTERING 648
#TOC> #TOC>
#TOC> ========================================================================== #TOC> ==========================================================================
# = 1 Review ============================================================== # = 1 REVIEW ==============================================================
# This tutorial covers basic concepts of graph theory and analysis in R. Make # This tutorial covers basic concepts of graph theory and analysis in R. Make
# sure you have pulled the latest version of the project from the GitHub # sure you have pulled the latest version of the project from the GitHub
# repository, and that you have typed init() to load some utility functions and # repository.
# data.
# Let's explore some of the basic ideas of graph theory by starting with a small # Let's explore some of the basic ideas of graph theory by starting with a small
# random graph. # random graph.
@ -90,11 +83,11 @@ set.seed(NULL) # reset the RNG
# intersection of a row and column contains a value/TRUE if there is an edge, # intersection of a row and column contains a value/TRUE if there is an edge,
# 0/FALSE otherwise. # 0/FALSE otherwise.
# Let's create an adjacency matrix for random graph: let's say a pair of nodes # Let's create an adjacency matrix for our random graph: let's say a pair
# has probability p <- 0.1 to have an edge, and our graph is symmetric , i.e. it # of nodes has probability p <- 0.09 to have an edge, and our graph is
# is an undirected graph, and it has neither self-edges, i.e. loops, nor # symmetric , i.e. it is an undirected graph, and it has neither self-edges,
# multiple edges between the same nodes, i.e. it is a "simple" graph. We use our # i.e. loops, nor multiple edges between the same nodes, i.e. it
# the Nnames vector as node labels. # is a "simple" graph. We use the Nnames vector as node labels.
makeRandomAM <- function(nam, p = 0.1) { makeRandomAM <- function(nam, p = 0.1) {
# Make a random adjacency matrix for a set of nodes with edge probability p # Make a random adjacency matrix for a set of nodes with edge probability p
@ -106,11 +99,11 @@ makeRandomAM <- function(nam, p = 0.1) {
# #
N <- length(nam) N <- length(nam)
AM <- matrix(numeric(N * N), ncol = N) # The adjacency matrix AM <- matrix(numeric(N * N), ncol = N) # Adjacency matrix, pre-filled with 0
rownames(AM) <- nam rownames(AM) <- nam
colnames(AM) <- nam colnames(AM) <- nam
for (iRow in 1:(N-1)) { # Note how we make sure iRow != iCol - this prevents for (iRow in 1:(N-1)) { # Note how we make sure iRow != iCol - this prevents
# loops # loops (self-edges)
for (iCol in (iRow+1):N) { for (iCol in (iRow+1):N) {
if (runif(1) < p) { # runif() creates uniform random numbers if (runif(1) < p) { # runif() creates uniform random numbers
# between 0 and 1. The expression is TRUE with # between 0 and 1. The expression is TRUE with
@ -128,8 +121,8 @@ set.seed(NULL) # reset the RNG
# Listing the matrix is not very informative - we should plot this graph. The # Listing the matrix is not very informative - we should plot this graph. The
# standard package for work with graphs in r is "igraph". We'll go into more # standard package for work with graphs in r is "igraph::". We'll go into more
# details of the igraph package a bit later, for now we just use it to plot: # details of the igraph:: package a bit later, for now we just use it to plot:
if (! requireNamespace("igraph", quietly = TRUE)) { if (! requireNamespace("igraph", quietly = TRUE)) {
install.packages("igraph") install.packages("igraph")
@ -149,7 +142,7 @@ myGxy <- igraph::layout_with_graphopt(myG,
set.seed(NULL) # reset the RNG set.seed(NULL) # reset the RNG
# The igraph package adds its own function to the collection of plot() # 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 # 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 # of the object that we request to plot. This plot function has parameters
# layout - the x,y coordinates of the nodes; # layout - the x,y coordinates of the nodes;
@ -160,14 +153,14 @@ set.seed(NULL) # reset the RNG
# See ?igraph.plotting for the complete list of parameters # See ?igraph.plotting for the complete list of parameters
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
plot(myG, plot(myG,
layout = myGxy, layout = myGxy,
rescale = FALSE, rescale = FALSE,
xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01), xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01), ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1], vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1],
vertex.size = 1600 + (300 * igraph::degree(myG)), vertex.size = 1000 + (900 * igraph::degree(myG)),
vertex.label = sprintf("%s(%i)", names(igraph::V(myG)), igraph::degree(myG)), vertex.label = sprintf("%s(%i)", names(igraph::V(myG)), igraph::degree(myG)),
vertex.label.family = "sans", vertex.label.family = "sans",
vertex.label.cex = 0.7) vertex.label.cex = 0.7)
@ -202,10 +195,13 @@ axis(side = 1, at = 0:7)
# quite well, automatically. But for this purpose I want the columns of the # quite well, automatically. But for this purpose I want the columns of the
# histogram to represent exactly one node-degree difference. # histogram to represent exactly one node-degree difference.
# A degree distribution is actually quite an important descriptor of graphs, # A degree distribution is actually a very informative descriptor of graphs,
# since it is very sensitive to the generating mechanism. For biological # 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 # networks, that is one of the key questions we are interested in: how was the
# network formed? # network formed? Asking about the mechanism, the way the interactions
# support function and shape the fitness landscape in which the cell evolves,
# connects our mathematical abstraction with biological insight.
# = 2 DEGREE DISTRIBUTIONS ================================================ # = 2 DEGREE DISTRIBUTIONS ================================================
@ -225,14 +221,14 @@ myG200 <- igraph::graph_from_adjacency_matrix(my200AM, mode = "undirected")
myGxy <- igraph::layout_with_graphopt(myG200, charge=0.0001) # calculate layout myGxy <- igraph::layout_with_graphopt(myG200, charge=0.0001) # calculate layout
# coordinates # coordinates
oPar <- par(mar= rep(0,4)) # Turn margins off, save graphics state oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off, save graphics state
plot(myG200, plot(myG200,
layout = myGxy, layout = myGxy,
rescale = FALSE, rescale = FALSE,
xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01), xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01), ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
vertex.color=heat.colors(max(igraph::degree(myG200)+1))[igraph::degree(myG200)+1], vertex.color=heat.colors(max(igraph::degree(myG200)+1))[igraph::degree(myG200)+1],
vertex.size = 150 + (60 * igraph::degree(myG200)), vertex.size = 150 + (70 * igraph::degree(myG200)),
vertex.label = NA) vertex.label = NA)
par(oPar) # restore graphics state par(oPar) # restore graphics state
@ -264,9 +260,9 @@ plot(log10(as.numeric(names(freqRank)) + 1),
# == 2.2 scale-free graph (Barabasi-Albert) ================================ # == 2.2 scale-free graph (Barabasi-Albert) ================================
# What does one of those intriguing "scale-free" distributions look like? The # What does one of those intriguing "scale-free" distributions look like? The
# iGraph package has a function to make random graphs according to 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 # Barabasi-Albert model of scale-free graphs. It is: sample_pa(), where pa
# stands for "preferential attachment". Preferential attachment is one type of # stands for "preferential attachment". Preferential attachment is _one_ type of
# process that will yield scale-free distributions. # process that will yield scale-free distributions.
N <- 200 N <- 200
@ -277,7 +273,7 @@ set.seed(NULL) # reset the RNG
GBAxy <- igraph::layout_with_graphopt(GBA, charge=0.0001) GBAxy <- igraph::layout_with_graphopt(GBA, charge=0.0001)
oPar <- par(mar= rep(0,4)) # Turn margins off, save graphics state oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off, save graphics state
plot(GBA, plot(GBA,
layout = GBAxy, layout = GBAxy,
rescale = FALSE, rescale = FALSE,
@ -314,9 +310,10 @@ plot(log10(as.numeric(names(freqRank)) + 1),
main = "200 nodes in a preferential-attachment network") main = "200 nodes in a preferential-attachment network")
# Sort-of linear, but many of the higher ranked nodes have a frequency of only # Sort-of linear, but many of the higher ranked nodes have a frequency of only
# one. That behaviour smooths out in larger graphs: # one. That behaviour smooths out in larger graphs - let's try 100000 nodes:
# #
X <- igraph::sample_pa(1e5, power = 0.8, directed = FALSE) # 100,000 nodes N <- 10000
X <- igraph::sample_pa(N, power = 0.8, directed = FALSE) # 100,000 nodes
freqRank <- table(igraph::degree(X)) freqRank <- table(igraph::degree(X))
plot(log10(as.numeric(names(freqRank)) + 1), plot(log10(as.numeric(names(freqRank)) + 1),
log10(as.numeric(freqRank)), type = "b", log10(as.numeric(freqRank)), type = "b",
@ -336,16 +333,16 @@ rm(X)
# probability for two nodes to have an edge to be a function of their Euclidian # probability for two nodes to have an edge to be a function of their Euclidian
# distance in the box. # distance in the box.
# Here is a function that makes an adjacency matrix for such graphs. iGraph has # Here is a function that makes an adjacency matrix for such graphs. iGraph::
# a similar function, sample_grg(), which connects nodes that are closer than a # has a similar function, sample_grg(), which connects nodes that are closer
# cutoff, the function I give you below is a bit more interesting since it # than a cutoff, the function I give you below is a bit more interesting since
# creates edges according to a probability that is determined by a generalized # it creates edges according to a probability that is determined by a
# logistic function of the distance. This sigmoidal function gives a smooth # generalized logistic function of the distance. This sigmoidal function gives
# cutoff and creates more "natural" graphs. Otherwise, the function is very # a smooth cutoff and creates more "natural" graphs. Otherwise, the function is
# similar to the random graph function, except that we output the "coordinates" # very similar to the random graph function, except that we output the
# of the nodes together with the adjacency matrix which we then use for the # "coordinates" of the nodes together with the adjacency matrix which we then
# layout. list() FTW. # use for the layout. list() FTW.
#
makeRandomGeometricAM <- function(nam, B = 25, Q = 0.001, t = 0.6) { makeRandomGeometricAM <- function(nam, B = 25, Q = 0.001, t = 0.6) {
# Make an adjacency matrix for an undirected random geometric graph from # Make an adjacency matrix for an undirected random geometric graph from
@ -400,7 +397,7 @@ makeRandomGeometricAM <- function(nam, B = 25, Q = 0.001, t = 0.6) {
# } # }
# #
# ... and this code plots p-values over the distances we could encouter between # ... 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 # our nodes: from 0 to sqrt(2) i.e. the diagonal of the unit square in which we
# will place our nodes. # will place our nodes.
# x <- seq(0, sqrt(2), length.out = 50) # x <- seq(0, sqrt(2), length.out = 50)
# plot(x, genLog(x), type="l", col="#AA0000", ylim = c(0, 1), # plot(x, genLog(x), type="l", col="#AA0000", ylim = c(0, 1),
@ -415,7 +412,7 @@ set.seed(NULL) # reset the RNG
myGRG <- igraph::graph_from_adjacency_matrix(rGAM$mat, mode = "undirected") myGRG <- igraph::graph_from_adjacency_matrix(rGAM$mat, mode = "undirected")
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
plot(myGRG, plot(myGRG,
layout = cbind(rGAM$x, rGAM$y), # use our node coordinates for layout, layout = cbind(rGAM$x, rGAM$y), # use our node coordinates for layout,
rescale = FALSE, rescale = FALSE,
@ -448,12 +445,12 @@ plot(log10(as.numeric(names(freqRank)) + 1),
# = 3 A CLOSER LOOK AT THE igraph PACKAGE ================================= # = 3 A CLOSER LOOK AT THE igraph:: PACKAGE ===============================
# == 3.1 Basics ============================================================ # == 3.1 Basics ============================================================
# The basic object of the igraph package is a graph object. Let's explore the # 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: # first graph some more, the one we built with our random gene names:
summary(myG) summary(myG)
@ -464,7 +461,7 @@ summary(myG)
mode(myG) mode(myG)
class(myG) class(myG)
# This means an igraph graph object is a special list object; it is opaque in # 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 # 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 # 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, # many ways to construct graphs - from adjacency matrices, as we have just done,
@ -494,12 +491,12 @@ plot(myG) # this is the result of default plot parameters
# See ?plot.igraph for details." # See ?plot.igraph for details."
# Plot with some customizing parameters # Plot with some customizing parameters
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
plot(myG, plot(myG,
layout = igraph::layout_with_fr(myG), layout = igraph::layout_with_fr(myG),
vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1], vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1],
vertex.size = 9 + (2 * igraph::degree(myG)), vertex.size = 12 + (5 * igraph::degree(myG)),
vertex.label.cex = 0.5 + (0.05 * igraph::degree(myG)), vertex.label.cex = 0.4 + (0.15 * igraph::degree(myG)),
edge.width = 2, edge.width = 2,
vertex.label = igraph::V(myG)$name, vertex.label = igraph::V(myG)$name,
vertex.label.family = "sans", vertex.label.family = "sans",
@ -507,25 +504,25 @@ plot(myG,
par(oPar) par(oPar)
# ... or with a different layout: # ... or with a different layout:
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
plot(myG, plot(myG,
layout = igraph::layout_in_circle(myG), layout = igraph::layout_in_circle(myG),
vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1], vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1],
vertex.size = 9 + (2 * igraph::degree(myG)), vertex.size = 12 + (5 * igraph::degree(myG)),
vertex.label.cex = 0.5 + (0.05 * igraph::degree(myG)), vertex.label.cex = 0.4 + (0.15 * igraph::degree(myG)),
edge.width = 2, edge.width = 2,
vertex.label = igraph::V(myG)$name, vertex.label = igraph::V(myG)$name,
vertex.label.family = "sans", vertex.label.family = "sans",
vertex.label.cex = 0.9) vertex.label.cex = 0.9)
par(oPar) par(oPar)
# igraph has a large number of graph-layout functions: see # igraph:: has a large number of graph-layout functions: see
# ?layout_ and try them all. # ?layout_ and try them all.
# == 3.2 Components ======================================================== # == 3.2 Components ========================================================
# The igraph function components() tells us whether there are components of the # The function igrap::components() tells us whether there are components of the
# graph in which there is no path to other components. # graph in which there is no path to other components.
igraph::components(myG) igraph::components(myG)
@ -548,7 +545,7 @@ names(c1)
# Let's explore some of the more interesting, topological graph measures. We # 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 # start by building a somewhat bigger graph. We aren't quite sure whether
# biological graphs are small-world, or random-geometric, or # biological graphs are small-world, or random-geometric, or
# preferential-attachment ... but igraph has ways to simulate the basic ones # preferential-attachment ... but igraph:: has ways to simulate the basic ones
# (and we could easily simulate our own). Look at the following help pages: # (and we could easily simulate our own). Look at the following help pages:
?igraph::sample_gnm # see also sample_gnp for the Erdös-Rényi models ?igraph::sample_gnm # see also sample_gnp for the Erdös-Rényi models
@ -573,10 +570,10 @@ plot(myG,
xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01), xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01), ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1], vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1],
vertex.size = 20 + (10 * igraph::degree(myG)), vertex.size = 40 + (15 * igraph::degree(myG)),
vertex.label.cex = 0.4 + (0.15 * igraph::degree(myG)),
vertex.label = igraph::V(myG)$name, vertex.label = igraph::V(myG)$name,
vertex.label.family = "sans", vertex.label.family = "sans")
vertex.label.cex = 0.8)
par(oPar) # restore graphics state par(oPar) # restore graphics state
# == 4.1 Diameter ========================================================== # == 4.1 Diameter ==========================================================
@ -600,17 +597,17 @@ bC <- igraph::centr_betw(myG) # calculate betweenness centrality
nodeBetw <- bC$res nodeBetw <- bC$res
nodeBetw <- round(log(nodeBetw +1)) + 1 nodeBetw <- round(log(nodeBetw +1)) + 1
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
plot(myG, plot(myG,
layout = myGxy, layout = myGxy,
rescale = FALSE, rescale = FALSE,
xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01), xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01), ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
vertex.color=heat.colors(max(nodeBetw))[nodeBetw], vertex.color=heat.colors(max(igraph::degree(myG)+1))[igraph::degree(myG)+1],
vertex.size = 20 + (10 * igraph::degree(myG)), vertex.size = 40 + (15 * igraph::degree(myG)),
vertex.label.cex = 0.4 + (0.15 * igraph::degree(myG)),
vertex.label = igraph::V(myG)$name, vertex.label = igraph::V(myG)$name,
vertex.label.family = "sans", vertex.label.family = "sans")
vertex.label.cex = 0.7)
par(oPar) par(oPar)
# Note that the betweenness - the number of shortest paths that pass through a # Note that the betweenness - the number of shortest paths that pass through a
@ -629,7 +626,7 @@ nodeBetw <- bCmyGRG$res
nodeBetw <- round((log(nodeBetw +1))^2.5) + 1 nodeBetw <- round((log(nodeBetw +1))^2.5) + 1
# colours and size proportional to betweenness # colours and size proportional to betweenness
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
plot(myGRG, plot(myGRG,
layout = cbind(rGAM$x, rGAM$y), # use our node coordinates for layout, layout = cbind(rGAM$x, rGAM$y), # use our node coordinates for layout,
rescale = FALSE, rescale = FALSE,
@ -677,18 +674,18 @@ commColors <- rep("#f1eef6", max(igraph::membership(myGRGclusters)))
commColors[1:5] <- c("#980043", "#dd1c77", "#df65b0", "#c994c7", "#d4b9da") commColors[1:5] <- c("#980043", "#dd1c77", "#df65b0", "#c994c7", "#d4b9da")
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= c(0, 0, 0, 0)) # Turn margins off
plot(myGRG, plot(myGRG,
layout = cbind(rGAM$x, rGAM$y), layout = cbind(rGAM$x, rGAM$y),
rescale = FALSE, rescale = FALSE,
xlim = c(min(rGAM$x) * 0.9, max(rGAM$x) * 1.1), xlim = c(min(rGAM$x) * 0.9, max(rGAM$x) * 1.1),
ylim = c(min(rGAM$y) * 0.9, max(rGAM$y) * 1.1), ylim = c(min(rGAM$y) * 0.9, max(rGAM$y) * 1.1),
vertex.color=commColors[igraph::membership(myGRGclusters)], 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) vertex.label = NA)
par(oPar) par(oPar)
# That's all.
# [END] # [END]