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FND-MAT-Graphs_and_networks.R
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@ -3,48 +3,67 @@
# 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.
# #
# Version: 0.1 # Version: 1.0
# #
# Date: 2017 08 28 # Date: 2017 10 06
# Author: Boris Steipe (boris.steipe@utoronto.ca) # Author: Boris Steipe (boris.steipe@utoronto.ca)
# #
# Versions: # Versions:
# 1.0 First final version for learning units.
# 0.1 First code copied from 2016 material. # 0.1 First code copied from 2016 material.
#
# #
# TODO: # TODO:
# #
# #
# == DO NOT SIMPLY source() THIS FILE! ======================================= # == DO NOT SIMPLY source() THIS FILE! =======================================
#
# If there are portions you don't understand, use R's help system, Google for an # If there are portions you don't understand, use R's help system, Google for an
# answer, or ask your instructor. Don't continue if you don't understand what's # answer, or ask your instructor. Don't continue if you don't understand what's
# going on. That's not how it works ... # going on. That's not how it works ...
#
# ============================================================================== # ==============================================================================
# = 1 ___Section___ #TOC> ==========================================================================
#TOC>
# This tutorial covers basic concepts of graph theory and analysis in R. You #TOC> Section Title Line
# should have typed init() to configure some utilities in the background. #TOC> ------------------------------------------------------
#TOC> 1 Review 48
#TOC> 2 DEGREE DISTRIBUTIONS 192
#TOC> 2.1 Random graph 198
#TOC> 2.2 scale-free graph (Barabasi-Albert) 242
#TOC> 2.3 Random geometric graph 304
#TOC> 3 A CLOSER LOOK AT THE igraph PACKAGE 424
#TOC> 3.1 Basics 427
#TOC> 3.2 Components 499
#TOC> 4 RANDOM GRAPHS AND GRAPH METRICS 518
#TOC> 4.1 Diameter 553
#TOC> 5 GRAPH CLUSTERING 621
#TOC>
#TOC> ==========================================================================
# ==============================================================================
# PART ONE: REVIEW
# ==============================================================================
# I assume you'll have read the Pavlopoulos review of graph theory concepts.
# Let's explore some of the ideas by starting with a small random graph." # = 1 Review ==============================================================
# 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
# repository, and that you have typed init() to load some utility functions and
# data.
# Let's explore some of the basic ideas of graph theory by starting with a small
# random graph.
# To begin let's write a little function that will create random "gene" names; # To begin let's write a little function that will create random "gene" names;
# there's no particular purpose to this other than to make our graphs look a # there's no particular purpose to this other than to make our graphs look a
# little more like what we would find in a publication ... # little more "biological ...
makeRandomGenenames <- function(N) { makeRandomGenenames <- function(N) {
nam <- character() nam <- character()
while (length(nam) < N) { while (length(nam) < N) {
a <- paste(c(sample(LETTERS, 1), sample(letters, 2)), a <- paste0(c(sample(LETTERS, 1), sample(letters, 2)),
sep="", collapse="") # three letters collapse="") # one uppercase, two lowercase letters
n <- sample(1:9, 1) # one number n <- sample(1:9, 1) # one number
nam[length(nam) + 1] <- paste(a, n, sep="") # store in vector nam[length(nam) + 1] <- paste(a, n, sep="") # store in vector
nam <- unique(nam) # delete if this was a duplicate nam <- unique(nam) # delete if this was a duplicate
@ -55,64 +74,61 @@ makeRandomGenenames <- function(N) {
N <- 20 N <- 20
set.seed(112358) set.seed(112358)
Nnames <- makeRandomGenenames(N) (Nnames <- makeRandomGenenames(N))
Nnames
# One way to represent graphs in a computer is as an "adjacency matrix". In this # One way to represent graphs in a computer is as an "adjacency matrix". In this
# matrix, each row and each column represents a node, and the cell at the # matrix, each row and each column represents a node, and the cell at the
# intersection of a row and column contains a value/TRUE if there is an edge, # intersection of a row and column contains a value/TRUE if there is an edge,
# 0/FALSE otherwise. It's easy to see that an undirected graph has a symmetric # 0/FALSE otherwise.
# adjacency matrix (i, j) == (j, i); and we can put values other than {1, 0}
# into a cell if we want to represent a weighted edge.
# At first, lets create a random graph: let's say a pair of nodes has # Let's create an adjacency matrix for random graph: let's say a pair of nodes
# probability p <- 0.1 to have an edge, and our graph is symmetric and has no # has probability p <- 0.1 to have an edge, and our graph is symmetric , i.e. it
# self-edges. We use our Nnames as node labels, but I've written the function so # is an undirected graph, and it has neither self-edges, i.e. loops, nor
# that we could also just ask for any number of un-named nodes, we'll use that later. # multiple edges between the same nodes, i.e. it is a "simple" graph. We use our
# the Nnames vector as node labels.
makeRandomGraph <- function(nam, p = 0.1) { makeRandomAM <- function(nam, p = 0.1) {
# nam: either a character vector of unique names, or a single # Make a random adjacency matrix for a set of nodes with edge probability p
# number that will be converted into a vector of integers. # Parameters:
# nam: a character vector of unique node names.
# p: probability that a random pair of nodes will have an edge. # p: probability that a random pair of nodes will have an edge.
# #
# Value: an adjacency matrix # Value: an adjacency matrix for a simple, undirected graph
# #
if (is.numeric(nam) && length(nam) == 1) { # if nam is a single number ...
nam <- as.character(1:nam)
}
N <- length(nam) N <- length(nam)
G <- matrix(numeric(N * N), ncol = N) # The adjacency matrix AM <- matrix(numeric(N * N), ncol = N) # The adjacency matrix
rownames(G) <- nam rownames(AM) <- nam
colnames(G) <- nam colnames(AM) <- nam
for (iRow in 1:(N-1)) { # Note how we make sure iRow != iCol for (iRow in 1:(N-1)) { # Note how we make sure iRow != iCol - this prevents
# loops
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 # between 0 and 1. The expression is TRUE with
G[iRow, iCol] <- 1 # row, col ! # probability p. if it is TRUE ...
G[iCol, iRow] <- 1 # col, row ! AM[iRow, iCol] <- 1 # ... record an edge for the pair (iRow, iCol)
} }
} }
} }
return(G) return(AM)
} }
set.seed(112358) set.seed(112358)
G <- makeRandomGraph(Nnames, p = 0.09) (myRandAM <- makeRandomAM(Nnames, p = 0.09))
G
# Listing the matrix is not very informative - we should plot this graph. We'll # Listing the matrix is not very informative - we should plot this graph. The
# go into more details of the igraph package a bit later, for now we just use it # standard package for work with graphs in r is "igraph". We'll go into more
# to plot: # details of the igraph package a bit later, for now we just use it to plot:
if (!require(igraph)) { if (!require(igraph)) {
install.packages("igraph") install.packages("igraph")
library(igraph) library(igraph)
} }
iG <- graph_from_adjacency_matrix(G) myG <- graph_from_adjacency_matrix(myRandAM, mode = "undirected")
iGxy <- layout_with_graphopt(iG, charge=0.001) # calculate layout coordinates set.seed(112358)
myGxy <- layout_with_graphopt(myG, charge=0.0012) # calculate layout coordinates
# The igraph package adds its own function to the collection of plot() # The igraph package adds its own function to the collection of plot()
@ -121,19 +137,22 @@ iGxy <- layout_with_graphopt(iG, charge=0.001) # calculate layout coordinates
# layout - the x,y coordinates of the nodes; # layout - the x,y coordinates of the nodes;
# vertex.color - which I define to color by node-degree # vertex.color - which I define to color by node-degree
# vertex size - which I define to increase with node-degree # vertex size - which I define to increase with node-degree
# vertex.label - which I set to use our Nnames vector # vertex.label - which I set to combine the names of the vertices of the
# graph - names(V(iG)) - with the node degree - degree(iG).
# See ?igraph.plotting for the complete list of parameters
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= rep(0,4)) # Turn margins off
plot(iG, plot(myG,
layout = iGxy, layout = myGxy,
rescale = FALSE, rescale = FALSE,
xlim = c(min(iGxy[,1]), max(iGxy[,1])) * 1.1, xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
ylim = c(min(iGxy[,2]), max(iGxy[,2])) * 1.1, ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
vertex.color=heat.colors(max(degree(iG)+1))[degree(iG)+1], vertex.color=heat.colors(max(degree(myG)+1))[degree(myG)+1],
vertex.size = 800 + (150 * degree(iG)), vertex.size = 1600 + (300 * degree(myG)),
vertex.label = as.character(degree(iG)/2), vertex.label = sprintf("%s(%i)", names(V(myG)), degree(myG)),
# vertex.label = Nnames, vertex.label.family = "sans",
edge.arrow.size = 0) vertex.label.cex = 0.7)
par(oPar) # reset plot window par(oPar) # reset plot window
@ -141,29 +160,22 @@ par(oPar) # reset plot window
# degree-distribution. In our example, the number of nodes was given: N; the # degree-distribution. In our example, the number of nodes was given: N; the
# number of edges can easily be calculated from the adjacency matrix. In our # number of edges can easily be calculated from the adjacency matrix. In our
# matrix, we have entered 1 for every edge. Thus we simply sum over the matrix: # matrix, we have entered 1 for every edge. Thus we simply sum over the matrix:
sum(G) sum(myRandAM)
# Is that correct? Is that what you see in the plot? # Is that what you expect?
# Yes and no: we entered every edge twice: once for a node [i,j], and again for
# the node [j, i]. Whether that is correct depends on what exactly we
# want to do with the matrix. If these were directed edges, we would need to
# keep track of them separately. Since we didn't intend them to be directed,
# we'll could divide the number of edges by 2. Why didn't we simply use an
# upper-triangular matrix? Because then we need to keep track of the ordering of
# edges if we want to know whether a particular edge exists or not. For example
# we could sort the nodes alphabetically, and make sure we always query a pair
# in alphabetical order. Then a triangular matrix would be efficient.
# What about the degree distribution? We can get that simply by summing over the # What about the degree distribution? We can get that simply by summing over the
# rows (or the columns):" # rows and summing over the columns and adding the two vectors.
rowSums(G) # check this against the plot! rowSums(myRandAM) + colSums(myRandAM) # check this against the plot!
# The function degree() gives the same values
degree(myG)
# Let's plot the degree distribution in a histogram: # Let's plot the degree distribution in a histogram:
rs <- rowSums(G) degG <- degree(myG)
brk <- seq(min(rs)-0.5, max(rs)+0.5, by=1) # define breaks for the histogram brk <- seq(min(degG)-0.5, max(degG)+0.5, by=1) # define histogram breaks
hist(rs, breaks=brk, col="#A5CCF5", hist(degG, breaks=brk, col="#A5CCF5",
xlim = c(-1,8), xaxt = "n", xlim = c(-1,8), xaxt = "n",
main = "Node degrees", xlab = "Degree", ylab = "Number") # plot histogram main = "Node degrees", xlab = "Degree", ylab = "Number") # plot histogram
axis(side = 1, at = 0:7) axis(side = 1, at = 0:7)
@ -177,111 +189,109 @@ axis(side = 1, at = 0:7)
# 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?
# ============================================================================== # = 2 DEGREE DISTRIBUTIONS ================================================
# PART TWO: DEGREE DISTRIBUTIONS
# ==============================================================================
# Let's simulate a few graphs that are a bit bigger to get a better sense of # Let's simulate a few graphs that are a bit bigger to get a better sense of
# their degree distributions: # their degree distributions:
# #
# === random graph # == 2.1 Random graph ======================================================
set.seed(31415927) set.seed(31415927)
G200 <- makeRandomGraph(200, p = 0.015) my200AM <- makeRandomAM(as.character(1:200), p = 0.015)
iG200 <- graph_from_adjacency_matrix(G200) myG200 <- graph_from_adjacency_matrix(my200AM, mode = "undirected")
iGxy <- layout_with_graphopt(iG200, charge=0.0001) # calculate layout coordinates myGxy <- layout_with_graphopt(myG200, charge=0.0001) # calculate layout coordinates
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= rep(0,4)) # Turn margins off
plot(iG200, plot(myG200,
layout = iGxy, layout = myGxy,
rescale = FALSE, rescale = FALSE,
xlim = c(min(iGxy[,1]), max(iGxy[,1])) * 1.1, xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
ylim = c(min(iGxy[,2]), max(iGxy[,2])) * 1.1, ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
vertex.color=heat.colors(max(degree(iG200)+1))[degree(iG200)+1], vertex.color=heat.colors(max(degree(myG200)+1))[degree(myG200)+1],
vertex.size = 200 + (30 * degree(iG200)), vertex.size = 150 + (60 * degree(myG200)),
vertex.label = "", vertex.label = NA)
edge.arrow.size = 0)
par(oPar) par(oPar)
# This graph has thirteen singletons and one large, connected component. Many # This graph has thirteen singletons and one large, connected component. Many
# biological graphs look approximately like this. # biological graphs look approximately like this.
# Calculate degree distributions # Calculate degree distributions
dg <- degree(iG200)/2 # here, we use the iGraph function degree() dg <- degree(myG200)
# not rowsums() from base R.
brk <- seq(min(dg)-0.5, max(dg)+0.5, by=1) brk <- seq(min(dg)-0.5, max(dg)+0.5, by=1)
hist(dg, breaks=brk, col="#A5CCF5", hist(dg, breaks=brk, col="#A5F5CC",
xlim = c(-1,11), xaxt = "n", xlim = c(-1,11), xaxt = "n",
main = "Node degrees", xlab = "Degree", ylab = "Number") # plot histogram main = "Node degrees", xlab = "Degree", ylab = "Number") # plot histogram
axis(side = 1, at = 0:10) axis(side = 1, at = 0:10)
# Note the pronounced peak of this distribution: this is not "scale-free".
# Note the characteristic peak of this distribution: this is not "scale-free". Here is a log-log plot of frequency vs. degree-rank: # Here is the log-log plot of frequency vs. degree-rank ...
(freqRank <- table(dg))
plot(log10(as.numeric(names(freqRank)) + 1),
log10(as.numeric(freqRank)), type = "b",
pch = 21, bg = "#A5CCF5",
xlab = "log(Rank)", ylab = "log(frequency)",
main = "200 nodes in a random network")
# === scale-free graph (Barabasi-Albert)
# What does one of those intriguing "scale-free" distributions look like? The
# iGraph package has a function to make random graphs according to the
# Barabasi-Albert model of scale-free graphs. It is: sample_pa(), where pa
# stands for "preferential attachment", one type of process that will yield
# scale-free distributions.
set.seed(31415927)
GBA <- sample_pa(200, power = 0.8)
iGxy <- layout_with_graphopt(GBA, charge=0.0001) # calculate layout coordinates
oPar <- par(mar= rep(0,4)) # Turn margins off
plot(GBA,
layout = iGxy,
rescale = FALSE,
xlim = c(min(iGxy[,1]), max(iGxy[,1])) * 1.1,
ylim = c(min(iGxy[,2]), max(iGxy[,2])) * 1.1,
vertex.color=heat.colors(max(degree(GBA)+1))[degree(GBA)+1],
vertex.size = 200 + (30 * degree(GBA)),
vertex.label = "",
edge.arrow.size = 0)
par(oPar)
# This is a very obviously different graph! Some biological networks have
# features that look like that - but in my experience the hub nodes are usually
# not that distinct. But then again, that really depends on the parameter
# "power". Feel encouraged to change "power" and get a sense for what difference
# this makes. Also: note that the graph has only a single component.
# What's the degree distribution of this graph?
(dg <- degree(GBA))
brk <- seq(min(dg)-0.5, max(dg)+0.5, by=1)
hist(dg, breaks=brk, col="#A5D5CC",
xlim = c(0,30), xaxt = "n",
main = "Node degrees 200 nodes PA graph",
xlab = "Degree", ylab = "Number")
axis(side = 1, at = seq(0, 30, by=5))
# Most nodes have a degree of 1, but one node has a degree of 28.
(freqRank <- table(dg)) (freqRank <- table(dg))
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",
pch = 21, bg = "#A5F5CC", pch = 21, bg = "#A5F5CC",
xlab = "log(Rank)", ylab = "log(frequency)", xlab = "log(Rank)", ylab = "log(frequency)",
main = "200 nodes in a random network")
# ... which shows us that this does NOT correspond to the single-slope linear
# relationship that we expect for a "scale-free" graph.
# == 2.2 scale-free graph (Barabasi-Albert) ================================
# What does one of those intriguing "scale-free" distributions look like? The
# iGraph package has a function to make random graphs according to the
# Barabasi-Albert model of scale-free graphs. It is: sample_pa(), where pa
# stands for "preferential attachment". Preferential attachment is one type of
# process that will yield scale-free distributions.
set.seed(31415927)
GBA <- sample_pa(200, power = 0.8, directed = FALSE)
GBAxy <- layout_with_graphopt(GBA, charge=0.0001) # calculate layout coordinates
oPar <- par(mar= rep(0,4)) # Turn margins off
plot(GBA,
layout = GBAxy,
rescale = FALSE,
xlim = c(min(GBAxy[,1]) * 0.99, max(GBAxy[,1]) * 1.01),
ylim = c(min(GBAxy[,2]) * 0.99, max(GBAxy[,2]) * 1.01),
vertex.color=heat.colors(max(degree(GBA)+1))[degree(GBA)+1],
vertex.size = 200 + (30 * degree(GBA)),
vertex.label = NA)
par(oPar)
# This is a very obviously different graph! Some biological networks have
# features that look like that - but in my experience the hub nodes are usually
# not that distinct. But then again, that really depends on the parameter
# "power". Play with the "power" parameter and get a sense for what difference
# this makes. Also: note that the graph has only a single component - no
# singletons.
# What's the degree distribution of this graph?
(dg <- degree(GBA))
brk <- seq(min(dg)-0.5, max(dg)+0.5, by=1)
hist(dg, breaks=brk, col="#DCF5B5",
xlim = c(0,max(dg)+1), xaxt = "n",
main = "Node degrees 200 nodes PA graph",
xlab = "Degree", ylab = "Number")
axis(side = 1, at = seq(0, max(dg)+1, by=5))
# Most nodes have a degree of 1, but one node has a degree of 19.
(freqRank <- table(dg))
plot(log10(as.numeric(names(freqRank)) + 1),
log10(as.numeric(freqRank)), type = "b",
pch = 21, bg = "#DCF5B5",
xlab = "log(Rank)", ylab = "log(frequency)",
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:
# #
X <- sample_pa(100000, power = 0.8) # 100,000 nodes X <- sample_pa(100000, power = 0.8, directed = FALSE) # 100,000 nodes
freqRank <- table(degree(X)) freqRank <- table(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",
@ -290,64 +300,66 @@ plot(log10(as.numeric(names(freqRank)) + 1),
main = "100,000 nodes in a random, scale-free network") main = "100,000 nodes in a random, scale-free network")
rm(X) rm(X)
# === Random geometric graph
# == 2.3 Random geometric graph ============================================
# Finally, let's simulate a random geometric graph and look at the degree # Finally, let's simulate a random geometric graph and look at the degree
# distribution. Remember: these graphs have a high probability to have edges # distribution. Remember: these graphs have a high probability to have edges
# between nodes that are "close" together - an entriely biological notion. # between nodes that are "close" together - an entirely biological notion.
# We'll randomly place our nodes in a box. Then we'll define the # We'll randomly place our nodes in a box. Then we'll define the
# probability for two nodes to have an edge to be a function of their distance. # probability for two nodes to have an edge to be a function of their Euclidian
# distance in the box.
# Here is a function that makes such graphs. iGraph has sample_grg(), which # Here is a function that makes an adjacency matrix for such graphs. iGraph has
# connects nodes that are closer than a cutoff, the function I give you below is # a similar function, sample_grg(), which connects nodes that are closer than a
# a bit more interesting since it creates edges according to a probability that # cutoff, the function I give you below is a bit more interesting since it
# is determined by a generalized logistic function of the distance. This # creates edges according to a probability that is determined by a generalized
# sigmoidal function gives a smooth cutoff and creates more "natural" graphs. # logistic function of the distance. This sigmoidal function gives a smooth
# Otherwise, the function is very similar to the random graph function, except # cutoff and creates more "natural" graphs. Otherwise, the function is very
# that we output the "coordinates" of the nodes together with the adjacency # similar to the random graph function, except that we output the "coordinates"
# matrix. Lists FTW. # of the nodes together with the adjacency matrix which we then use for the
# layout. list() FTW.
# #
makeRandomGeometricGraph <- function(nam, B = 25, Q = 0.001, t = 0.6) {
# nam: either a character vector of unique names, or a single makeRandomGeometricAM <- function(nam, B = 25, Q = 0.001, t = 0.6) {
# number that will be converted into a vector of integers. # Make an adjacency matrix for an undirected random geometric graph from
# edges connected with probabilities according to a generalized logistic
# function.
# Parameters:
# nam: a character vector of unique names
# B, Q, t: probability that a random pair (i, j) of nodes gets an # B, Q, t: probability that a random pair (i, j) of nodes gets an
# edge determined by a generalized logistic function # edge determined by a generalized logistic function
# p <- 1 - 1/((1 + (Q * (exp(-B * (x-t)))))^(1 / 0.9))) # p <- 1 - 1/((1 + (Q * (exp(-B * (x-t)))))^(1 / 0.9)))
# #
# Value: a list with the following components: # Value: a list with the following components:
# G$mat : an adjacency matrix # AM$mat : an adjacency matrix
# G$nam : labels for the nodes # AM$nam : labels for the nodes
# G$x : x-coordinates for the nodes # AM$x : x-coordinates for the nodes
# G$y : y-coordinates for the nodes # AM$y : y-coordinates for the nodes
# #
nu <- 1 # probably not useful to change nu <- 1 # probably not useful to change
G <- list() AM <- list()
AM$nam <- nam
if (is.numeric(nam) && length(nam) == 1) { N <- length(AM$nam)
nam <- as.character(1:nam) AM$mat <- matrix(numeric(N * N), ncol = N) # The adjacency matrix
} rownames(AM$mat) <- AM$nam
G$nam <- nam colnames(AM$mat) <- AM$nam
N <- length(G$nam) AM$x <- runif(N) # Randomly place nodes into the unit square
G$mat <- matrix(numeric(N * N), ncol = N) # The adjacency matrix AM$y <- runif(N)
rownames(G$mat) <- G$nam
colnames(G$mat) <- G$nam
G$x <- runif(N)
G$y <- runif(N)
for (iRow in 1:(N-1)) { # Same principles as in makeRandomGraph() for (iRow in 1:(N-1)) { # Same principles as in makeRandomGraph()
for (iCol in (iRow+1):N) { for (iCol in (iRow+1):N) {
# geometric distance ... # geometric distance ...
d <- sqrt((G$x[iRow] - G$x[iCol])^2 + d <- sqrt((AM$x[iRow] - AM$x[iCol])^2 +
(G$y[iRow] - G$y[iCol])^2) # Pythagoras (AM$y[iRow] - AM$y[iCol])^2) # Pythagoras
# distance dependent probability # distance dependent probability
p <- 1 - 1/((1 + (Q * (exp(-B * (d-t)))))^(1 / nu)) p <- 1 - 1/((1 + (Q * (exp(-B * (d-t)))))^(1 / nu))
if (runif(1) < p) { if (runif(1) < p) {
G$mat[iRow, iCol] <- 1 AM$mat[iRow, iCol] <- 1
G$mat[iCol, iRow] <- 1
} }
} }
} }
return(G) return(AM)
} }
# Getting the parameters of a generalized logistic right takes a bit of # Getting the parameters of a generalized logistic right takes a bit of
@ -371,28 +383,26 @@ makeRandomGeometricGraph <- function(nam, B = 25, Q = 0.001, t = 0.6) {
# 200 node random geomteric graph # 200 node random geomteric graph
set.seed(112358) set.seed(112358)
GRG <- makeRandomGeometricGraph(200, t=0.4) rGAM <- makeRandomGeometricAM(as.character(1:200), t=0.4)
iGRG <- graph_from_adjacency_matrix(GRG$mat) myGRG <- graph_from_adjacency_matrix(rGAM$mat, mode = "undirected")
iGRGxy <- cbind(GRG$x, GRG$y) # use our node coordinates for layout
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= rep(0,4)) # Turn margins off
plot(iGRG, plot(myGRG,
layout = iGRGxy, layout = cbind(rGAM$x, rGAM$y), # use our node coordinates for layout,
rescale = FALSE, rescale = FALSE,
xlim = c(min(iGRGxy[,1]), max(iGRGxy[,1])) * 1.1, xlim = c(min(rGAM$x) * 0.9, max(rGAM$x) * 1.1),
ylim = c(min(iGRGxy[,2]), max(iGRGxy[,2])) * 1.1, ylim = c(min(rGAM$y) * 0.9, max(rGAM$y) * 1.1),
vertex.color=heat.colors(max(degree(iGRG)+1))[degree(iGRG)+1], vertex.color=heat.colors(max(degree(myGRG)+1))[degree(myGRG)+1],
vertex.size = 0.1 + (0.1 * degree(iGRG)), vertex.size = 0.1 + (0.2 * degree(myGRG)),
vertex.label = "", vertex.label = NA)
edge.arrow.size = 0)
par(oPar) par(oPar)
# degree distribution: # degree distribution:
(dg <- degree(iGRG)/2) (dg <- degree(myGRG))
brk <- seq(min(dg)-0.5, max(dg)+0.5, by=1) brk <- seq(min(dg) - 0.5, max(dg) + 0.5, by = 1)
hist(dg, breaks=brk, col="#FCD6E2", hist(dg, breaks = brk, col = "#FCC6D2",
xlim = c(0, 25), xaxt = "n", xlim = c(0, 25), xaxt = "n",
main = "Node degrees: 200 nodes RG graph", main = "Node degrees: 200 nodes RG graph",
xlab = "Degree", ylab = "Number") xlab = "Degree", ylab = "Number")
@ -405,29 +415,27 @@ axis(side = 1, at = c(0, min(dg):max(dg)))
(freqRank <- table(dg)) (freqRank <- table(dg))
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",
pch = 21, bg = "#FCD6E2", pch = 21, bg = "#FCC6D2",
xlab = "log(Rank)", ylab = "log(frequency)", xlab = "log(Rank)", ylab = "log(frequency)",
main = "200 nodes in a random geometric network") main = "200 nodes in a random geometric network")
# ==================================================================== # = 3 A CLOSER LOOK AT THE igraph PACKAGE =================================
# PART THREE: A CLOSER LOOK AT THE igraph PACKAGE
# ====================================================================
# == 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(iG) summary(myG)
# This output means: this is an IGRAPH graph, with D = directed edges and N = # This output means: this is an IGRAPH graph, with U = UN-directed edges
# named nodes, that has 20 nodes and 40 edges. For details, see # and N = named nodes, that has 20 nodes and 20 edges. For details, see
?print.igraph ?print.igraph
mode(iG) mode(myG)
class(iG) 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
@ -437,14 +445,18 @@ class(iG)
# recipes, called _games_ in this package. # recipes, called _games_ in this package.
# Two basic functions retrieve nodes "Vertices", and "Edges": # Two basic functions retrieve nodes "Vertices", and "Edges":
V(iG) V(myG)
E(iG) E(myG)
# additional properties can be retrieved from the Vertices ...
V(myG)$name
# As with many R objects, loading the package provides special functions that # As with many R objects, loading the package provides special functions that
# can be accessed via the same name as the basic R functions, for example: # can be accessed via the same name as the basic R functions, for example:
print(iG) print(myG)
plot(iG) plot(myG) # this is the result of default plot parameters
# ... where plot() allows the usual flexibility of fine-tuning the plot. We # ... where plot() allows the usual flexibility of fine-tuning the plot. We
# first layout the node coordinates with the Fruchtermann-Reingold algorithm - a # first layout the node coordinates with the Fruchtermann-Reingold algorithm - a
@ -454,41 +466,56 @@ plot(iG)
# labels by degree and the use of the V() function to retrieve the vertex names. # labels by degree and the use of the V() function to retrieve the vertex names.
# See ?plot.igraph for details." # See ?plot.igraph for details."
iGxy <- layout_with_fr(iG) # calculate layout coordinates
# Plot with some customizing parameters # Plot with some customizing parameters
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= rep(0,4)) # Turn margins off
plot(iG, plot(myG,
layout = iGxy, layout = layout_with_fr(myG),
vertex.color=heat.colors(max(degree(iG)+1))[degree(iG)+1], vertex.color=heat.colors(max(degree(myG)+1))[degree(myG)+1],
vertex.size = 9 + (2 * degree(iG)), vertex.size = 9 + (2 * degree(myG)),
vertex.label.cex = 0.5 + (0.05 * degree(iG)), vertex.label.cex = 0.5 + (0.05 * degree(myG)),
edge.arrow.size = 0,
edge.width = 2, edge.width = 2,
vertex.label = toupper(V(iG)$name)) vertex.label = V(myG)$name,
vertex.label.family = "sans",
vertex.label.cex = 0.9)
par(oPar) par(oPar)
# ... or with a different layout:
oPar <- par(mar= rep(0,4)) # Turn margins off
plot(myG,
layout = layout_in_circle(myG),
vertex.color=heat.colors(max(degree(myG)+1))[degree(myG)+1],
vertex.size = 9 + (2 * degree(myG)),
vertex.label.cex = 0.5 + (0.05 * degree(myG)),
edge.width = 2,
vertex.label = V(myG)$name,
vertex.label.family = "sans",
vertex.label.cex = 0.9)
par(oPar)
# == Components # igraph has a large number of graph-layout functions: see
# ?layout_ and try them all.
# == 3.2 Components ========================================================
# The igraph function components() tells us whether there are components of the # The igraph function components() tells us whether there are components of the
# graph in which there is no path to other components. # graph in which there is no path to other components.
components(iG) components(myG)
# In the _membership_ vector, nodes are annotatd with the index of the component # In the _membership_ vector, nodes are annotated with the index of the
# they are part of. Sui7 is the only node of component 2, Cyj1 is in the third # component they are part of. Sui7 is the only node of component 2, Cyj1 is in
# component etc. This is perhaps more clear if we sort by component index # the third component etc. This is perhaps more clear if we sort by component
sort(components(iG)$membership) # index
sort(components(myG)$membership, decreasing = TRUE)
# Retrieving e.g. the members of the first component from the list can be done by subsetting: # Retrieving e.g. the members of the first component from the list can be done by subsetting:
components(iG)$membership == 1 # logical .. (sel <- components(myG)$membership == 1) # boolean vector ..
components(iG)$membership[components(iG)$membership == 1] (c1 <- components(myG)$membership[sel])
names(components(iG)$membership)[components(iG)$membership == 1] names(c1)
# = 4 RANDOM GRAPHS AND GRAPH METRICS =====================================
# == RANDOM GRAPHS AND GRAPH METRICS =================================
# Let's explore some of the more interesting, topological graph measures. We # Let's explore some of the more interesting, topological graph measures. We
@ -503,55 +530,57 @@ names(components(iG)$membership)[components(iG)$membership == 1]
# But note that there are many more sample_ functions. Check out the docs! # 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 # 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 # 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 # considered to be more central. And that's also the way the force-directed
# layout drawas them, obviously. # layout drawas them, obviously.
set.seed(112358) set.seed(112358)
iGxy <- layout_with_fr(iG) # calculate layout coordinates myGxy <- layout_with_fr(myG) # calculate layout coordinates
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= rep(0,4)) # Turn margins off
plot(iG, plot(myG,
layout = iGxy, layout = myGxy,
rescale = FALSE, rescale = FALSE,
xlim = c(min(iGxy[,1]), max(iGxy[,1])) * 1.1, xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
ylim = c(min(iGxy[,2]), max(iGxy[,2])) * 1.1, ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
vertex.color=heat.colors(max(degree(iG)+1))[degree(iG)+1], vertex.color=heat.colors(max(degree(myG)+1))[degree(myG)+1],
vertex.size = 20 + (10 * degree(iG)), vertex.size = 20 + (10 * degree(myG)),
vertex.label = Nnames, vertex.label = V(myG)$name,
edge.arrow.size = 0) vertex.label.family = "sans",
vertex.label.cex = 0.8)
par(oPar) par(oPar)
# == Diameter # == 4.1 Diameter ==========================================================
diameter(iG) # The diameter of a graph is its maximum length shortest path. diameter(myG) # The diameter of a graph is its maximum length shortest path.
# let's plot this path: here are the nodes ... # let's plot this path: here are the nodes ...
get_diameter(iG) get_diameter(myG)
# ... and we can get the x, y coordinates from iGxy by subsetting with the node # ... and we can get the x, y coordinates from iGxy by subsetting with the node
# names. The we draw the diameter-path with a transparent, thick pink line: # names. The we draw the diameter-path with a transparent, thick pink line:
lines(iGxy[get_diameter(iG),], lwd=10, col="#ff63a788") lines(myGxy[get_diameter(myG),], lwd=10, col="#ff63a788")
# == Centralization scores # == Centralization scores
?centralize ?centralize
# replot our graph, and color by log_betweenness: # replot our graph, and color by log_betweenness:
bC <- centr_betw(iG) # calculate betweenness centrality bC <- centr_betw(myG) # calculate betweenness centrality
nodeBetw <- bC$res nodeBetw <- bC$res
nodeBetw <- round(log(nodeBetw +1)) + 1 nodeBetw <- round(log(nodeBetw +1)) + 1
oPar <- par(mar= rep(0,4)) # Turn margins off oPar <- par(mar= rep(0,4)) # Turn margins off
plot(iG, plot(myG,
layout = iGxy, layout = myGxy,
rescale = FALSE, rescale = FALSE,
xlim = c(min(iGxy[,1]), max(iGxy[,1])) * 1.1, xlim = c(min(myGxy[,1]) * 0.99, max(myGxy[,1]) * 1.01),
ylim = c(min(iGxy[,2]), max(iGxy[,2])) * 1.1, ylim = c(min(myGxy[,2]) * 0.99, max(myGxy[,2]) * 1.01),
vertex.color=heat.colors(max(nodeBetw))[nodeBetw], vertex.color=heat.colors(max(nodeBetw))[nodeBetw],
vertex.size = 20 + (10 * degree(iG)), vertex.size = 20 + (10 * degree(myG)),
vertex.label = Nnames, vertex.label = V(myG)$name,
edge.arrow.size = 0) 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
@ -564,31 +593,33 @@ par(oPar)
# #
# Lets plot betweenness centrality for our random geometric graph: # Lets plot betweenness centrality for our random geometric graph:
bCiGRG <- centr_betw(iGRG) # calculate betweenness centrality bCmyGRG <- centr_betw(myGRG) # calculate betweenness centrality
nodeBetw <- bCiGRG$res nodeBetw <- bCmyGRG$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= rep(0,4)) # Turn margins off
plot(iGRG, plot(myGRG,
layout = iGRGxy, layout = cbind(rGAM$x, rGAM$y), # use our node coordinates for layout,
rescale = FALSE, rescale = FALSE,
xlim = c(min(iGRGxy[,1]), max(iGRGxy[,1])), xlim = c(min(rGAM$x) * 0.9, max(rGAM$x) * 1.1),
ylim = c(min(iGRGxy[,2]), max(iGRGxy[,2])), ylim = c(min(rGAM$y) * 0.9, max(rGAM$y) * 1.1),
vertex.color=heat.colors(max(nodeBetw))[nodeBetw], vertex.color=heat.colors(max(nodeBetw))[nodeBetw],
vertex.size = 0.1 + (0.03 * nodeBetw), vertex.size = 0.1 + (0.03 * nodeBetw),
vertex.label = "", vertex.label = NA)
edge.arrow.size = 0)
par(oPar) par(oPar)
diameter(iGRG) diameter(myGRG)
lines(iGRGxy[get_diameter(iGRG),], lwd=10, col="#ff335533") lines(rGAM$x[get_diameter(myGRG)],
rGAM$y[get_diameter(myGRG)],
lwd = 10,
col = "#ff335533")
# == CLUSTERING ====================================================== # = 5 GRAPH CLUSTERING ====================================================
# Clustering finds "communities" in graphs - and depending what the edges # Clustering finds "communities" in graphs - and depending what the edges
# represent, these could be complexes, pathways, biological systems or similar. # represent, these could be complexes, pathways, biological systems or similar.
@ -597,11 +628,11 @@ lines(iGRGxy[get_diameter(iGRG),], lwd=10, col="#ff335533")
# http://www.ncbi.nlm.nih.gov/pubmed/18216267 and htttp://www.mapequation.org # http://www.ncbi.nlm.nih.gov/pubmed/18216267 and htttp://www.mapequation.org
iGRGclusters <- cluster_infomap(iGRG) myGRGclusters <- cluster_infomap(myGRG)
modularity(iGRGclusters) # ... measures how separated the different membership modularity(myGRGclusters) # ... measures how separated the different membership
# types are from each other # types are from each other
membership(iGRGclusters) # which nodes are in what cluster? membership(myGRGclusters) # which nodes are in what cluster?
table(membership(iGRGclusters)) # how large are the clusters? table(membership(myGRGclusters)) # how large are the clusters?
# The largest cluster has 48 members, the second largest has 25, etc. # The largest cluster has 48 members, the second largest has 25, etc.
@ -610,29 +641,24 @@ table(membership(iGRGclusters)) # how large are the clusters?
# their cluster membership: # their cluster membership:
# first, make a vector with as many grey colors as we have communities ... # first, make a vector with as many grey colors as we have communities ...
commColors <- rep("#f1eef6", max(membership(iGRGclusters))) commColors <- rep("#f1eef6", max(membership(myGRGclusters)))
# ... then overwrite the first five with "real colors" - something like rust, # ... then overwrite the first five with "real colors" - something like rust,
# lilac, pink, and mauve or so. # lilac, pink, and mauve or so.
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= rep(0,4)) # Turn margins off
plot(iGRG, plot(myGRG,
layout = iGRGxy, layout = cbind(rGAM$x, rGAM$y),
rescale = FALSE, rescale = FALSE,
xlim = c(min(iGRGxy[,1]), max(iGRGxy[,1])), xlim = c(min(rGAM$x) * 0.9, max(rGAM$x) * 1.1),
ylim = c(min(iGRGxy[,2]), max(iGRGxy[,2])), ylim = c(min(rGAM$y) * 0.9, max(rGAM$y) * 1.1),
vertex.color=commColors[membership(iGRGclusters)], vertex.color=commColors[membership(myGRGclusters)],
vertex.size = 0.1 + (0.1 * degree(iGRG)), vertex.size = 0.1 + (0.1 * degree(myGRG)),
vertex.label = "", vertex.label = NA)
edge.arrow.size = 0)
par(oPar) par(oPar)
# = 1 Tasks
# [END] # [END]