Network Analysis and Visualization with R and igraph
Katherine Ognyanova, www.kateto.net
NetSciX 2016 School of Code Workshop, Wroclaw, Poland
Contents
1. A quick reminder of R basics 3
1.1 Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Value comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Special constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Matrces & Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 Lists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 Data Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.9 Flow Control and loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.10 R plots and colors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.11 R troubleshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2. Networks in igraph 14
2.1 Create networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Edge, vertex, and network attributes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Specific graphs and graph models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3. Reading network data from files 27
3.1 DATASET 1: edgelist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 DATASET 2: matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4. Turning networks into igraph objects 28
4.1 Dataset 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Dataset 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1
5. Plotting networks with igraph 32
5.1 Plotting parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5.2 Network layouts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Improving network plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4 Interactive plotting with tkplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.5 Other ways to represent a network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.6 Plotting two-mode networks with igraph . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6. Network and node descriptives 50
6.1 Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.2 Reciprocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3 Transitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.4 Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.5 Node degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.6 Degree distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.7 Centrality & centralization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.8 Hubs and authorities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
7. Distances and paths 56
8. Subgroups and communities 59
8.1 Cliques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8.2 Community detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
8.3 K-core decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
9. Assortativity and Homophily 64
2
Note: You can download all workshop materials here, or visit kateto.net/netscix2016.
This tutorial covers basics of network analysis and visualization with the R package igraph (main-
tained by Gabor Csardi and Tamas Nepusz). The igraph library provides versatile options for
descriptive network analysis and visualization in R, Python, and C/C++. This workshop will focus
on the R implementation. You will need an R installation, and RStudio. You should also install the
latest version of igraph for R:
install.packages("igraph")
1. A quick reminder of R basics
Before we start working with networks, we will go through a quick introduction/reminder of some
simple tasks and principles in R.
1.1 Assignment
You can assign a value to an object using assign(), <-, or =.
x <- 3 # Assignment
x # Evaluate the expression and print result
y <- 4 # Assignment
y + 5 # Evaluation, y remains 4
z <- x + 17*y # Assignment
z # Evaluation
3
rm(z) # Remove z: deletes the object.
z # Error!
1.2 Value comparisons
We can use the standard operators <, >, <=, >=, ==(equality) and != (inequality). Comparisons
return Boolean values: TRUE or FALSE (often abbreviated to just T and F).
2==2 # Equality
2!=2 # Inequality
x <= y # less than or equal: "<", ">", and ">=" also work
1.3 Special constants
Special constants include:
• NA for missing or undefined data
• NULL for empty object (e.g. null/empty lists)
• Inf and -Inf for positive and negative infinity
• NaN for results that cannot be reasonably defined
# NA - missing or undefined data
5 + NA # When used in an expression, the result is generally NA
is.na(5+NA) # Check if missing
# NULL - an empty object, e.g. a null/empty list
10 + NULL # use returns an empty object (length zero)
is.null(NULL) # check if NULL
Inf and -Inf represent positive and negative infinity. They can be returned by mathematical
operations like division of a number by zero:
5/0
is.finite(5/0) # Check if a number is finite (it is not).
NaN (Not a Number) - the result of an operation that cannot be reasonably defined, such as dividing
zero by zero.
0/0
is.nan(0/0)
1.4 Vectors
Vectors can be constructed by combining their elements with the important R function c().
4
v1 <- c(1, 5, 11, 33) # Numeric vector, length 4
v2 <- c("hello","world") # Character vector, length 2 (a vector of strings)
v3 <- c(TRUE, TRUE, FALSE) # Logical vector, same as c(T, T, F)
Combining different types of elements in one vector will coerce the elements to the least restrictive
type:
v4 <- c(v1,v2,v3,"boo") # All elements turn into strings
Other ways to create vectors include:
v <- 1:7 # same as c(1,2,3,4,5,6,7)
v <- rep(0, 77) # repeat zero 77 times: v is a vector of 77 zeroes
v <- rep(1:3, times=2) # Repeat 1,2,3 twice
v <- rep(1:10, each=2) # Repeat each element twice
v <- seq(10,20,2) # sequence: numbers between 10 and 20, in jumps of 2
v1 <- 1:5 # 1,2,3,4,5
v2 <- rep(1,5) # 1,1,1,1,1
Check the length of a vector:
length(v1)
length(v2)
Element-wise operations:
v1 + v2 # Element-wise addition
v1 + 1 # Add 1 to each element
v1 * 2 # Multiply each element by 2
v1 + c(1,7) # This doesn't work: (1,7) is a vector of different length
Mathematical operations:
sum(v1) # The sum of all elements
mean(v1) # The average of all elements
sd(v1) # The standard deviation
cor(v1,v1*5) # Correlation between v1 and v1*5
Logical operations:
v1 > 2 # Each element is compared to 2, returns logical vector
v1==v2 # Are corresponding elements equivalent, returns logical vector.
v1!=v2 # Are corresponding elements *not* equivalent? Same as !(v1==v2)
(v1>2) | (v2>0) # | is the boolean OR, returns a vector.
(v1>2) & (v2>0) # & is the boolean AND, returns a vector.
(v1>2) || (v2>0) # || is the boolean OR, returns a single value
(v1>2) && (v2>0) # && is the boolean AND, ditto
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Vector elements:
v1[3] # third element of v1
v1[2:4] # elements 2, 3, 4 of v1
v1[c(1,3)] # elements 1 and 3 - note that your indexes are a vector
v1[c(T,T,F,F,F)] # elements 1 and 2 - only the ones that are TRUE
v1[v1>3] # v1>3 is a logical vector TRUE for elements >3
Note that the indexing in R starts from 1, a fact known to confuse and upset people used to
languages that index from 0.
To add more elements to a vector, simply assign them values.
v1[6:10] <- 6:10
We can also directly assign the vector a length:
length(v1) <- 15 # the last 5 elements are added as missing data: NA
1.5 Factors
Factors are used to store categorical data.
eye.col.v <- c("brown", "green", "brown", "blue", "blue", "blue") #vector
eye.col.f <- factor(c("brown", "green", "brown", "blue", "blue", "blue")) #factor
eye.col.v
## [1] "brown" "green" "brown" "blue" "blue" "blue"
eye.col.f
## [1] brown green brown blue blue blue
## Levels: blue brown green
R will identify the different levels of the factor - e.g. all distinct values. The data is stored internally
as integers - each number corresponding to a factor level.
levels(eye.col.f) # The levels (distinct values) of the factor (categorical var)
## [1] "blue" "brown" "green"
as.numeric(eye.col.f) # As numeric values: 1 is blue, 2 is brown, 3 is green
## [1] 2 3 2 1 1 1
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as.numeric(eye.col.v) # The character vector can not be coerced to numeric
## Warning: NAs introduced by coercion
## [1] NA NA NA NA NA NA
as.character(eye.col.f)
## [1] "brown" "green" "brown" "blue" "blue" "blue"
as.character(eye.col.v)
## [1] "brown" "green" "brown" "blue" "blue" "blue"
1.6 Matrces & Arrays
A matrix is a vector with dimensions:
m <- rep(1, 20) # A vector of 20 elements, all 1
dim(m) <- c(5,4) # Dimensions set to 5 & 4, so m is now a 5x4 matrix
Creating a matrix using matrix():
m <- matrix(data=1, nrow=5, ncol=4) # same matrix as above, 5x4, full of 1s
m <- matrix(1,5,4) # same matrix as above
dim(m) # What are the dimensions of m?
## [1] 5 4
Creating a matrix by combining vectors:
m <- cbind(1:5, 5:1, 5:9) # Bind 3 vectors as columns, 5x3 matrix
m <- rbind(1:5, 5:1, 5:9) # Bind 3 vectors as rows, 3x5 matrix
Selecting matrix elements:
m <- matrix(1:10,10,10)
m[2,3] # Matrix m, row 2, column 3 - a single cell
m[2,] # The whole second row of m as a vector
m[,2] # The whole second column of m as a vector
m[1:2,4:6] # submatrix: rows 1 and 2, columns 4, 5 and 6
m[-1,] # all rows *except* the first one
Other operations with matrices:
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# Are elements in row 1 equivalent to corresponding elements from column 1:
m[1,]==m[,1]
# A logical matrix: TRUE for m elements >3, FALSE otherwise:
m>3
# Selects only TRUE elements - that is ones greater than 3:
m[m>3]
t(m) # Transpose m
m <- t(m) # Assign m the transposed m
m %*% t(m) # %*% does matrix multiplication
m * m # * does element-wise multiplication
Arrays are used when we have more than 2 dimensions. We can create them using the array()
function:
a <- array(data=1:18,dim=c(3,3,2)) # 3d with dimensions 3x3x2
a <- array(1:18,c(3,3,2)) # the same array
1.7 Lists
Lists are collections of objects. A single list can contain all kinds of elements - character strings,
numeric vectors, matrices, other lists, and so on. The elements of lists are often named for easier
access.
l1 <- list(boo=v1,foo=v2,moo=v3,zoo="Animals!") # A list with four components
l2 <- list(v1,v2,v3,"Animals!")
Create an empty list:
l3 <- list()
l4 <- NULL
Accessing list elements:
l1["boo"] # Access boo with single brackets: this returns a list.
l1[["boo"]] # Access boo with double brackets: this returns the numeric vector
l1[[1]] # Returns the first component of the list, equivalent to above.
l1$boo # Named elements can be accessed with the $ operator, as with [[]]
Adding more elements to a list:
l3[[1]] <- 11 # add an element to the empty list l3
l4[[3]] <- c(22, 23) # add a vector as element 3 in the empty list l4.
Since we added element 3 to the list l4above, elements 1 and 2 will be generated and empty (NULL).
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l1[[5]] <- "More elements!" # The list l1 had 4 elements, we're adding a 5th here.
l1[[8]] <- 1:11
We added an 8th element, but not 6th and 7th to the listl1 above. Elements number 6 and 7 will
be created empty (NULL).
l1$Something <- "A thing" # Adds a ninth element - "A thing", named "Something"
1.8 Data Frames
The data frame is a special kind of list used for storing dataset tables. Think of rows as cases,
columns as variables. Each column is a vector or factor.
Creating a dataframe:
dfr1 <- data.frame( ID=1:4,
FirstName=c("John","Jim","Jane","Jill"),
Female=c(F,F,T,T),
Age=c(22,33,44,55) )
dfr1$FirstName # Access the second column of dfr1.
## [1] John Jim Jane Jill
## Levels: Jane Jill Jim John
Notice that R thinks that dfr1$FirstName is a categorical variable and so it’s treating it like a
factor, not a character vector. Let’s get rid of the factor by telling R to treat ‘FirstName’ as a
vector:
dfr1$FirstName <- as.vector(dfr1$FirstName)
Alternatively, you can tell R you don’t like factors from the start using stringsAsFactors=FALSE
dfr2 <- data.frame(FirstName=c("John","Jim","Jane","Jill"), stringsAsFactors=F)
dfr2$FirstName # Success: not a factor.
## [1] "John" "Jim" "Jane" "Jill"
Access elements of the data frame:
dfr1[1,] # First row, all columns
dfr1[,1] # First column, all rows
dfr1$Age # Age column, all rows
dfr1[1:2,3:4] # Rows 1 and 2, columns 3 and 4 - the gender and age of John & Jim
dfr1[c(1,3),] # Rows 1 and 3, all columns
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Find the names of everyone over the age of 30 in the data:
dfr1[dfr1$Age>30,2]
## [1] "Jim" "Jane" "Jill"
Find the average age of all females in the data:
mean ( dfr1[dfr1$Female==TRUE,4] )
## [1] 49.5
1.9 Flow Control and loops
The controls and loops in R are fairly straightforward (see below). They determine if a block of
code will be executed, and how many times. Blocks of code in R are enclosed in curly brackets {}.
# if (condition) expr1 else expr2
x <- 5; y <- 10
if (x==0) y <- 0 else y <- y/x #
y
## [1] 2
# for (variable in sequence) expr
ASum <- 0; AProd <- 1
for (i in 1:x)
{
ASum <- ASum + i
AProd <- AProd * i
}
ASum # equivalent to sum(1:x)
## [1] 15
AProd # equivalemt to prod(1:x)
## [1] 120
# while (condintion) expr
while (x > 0) {print(x); x <- x-1;}
# repeat expr, use break to exit the loop
repeat { print(x); x <- x+1; if (x>10) break}
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1.10 R plots and colors
In most R functions, you can use named colors, hex, or RGB values. In the simple base R plot chart
below, x and y are the point coordinates, pch is the point symbol shape, cex is the point size, and
col is the color. To see the parameters for plotting in base R, check out ?par
plot(x=1:10, y=rep(5,10), pch=19, cex=3, col="dark red")
points(x=1:10, y=rep(6, 10), pch=19, cex=3, col="557799")
points(x=1:10, y=rep(4, 10), pch=19, cex=3, col=rgb(.25, .5, .3))
You may notice that RGB here ranges from 0 to 1. While this is the R default, you can also set it
for to the 0-255 range using something like rgb(10, 100, 100, maxColorValue=255).
We can set the opacity/transparency of an element using the parameter alpha (range 0-1):
plot(x=1:5, y=rep(5,5), pch=19, cex=12, col=rgb(.25, .5, .3, alpha=.5), xlim=c(0,6))
If we have a hex color representation, we can set the transparency alpha using adjustcolor from
package grDevices. For fun, let’s also set the plot background to gray using the par() function for
graphical parameters.
par(bg="gray40")
col.tr <- grDevices::adjustcolor("557799", alpha=0.7)
plot(x=1:5, y=rep(5,5), pch=19, cex=12, col=col.tr, xlim=c(0,6))
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If you plan on using the built-in color names, here’s how to list all of them:
colors() # List all named colors
grep("blue", colors(), value=T) # Colors that have "blue" in the name
In many cases, we need a number of contrasting colors, or multiple shades of a color. R comes with
some predefined palette function that can generate those for us. For example:
pal1 <- heat.colors(5, alpha=1) # 5 colors from the heat palette, opaque
pal2 <- rainbow(5, alpha=.5) # 5 colors from the heat palette, transparent
plot(x=1:10, y=1:10, pch=19, cex=5, col=pal1)
plot(x=1:10, y=1:10, pch=19, cex=5, col=pal2)
We can also generate our own gradients using colorRampPalette. Note that colorRampPalette
returns a function that we can use to generate as many colors from that palette as we need.
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palf <- colorRampPalette(c("gray80", "dark red"))
plot(x=10:1, y=1:10, pch=19, cex=5, col=palf(10))
To add transparency to colorRampPalette, you need to use a parameter alpha=TRUE:
palf <- colorRampPalette(c(rgb(1,1,1, .2),rgb(.8,0,0, .7)), alpha=TRUE)
plot(x=10:1, y=1:10, pch=19, cex=5, col=palf(10))
1.11 R troubleshooting
While I generate many (and often very creative) errors in R, there are three simple things that will
most often go wrong for me. Those include:
1) Capitalization. R is case sensitive - a graph vertex named “Jack” is not the same as one
named “jack”. The function rowSums won’t work if spelled as rowsums or RowSums.
2) Object class. While many functions are willing to take anything you throw at them, some will
still surprisingly require character vector or a factor instead of a numeric vector, or a matrix
instead of a data frame. Functions will also occasionally return results in an unexpected
formats.
3) Package namespaces. Occasionally problems will arise when different packages contain
functions with the same name. R may warn you about this by saying something like “The
following object(s) are masked from ‘package:igraph’ as you load a package. One way to deal
with this is to call functions from a package explicitly using ::. For instance, if function
blah() is present in packages A and B, you can call A::blah and B::blah. In other cases
the problem is more complicated, and you may have to load packages in certain order, or not
use them together at all. For example (and pertinent to this workshop), igraph and Statnet
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packages cause some problems when loaded at the same time. It is best to detach one before
loading the other.
library(igraph) # load a package
detach(package:igraph) # detach a package
For more advanced troubleshooting, check out try(), tryCatch(), and debug().
2. Networks in igraph
rm(list = ls()) # Remove all the objects we created so far.
library(igraph) # Load the igraph package
2.1 Create networks
The code below generates an undirected graph with three edges. The numbers are interpreted as
vertex IDs, so the edges are 1–>2, 2–>3, 3–>1.
g1 <- graph( edges=c(1,2, 2,3, 3, 1), n=3, directed=F )
plot(g1) # A simple plot of the network - we'll talk more about plots later
1
2
3
class(g1)
## [1] "igraph"
g1
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## IGRAPH U--- 3 3 --
## + edges:
## [1] 1--2 2--3 1--3
# Now with 10 vertices, and directed by default:
g2 <- graph( edges=c(1,2, 2,3, 3, 1), n=10 )
plot(g2)
1
2
3
4
5
6
7
8
910
g2
## IGRAPH D--- 10 3 --
## + edges:
## [1] 1->2 2->3 3->1
g3 <- graph( c("John", "Jim", "Jim", "Jill", "Jill", "John")) # named vertices
# When the edge list has vertex names, the number of nodes is not needed
plot(g3)
John
Jim
Jill
g3
## IGRAPH DN-- 3 3 --
## + attr: name (v/c)
## + edges (vertex names):
## [1] John->Jim Jim ->Jill Jill->John
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g4 <- graph( c("John", "Jim", "Jim", "Jack", "Jim", "Jack", "John", "John"),
isolates=c("Jesse", "Janis", "Jennifer", "Justin") )
# In named graphs we can specify isolates by providing a list of their names.
plot(g4, edge.arrow.size=.5, vertex.color="gold", vertex.size=15,
vertex.frame.color="gray", vertex.label.color="black",
vertex.label.cex=0.8, vertex.label.dist=2, edge.curved=0.2)
John
Jim
Jack
Jesse
Janis
Justin
Small graphs can also be generated with a description of this kind: - for undirected tie, +- or -+
for directed ties pointing left & right, ++ for a symmetric tie, and “:” for sets of vertices.
plot(graph_from_literal(a---b, b---c)) # the number of dashes doesn't matter
a
b
c
plot(graph_from_literal(a--+b, b+--c))
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ab
c
plot(graph_from_literal(a+-+b, b+-+c))
a
b
c
plot(graph_from_literal(a:b:c---c:d:e))
a
b
c
d
e
gl <- graph_from_literal(a-b-c-d-e-f, a-g-h-b, h-e:f:i, j)
plot(gl)
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ab
c
d
e
f
g
h
i
j
2.2 Edge, vertex, and network attributes
Access vertices and edges:
E(g4) # The edges of the object
## + 4/4 edges (vertex names):
## [1] John->Jim Jim ->Jack Jim ->Jack John->John
V(g4) # The vertices of the object
## + 7/7 vertices, named:
## [1] John Jim Jack Jesse Janis Jennifer Justin
You can also examine the network matrix directly:
g4[]
## 7 x 7 sparse Matrix of class "dgCMatrix"
## John Jim Jack Jesse Janis Jennifer Justin
## John 1 1 . . . . .
## Jim . . 2 . . . .
## Jack . . . . . . .
## Jesse . . . . . . .
## Janis . . . . . . .
## Jennifer . . . . . . .
## Justin . . . . . . .
g4[1,]
## John Jim Jack Jesse Janis Jennifer Justin
## 1 1 0 0 0 0 0
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Add attributes to the network, vertices, or edges:
V(g4)$name # automatically generated when we created the network.
## [1] "John" "Jim" "Jack" "Jesse" "Janis" "Jennifer"
## [7] "Justin"
V(g4)$gender <- c("male", "male", "male", "male", "female", "female", "male")
E(g4)$type <- "email" # Edge attribute, assign "email" to all edges
E(g4)$weight <- 10 # Edge weight, setting all existing edges to 10
Examine attributes:
edge_attr(g4)
## $type
## [1] "email" "email" "email" "email"
##
## $weight
## [1] 10 10 10 10
vertex_attr(g4)
## $name
## [1] "John" "Jim" "Jack" "Jesse" "Janis" "Jennifer"
## [7] "Justin"
##
## $gender
## [1] "male" "male" "male" "male" "female" "female" "male"
graph_attr(g4)
## named list()
Another way to set attributes (you can similarly use set_edge_attr(), set_vertex_attr(), etc.):
g4 <- set_graph_attr(g4, "name", "Email Network")
g4 <- set_graph_attr(g4, "something", "A thing")
graph_attr_names(g4)
## [1] "name" "something"
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graph_attr(g4, "name")
## [1] "Email Network"
graph_attr(g4)
## $name
## [1] "Email Network"
##
## $something
## [1] "A thing"
g4 <- delete_graph_attr(g4, "something")
graph_attr(g4)
## $name
## [1] "Email Network"
plot(g4, edge.arrow.size=.5, vertex.label.color="black", vertex.label.dist=1.5,
vertex.color=c( "pink", "skyblue")[1+(V(g4)$gender=="male")] )
John
Jim
Jack
Jesse
Janis
Jennifer
Justin
The graph g4 has two edges going from Jim to Jack, and a loop from John to himself. We can
simplify our graph to remove loops & multiple edges between the same nodes. Use edge.attr.comb
to indicate how edge attributes are to be combined - possible options include sum, mean, prod
(product), min, max, first/last (selects the first/last edge’s attribute). Option “ignore” says the
attribute should be disregarded and dropped.
g4s <- simplify( g4, remove.multiple = T, remove.loops = F,
edge.attr.comb=c(weight="sum", type="ignore") )
plot(g4s, vertex.label.dist=1.5)
20
John
Jim
Jack
Jesse
Janis
Jennifer
Justin
g4s
## IGRAPH DNW- 7 3 -- Email Network
## + attr: name (g/c), name (v/c), gender (v/c), weight (e/n)
## + edges (vertex names):
## [1] John->John John->Jim Jim ->Jack
The description of an igraph object starts with up to four letters:
1. D or U, for a directed or undirected graph
2. N for a named graph (where nodes have a name attribute)
3. W for a weighted graph (where edges have a weight attribute)
4. B for a bipartite (two-mode) graph (where nodes have a type attribute)
The two numbers that follow (7 5) refer to the number of nodes and edges in the graph. The
description also lists node & edge attributes, for example:
• (g/c) - graph-level character attribute
• (v/c) - vertex-level character attribute
• (e/n) - edge-level numeric attribute
2.3 Specific graphs and graph models
Empty graph
eg <- make_empty_graph(40)
plot(eg, vertex.size=10, vertex.label=NA)
21
Full graph
fg <- make_full_graph(40)
plot(fg, vertex.size=10, vertex.label=NA)
Simple star graph
st <- make_star(40)
plot(st, vertex.size=10, vertex.label=NA)
Tree graph
22
tr <- make_tree(40, children = 3, mode = "undirected")
plot(tr, vertex.size=10, vertex.label=NA)
Ring graph
rn <- make_ring(40)
plot(rn, vertex.size=10, vertex.label=NA)
Erdos-Renyi random graph model
(‘n’ is number of nodes, ‘m’ is the number of edges).
er <- sample_gnm(n=100, m=40)
plot(er, vertex.size=6, vertex.label=NA)
23
Watts-Strogatz small-world model
Creates a lattice (with dim dimensions and size nodes across dimension) and rewires edges randomly
with probability p. The neighborhood in which edges are connected is nei. You can allow loops
and multiple edges.
sw <- sample_smallworld(dim=2, size=10, nei=1, p=0.1)
plot(sw, vertex.size=6, vertex.label=NA, layout=layout_in_circle)
Barabasi-Albert preferential attachment model for scale-free graphs
(n is number of nodes, power is the power of attachment (1 is linear); m is the number of edges
added on each time step)
ba <- sample_pa(n=100, power=1, m=1, directed=F)
plot(ba, vertex.size=6, vertex.label=NA)
igraph can also give you some notable historical graphs. For instance:
zach <- graph("Zachary") # the Zachary carate club
plot(zach, vertex.size=10, vertex.label=NA)
24
Rewiring a graph
each_edge() is a rewiring method that changes the edge endpoints uniformly randomly with a
probability prob.
rn.rewired <- rewire(rn, each_edge(prob=0.1))
plot(rn.rewired, vertex.size=10, vertex.label=NA)
Rewire to connect vertices to other vertices at a certain distance.
rn.neigh = connect.neighborhood(rn, 5)
plot(rn.neigh, vertex.size=8, vertex.label=NA)
25
Combine graphs (disjoint union, assuming separate vertex sets): %du%
plot(rn, vertex.size=10, vertex.label=NA)
plot(tr, vertex.size=10, vertex.label=NA)
plot(rn %du% tr, vertex.size=10, vertex.label=NA)
26
3. Reading network data from files
In the following sections of the tutorial, we will work primarily with two small example data sets.
Both contain data about media organizations. One involves a network of hyperlinks and mentions
among news sources. The second is a network of links between media venues and consumers. While
the example data used here is small, many of the ideas behind the analyses and visualizations we
will generate apply to medium and large-scale networks.
3.1 DATASET 1: edgelist
The first data set we are going to work with consists of two files, “Media-Example-NODES.csv” and
“Media-Example-EDGES.csv” (download here).
nodes <- read.csv("Dataset1-Media-Example-NODES.csv", header=T, as.is=T)
links <- read.csv("Dataset1-Media-Example-EDGES.csv", header=T, as.is=T)
Examine the data:
head(nodes)
head(links)
nrow(nodes); length(unique(nodes$id))
nrow(links); nrow(unique(links[,c("from", "to")]))
Notice that there are more links than unique from-to combinations. That means we have cases
in the data where there are multiple links between the same two nodes. We will collapse all links
of the same type between the same two nodes by summing their weights, using aggregate() by
“from”, “to”, & “type”. We don’t use simplify() here so as not to collapse different link types.
links <- aggregate(links[,3], links[,-3], sum)
links <- links[order(links$from, links$to),]
colnames(links)[4] <- "weight"
rownames(links) <- NULL
3.2 DATASET 2: matrix
Two-mode or bipartite graphs have two different types of actors and links that go across, but not
within each type. Our second media example is a network of that kind, examining links between
news sources and their consumers.
nodes2 <- read.csv("Dataset2-Media-User-Example-NODES.csv", header=T, as.is=T)
links2 <- read.csv("Dataset2-Media-User-Example-EDGES.csv", header=T, row.names=1)
Examine the data:
27
head(nodes2)
head(links2)
We can see that links2 is an adjacency matrix for a two-mode network:
links2 <- as.matrix(links2)
dim(links2)
dim(nodes2)
———————————–
4. Turning networks into igraph objects
We start by converting the raw data to an igraph network object. Here we use igraph’s
graph.data.frame function, which takes two data frames: d and vertices.
• d describes the edges of the network. Its first two columns are the IDs of the source and the
target node for each edge. The following columns are edge attributes (weight, type, label, or
anything else).
• vertices starts with a column of node IDs. Any following columns are interpreted as node
attributes.
4.1 Dataset 1
library(igraph)
net <- graph_from_data_frame(d=links, vertices=nodes, directed=T)
class(net)
## [1] "igraph"
net
## IGRAPH DNW- 17 49 --
## + attr: name (v/c), media (v/c), media.type (v/n), type.label
## | (v/c), audience.size (v/n), type (e/c), weight (e/n)
## + edges (vertex names):
## [1] s01->s02 s01->s03 s01->s04 s01->s15 s02->s01 s02->s03 s02->s09
## [8] s02->s10 s03->s01 s03->s04 s03->s05 s03->s08 s03->s10 s03->s11
## [15] s03->s12 s04->s03 s04->s06 s04->s11 s04->s12 s04->s17 s05->s01
## [22] s05->s02 s05->s09 s05->s15 s06->s06 s06->s16 s06->s17 s07->s03
## [29] s07->s08 s07->s10 s07->s14 s08->s03 s08->s07 s08->s09 s09->s10
## [36] s10->s03 s12->s06 s12->s13 s12->s14 s13->s12 s13->s17 s14->s11
## [43] s14->s13 s15->s01 s15->s04 s15->s06 s16->s06 s16->s17 s17->s04
28
We also have easy access to nodes, edges, and their attributes with:
E(net) # The edges of the "net" object
V(net) # The vertices of the "net" object
E(net)$type # Edge attribute "type"
V(net)$media # Vertex attribute "media"
Now that we have our igraph network object, let’s make a first attempt to plot it.
plot(net, edge.arrow.size=.4,vertex.label=NA)
That doesn’t look very good. Let’s start fixing things by removing the loops in the graph.
net <- simplify(net, remove.multiple = F, remove.loops = T)
You might notice that we could have used simplify to combine multiple edges by summing their
weights with a command like simplify(net, edge.attr.comb=list(weight="sum","ignore")).
The problem is that this would also combine multiple edge types (in our data: “hyperlinks” and
“mentions”).
If you need them, you can extract an edge list or a matrix from igraph networks.
as_edgelist(net, names=T)
as_adjacency_matrix(net, attr="weight")
Or data frames describing nodes and edges:
as_data_frame(net, what="edges")
as_data_frame(net, what="vertices")
29
4.2 Dataset 2
As we have seen above, this time the edges of the network are in a matrix format. We can read
those into a graph object using graph_from_incidence_matrix(). In igraph, bipartite networks
have a node attribute called type that is FALSE (or 0) for vertices in one mode and TRUE (or 1)
for those in the other mode.
head(nodes2)
## id media media.type media.name audience.size
## 1 s01 NYT 1 Newspaper 20
## 2 s02 WaPo 1 Newspaper 25
## 3 s03 WSJ 1 Newspaper 30
## 4 s04 USAT 1 Newspaper 32
## 5 s05 LATimes 1 Newspaper 20
## 6 s06 CNN 2 TV 56
head(links2)
## U01 U02 U03 U04 U05 U06 U07 U08 U09 U10 U11 U12 U13 U14 U15 U16 U17
## s01 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## s02 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0
## s03 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0
## s04 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0
## s05 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0
## s06 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 1
## U18 U19 U20
## s01 0 0 0
## s02 0 0 1
## s03 0 0 0
## s04 0 0 0
## s05 0 0 0
## s06 0 0 0
net2 <- graph_from_incidence_matrix(links2)
table(V(net2)$type)
##
## FALSE TRUE
## 10 20
To transform a one-mode network matrix into an igraph object, use instead graph_from_adjacency_matrix().
We can also easily generate bipartite projections for the two-mode network: (co-memberships are
easy to calculate by multiplying the network matrix by its transposed matrix, or using igraph’s
bipartite.projection() function).
30
net2.bp <- bipartite.projection(net2)
We can calculate the projections manually as well:
as_incidence_matrix(net2) %*% t(as_incidence_matrix(net2))
t(as_incidence_matrix(net2)) %*% as_incidence_matrix(net2)
plot(net2.bp$proj1, vertex.label.color="black", vertex.label.dist=1,
vertex.size=7, vertex.label=nodes2$media[!is.na(nodes2$media.type)])
NYT
WaPo
WSJ
USAT
LATimes
CNN
MSNBC
FOX
ABC
BBC
plot(net2.bp$proj2, vertex.label.color="black", vertex.label.dist=1,
vertex.size=7, vertex.label=nodes2$media[ is.na(nodes2$media.type)])
John
Mary
Paul
Ted
Tom
Kate
Ed
Anna
Dan
Nancy
Sandra
RondaSheila
Jim
Jill
Jo
BrianJason
Lisa
Dave
31
5. Plotting networks with igraph
Plotting with igraph: the network plots have a wide set of parameters you can set. Those include
node options (starting with vertex.) and edge options (starting with edge.). A list of selected
options is included below, but you can also check out ?igraph.plotting for more information.
The igraph plotting parameters include (among others):
5.1 Plotting parameters
NODES
vertex.color Node color
vertex.frame.color Node border color
vertex.shape One of “none”, “circle”, “square”, “csquare”, “rectangle”
“crectangle”, “vrectangle”, “pie”, “raster”, or “sphere”
vertex.size Size of the node (default is 15)
vertex.size2 The second size of the node (e.g. for a rectangle)
vertex.label Character vector used to label the nodes
vertex.label.family Font family of the label (e.g.“Times”, “Helvetica”)
vertex.label.font Font: 1 plain, 2 bold, 3, italic, 4 bold italic, 5 symbol
vertex.label.cex Font size (multiplication factor, device-dependent)
vertex.label.dist Distance between the label and the vertex
vertex.label.degree The position of the label in relation to the vertex,
where 0 right, “pi” is left, “pi/2” is below, and “-pi/2” is above
EDGES
edge.color Edge color
edge.width Edge width, defaults to 1
edge.arrow.size Arrow size, defaults to 1
edge.arrow.width Arrow width, defaults to 1
edge.lty Line type, could be 0 or “blank”, 1 or “solid”, 2 or “dashed”,
3 or “dotted”, 4 or “dotdash”, 5 or “longdash”, 6 or “twodash”
edge.label Character vector used to label edges
edge.label.family Font family of the label (e.g.“Times”, “Helvetica”)
edge.label.font Font: 1 plain, 2 bold, 3, italic, 4 bold italic, 5 symbol
edge.label.cex Font size for edge labels
edge.curved Edge curvature, range 0-1 (FALSE sets it to 0, TRUE to 0.5)
arrow.mode Vector specifying whether edges should have arrows,
possible values: 0 no arrow, 1 back, 2 forward, 3 both
OTHER
margin Empty space margins around the plot, vector with length 4
frame if TRUE, the plot will be framed
main If set, adds a title to the plot
sub If set, adds a subtitle to the plot
32
We can set the node & edge options in two ways - the first one is to specify them in the plot()
function, as we are doing below.
# Plot with curved edges (edge.curved=.1) and reduce arrow size:
plot(net, edge.arrow.size=.4, edge.curved=.1)
s01
s02
s03
s04
s05
s06
s07
s08
s09
s10
s11
s12
s13
s14
s15
s16s17
# Set edge color to gray, and the node color to orange.
# Replace the vertex label with the node names stored in "media"
plot(net, edge.arrow.size=.2, edge.curved=0,
vertex.color="orange", vertex.frame.color="#555555",
vertex.label=V(net)$media, vertex.label.color="black",
vertex.label.cex=.7)
NY Times
Washington Post
Wall Street Journal
USA Today
LA Times
New York Post
CNNMSNBC
FOX News
ABC
BBC
Yahoo NewsGoogle News
Reuters.com
NYTimes.com
WashingtonPost.com
AOL.com
The second way to set attributes is to add them to the igraph object. Let’s say we want to color
our network nodes based on type of media, and size them based on audience size (larger audience
-> larger node). We will also change the width of the edges based on their weight.
33
# Generate colors based on media type:
colrs <- c("gray50", "tomato", "gold")
V(net)$color <- colrs[V(net)$media.type]
# Set node size based on audience size:
V(net)$size <- V(net)$audience.size*0.7
# The labels are currently node IDs.
# Setting them to NA will render no labels:
V(net)$label.color <- "black"
V(net)$label <- NA
# Set edge width based on weight:
E(net)$width <- E(net)$weight/6
#change arrow size and edge color:
E(net)$arrow.size <- .2
E(net)$edge.color <- "gray80"
E(net)$width <- 1+E(net)$weight/12
We can also override the attributes explicitly in the plot:
plot(net, edge.color="orange", vertex.color="gray50")
34
It helps to add a legend explaining the meaning of the colors we used:
plot(net)
legend(x=-1.5, y=-1.1, c("Newspaper","Television", "Online News"), pch=21,
col="#777777", pt.bg=colrs, pt.cex=2, cex=.8, bty="n", ncol=1)
Newspaper
Television
Online News
Sometimes, especially with semantic networks, we may be interested in plotting only the labels of
the nodes:
plot(net, vertex.shape="none", vertex.label=V(net)$media,
vertex.label.font=2, vertex.label.color="gray40",
vertex.label.cex=.7, edge.color="gray85")
35
NY Times
Washington Post
Wall Street Journal
USA Today
LA Times
New York Post
CNN
MSNBCFOX News
ABC
BBC
Yahoo News
Google News
Reuters.com
NYTimes.com
WashingtonPost.com
AOL.com
Let’s color the edges of the graph based on their source node color. We can get the starting node
for each edge with the ends() igraph function.
edge.start <- ends(net, es=E(net), names=F)[,1]
edge.col <- V(net)$color[edge.start]
plot(net, edge.color=edge.col, edge.curved=.1)
5.2 Network layouts
Network layouts are simply algorithms that return coordinates for each node in a network.
For the purposes of exploring layouts, we will generate a slightly larger 80-node graph. We use the
sample_pa() function which generates a simple graph starting from one node and adding more
nodes and links based on a preset level of preferential attachment (Barabasi-Albert model).
36
net.bg <- sample_pa(80)
V(net.bg)$size <- 8
V(net.bg)$frame.color <- "white"
V(net.bg)$color <- "orange"
V(net.bg)$label <- ""
E(net.bg)$arrow.mode <- 0
plot(net.bg)
You can set the layout in the plot function:
plot(net.bg, layout=layout_randomly)
Or you can calculate the vertex coordinates in advance:
l <- layout_in_circle(net.bg)
plot(net.bg, layout=l)
37
l is simply a matrix of x, y coordinates (N x 2) for the N nodes in the graph. You can easily
generate your own:
l <- cbind(1:vcount(net.bg), c(1, vcount(net.bg):2))
plot(net.bg, layout=l)
This layout is just an example and not very helpful - thankfully igraph has a number of built-in
layouts, including:
# Randomly placed vertices
l <- layout_randomly(net.bg)
plot(net.bg, layout=l)
38
# Circle layout
l <- layout_in_circle(net.bg)
plot(net.bg, layout=l)
# 3D sphere layout
l <- layout_on_sphere(net.bg)
plot(net.bg, layout=l)
39
Fruchterman-Reingold is one of the most used force-directed layout algorithms out there.
Force-directed layouts try to get a nice-looking graph where edges are similar in length and cross
each other as little as possible. They simulate the graph as a physical system. Nodes are electrically
charged particles that repulse each other when they get too close. The edges act as springs that
attract connected nodes closer together. As a result, nodes are evenly distributed through the chart
area, and the layout is intuitive in that nodes which share more connections are closer to each
other. The disadvantage of these algorithms is that they are rather slow and therefore less often
used in graphs larger than ~1000 vertices. You can set the “weight” parameter which increases the
attraction forces among nodes connected by heavier edges.
l <- layout_with_fr(net.bg)
plot(net.bg, layout=l)
You will notice that the layout is not deterministic - different runs will result in slightly different
configurations. Saving the layout in l allows us to get the exact same result multiple times, which
can be helpful if you want to plot the time evolution of a graph, or different relationships – and
want nodes to stay in the same place in multiple plots.
par(mfrow=c(2,2), mar=c(0,0,0,0)) # plot four figures - 2 rows, 2 columns
plot(net.bg, layout=layout_with_fr)
plot(net.bg, layout=layout_with_fr)
plot(net.bg, layout=l)
plot(net.bg, layout=l)
40
dev.off()
By default, the coordinates of the plots are rescaled to the [-1,1] interval for both x and y. You can
change that with the parameter rescale=FALSE and rescale your plot manually by multiplying the
coordinates by a scalar. You can use norm_coords to normalize the plot with the boundaries you
want.
l <- layout_with_fr(net.bg)
l <- norm_coords(l, ymin=-1, ymax=1, xmin=-1, xmax=1)
par(mfrow=c(2,2), mar=c(0,0,0,0))
plot(net.bg, rescale=F, layout=l*0.4)
plot(net.bg, rescale=F, layout=l*0.6)
plot(net.bg, rescale=F, layout=l*0.8)
plot(net.bg, rescale=F, layout=l*1.0)
41
dev.off()
Another popular force-directed algorithm that produces nice results for connected graphs is Kamada
Kawai. Like Fruchterman Reingold, it attempts to minimize the energy in a spring system.
l <- layout_with_kk(net.bg)
plot(net.bg, layout=l)
The LGL algorithm is meant for large, connected graphs. Here you can also specify a root: a node
that will be placed in the middle of the layout.
42
plot(net.bg, layout=layout_with_lgl)
Let’s take a look at all available layouts in igraph:
layouts <- grep("^layout_", ls("package:igraph"), value=TRUE)[-1]
# Remove layouts that do not apply to our graph.
layouts <- layouts[!grepl("bipartite|merge|norm|sugiyama|tree", layouts)]
par(mfrow=c(3,3), mar=c(1,1,1,1))
for (layout in layouts) {
print(layout)
l <- do.call(layout, list(net))
plot(net, edge.arrow.mode=0, layout=l, main=layout) }
layout_as_star layout_components layout_in_circle
layout_nicely layout_on_grid layout_on_sphere
43
layout_randomly layout_with_dh layout_with_drl
layout_with_fr layout_with_gem layout_with_graphopt
layout_with_kk layout_with_lgl layout_with_mds
5.3 Improving network plots
Notice that our network plot is still not too helpful. We can identify the type and size of nodes,
but cannot see much about the structure since the links we’re examining are so dense. One way to
approach this is to see if we can sparsify the network, keeping only the most important ties and
discarding the rest.
hist(links$weight)
mean(links$weight)
sd(links$weight)
There are more sophisticated ways to extract the key edges, but for the purposes of this exercise
we’ll only keep ones that have weight higher than the mean for the network. In igraph, we can
delete edges using delete_edges(net, edges):
44
cut.off <- mean(links$weight)
net.sp <- delete_edges(net, E(net)[weightplot(net.sp)
Another way to think about this is to plot the two tie types (hyperlink & mention) separately.
E(net)$width <- 1.5
plot(net, edge.color=c("dark red", "slategrey")[(E(net)$type=="hyperlink")+1],
vertex.color="gray40", layout=layout.circle)
net.m <- net - E(net)[E(net)$type=="hyperlink"] # another way to delete edges
net.h <- net - E(net)[E(net)$type=="mention"]
# Plot the two links separately:
par(mfrow=c(1,2))
plot(net.h, vertex.color="orange", main="Tie: Hyperlink")
plot(net.m, vertex.color="lightsteelblue2", main="Tie: Mention")
45
Tie: Hyperlink Tie: Mention
# Make sure the nodes stay in place in both plots:
l <- layout_with_fr(net)
plot(net.h, vertex.color="orange", layout=l, main="Tie: Hyperlink")
plot(net.m, vertex.color="lightsteelblue2", layout=l, main="Tie: Mention")
Tie: Hyperlink Tie: Mention
dev.off()
5.4 Interactive plotting with tkplot
R and igraph allow for interactive plotting of networks. This might be a useful option for you if you
want to tweak slightly the layout of a small graph. After adjusting the layout manually, you can get
the coordinates of the nodes and use them for other plots.
tkid <- tkplot(net) #tkid is the id of the tkplot that will open
l <- tkplot.getcoords(tkid) # grab the coordinates from tkplot
tk_close(tkid, window.close = T)
plot(net, layout=l)
46
5.5 Other ways to represent a network
At this point it might be useful to provide a quick reminder that there are many ways to represent
a network not limited to a hairball plot.
For example, here is a quick heatmap of the network matrix:
netm <- get.adjacency(net, attr="weight", sparse=F)
colnames(netm) <- V(net)$media
rownames(netm) <- V(net)$media
palf <- colorRampPalette(c("gold", "dark orange"))
heatmap(netm[,17:1], Rowv = NA, Colv = NA, col = palf(100),
scale="none", margins=c(10,10) )
47
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NY Times
Washington Post
Wall Street Journal
USA Today
LA Times
New York Post
CNN
MSNBC
FOX News
ABC
BBC
Yahoo News
Google News
Reuters.com
NYTimes.com
WashingtonPost.com
AOL.com
5.6 Plotting two-mode networks with igraph
As with one-mode networks, we can modify the network object to include the visual properties that
will be used by default when plotting the network. Notice that this time we will also change the
shape of the nodes - media outlets will be squares, and their users will be circles.
V(net2)$color <- c("steel blue", "orange")[V(net2)$type+1]
V(net2)$shape <- c("square", "circle")[V(net2)$type+1]
V(net2)$label <- ""
V(net2)$label[V(net2)$type==F] <- nodes2$media[V(net2)$type==F]
V(net2)$label.cex=.4
V(net2)$label.font=2
plot(net2, vertex.label.color="white", vertex.size=(2-V(net2)$type)*8)
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NYT
WaPo
WSJ
USAT
LATimes
CNN
MSNBCFOX
ABC
BBC
Igraph also has a special layout for bipartite networks (though it doesn’t always work great, and
you might be better off generating your own two-mode layout).
plot(net2, vertex.label=NA, vertex.size=7, layout=layout_as_bipartite)
Using text as nodes may be helpful at times:
plot(net2, vertex.shape="none", vertex.label=nodes2$media,
vertex.label.color=V(net2)$color, vertex.label.font=2.5,
vertex.label.cex=.6, edge.color="gray70", edge.width=2)
49
NYT
WaPo
WSJ
USAT
LATimes
CNN
MSNBC
FOX
ABC
BBC
John
Mary
Paul
Ted
Tom
Kate
Ed
Anna
Dan
Nancy
Sandra
Ronda
Sheila
Jim
Jill
Jo Brian
Jason
Lisa
Dave
6. Network and node descriptives
6.1 Density
The proportion of present edges from all possible edges in the network.
edge_density(net, loops=F)
## [1] 0.1764706
ecount(net)/(vcount(net)*(vcount(net)-1)) #for a directed network
## [1] 0.1764706
6.2 Reciprocity
The proportion of reciprocated ties (for a directed network).
reciprocity(net)
dyad_census(net) # Mutual, asymmetric, and null node pairs
2*dyad_census(net)$mut/ecount(net) # Calculating reciprocity
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6.3 Transitivity
• global - ratio of triangles (direction disregarded) to connected triples.
• local - ratio of triangles to connected triples each vertex is part of.
transitivity(net, type="global") # net is treated as an undirected network
transitivity(as.undirected(net, mode="collapse")) # same as above
transitivity(net, type="local")
triad_census(net) # for directed networks
Triad types (per Davis & Leinhardt):
• 003 A, B, C, empty triad.
• 012 A->B, C
• 102 A<->B, C
• 021D A<-B->C
• 021U A->B<-C
• 021C A->B->C
• 111D A<->B<-C
• 111U A<->B->C
• 030T A->B<-C, A->C
• 030C A<-B<-C, A->C.
• 201 A<->B<->C.
• 120D A<-B->C, A<->C.
• 120U A->B<-C, A<->C.
• 120C A->B->C, A<->C.
• 210 A->B<->C, A<->C.
• 300 A<->B<->C, A<->C, completely connected.
6.4 Diameter
A network diameter is the longest geodesic distance (length of the shortest path between two nodes)
in the network. In igraph, diameter() returns the distance, while get_diameter() returns the
nodes along the first found path of that distance.
Note that edge weights are used by default, unless set to NA.
diameter(net, directed=F, weights=NA)
## [1] 4
diameter(net, directed=F)
## [1] 28
51
diam <- get_diameter(net, directed=T)
diam
## + 7/17 vertices, named:
## [1] s12 s06 s17 s04 s03 s08 s07
Note that get_diameter() returns a vertex sequence. Note though that when asked to behaved as
a vector, a vertex sequence will produce the numeric indexes of the nodes in it. The same applies
for edge sequences.
class(diam)
## [1] "igraph.vs"
as.vector(diam)
## [1] 12 6 17 4 3 8 7
Color nodes along the diameter:
vcol <- rep("gray40", vcount(net))
vcol[diam] <- "gold"
ecol <- rep("gray80", ecount(net))
ecol[E(net, path=diam)] <- "orange"
# E(net, path=diam) finds edges along a path, here 'diam'
plot(net, vertex.color=vcol, edge.color=ecol, edge.arrow.mode=0)
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6.5 Node degrees
The function degree() has a mode of in for in-degree, out for out-degree, and all or total for
total degree.
deg <- degree(net, mode="all")
plot(net, vertex.size=deg*3)
hist(deg, breaks=1:vcount(net)-1, main="Histogram of node degree")
Histogram of node degree
deg
Fr
eq
ue
nc
y
0 5 10 15
0
1
2
3
4
5
6
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6.6 Degree distribution
deg.dist <- degree_distribution(net, cumulative=T, mode="all")
plot( x=0:max(deg), y=1-deg.dist, pch=19, cex=1.2, col="orange",
xlab="Degree", ylab="Cumulative Frequency")
0 2 4 6 8 10 12
0.
0
0.
2
0.
4
0.
6
0.
8
Degree
Cu
m
u
la
tiv
e
F
re
qu
en
cy
6.7 Centrality & centralization
Centrality functions (vertex level) and centralization functions (graph level). The centralization
functions return res - vertex centrality, centralization, and theoretical_max - maximum
centralization score for a graph of that size. The centrality function can run on a subset of nodes
(set with the vids parameter). This is helpful for large graphs where calculating all centralities may
be a resource-intensive and time-consuming task.
Degree (number of ties)
degree(net, mode="in")
centr_degree(net, mode="in", normalized=T)
Closeness (centrality based on distance to others in the graph)
Inverse of the node’s average geodesic distance to others in the network.
closeness(net, mode="all", weights=NA)
centr_clo(net, mode="all", normalized=T)
54
Eigenvector (centrality proportional to the sum of connection centralities)
Values of the first eigenvector of the graph matrix.
eigen_centrality(net, directed=T, weights=NA)
centr_eigen(net, directed=T, normalized=T)
Betweenness (centrality based on a broker position connecting others)
Number of geodesics that pass through the node or the edge.
betweenness(net, directed=T, weights=NA)
edge_betweenness(net, directed=T, weights=NA)
centr_betw(net, directed=T, normalized=T)
6.8 Hubs and authorities
The hubs and authorities algorithm developed by Jon Kleinberg was initially used to examine
web pages. Hubs were expected to contain catalogs with a large number of outgoing links; while
authorities would get many incoming links from hubs, presumably because of their high-quality
relevant information.
hs <- hub_score(net, weights=NA)$vector
as <- authority_score(net, weights=NA)$vector
par(mfrow=c(1,2))
plot(net, vertex.size=hs*50, main="Hubs")
plot(net, vertex.size=as*30, main="Authorities")
Hubs Authorities
dev.off()
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7. Distances and paths
Average path length: the mean of the shortest distance between each pair of nodes in the network
(in both directions for directed graphs).
mean_distance(net, directed=F)
## [1] 2.058824
mean_distance(net, directed=T)
## [1] 2.742188
We can also find the length of all shortest paths in the graph:
distances(net) # with edge weights
distances(net, weights=NA) # ignore weights
We can extract the distances to a node or set of nodes we are interested in. Here we will get the
distance of every media from the New York Times.
dist.from.NYT <- distances(net, v=V(net)[media=="NY Times"], to=V(net), weights=NA)
# Set colors to plot the distances:
oranges <- colorRampPalette(c("dark red", "gold"))
col <- oranges(max(dist.from.NYT)+1)
col <- col[dist.from.NYT+1]
plot(net, vertex.color=col, vertex.label=dist.from.NYT, edge.arrow.size=.6,
vertex.label.color="white")
0
1 1
1
1
2
22
2 2
2
2 3
3
1
3
2
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We can also find the shortest path between specific nodes. Say here between MSNBC and the New
York Post:
news.path <- shortest_paths(net,
from = V(net)[media=="MSNBC"],
to = V(net)[media=="New York Post"],
output = "both") # both path nodes and edges
# Generate edge color variable to plot the path:
ecol <- rep("gray80", ecount(net))
ecol[unlist(news.path$epath)] <- "orange"
# Generate edge width variable to plot the path:
ew <- rep(2, ecount(net))
ew[unlist(news.path$epath)] <- 4
# Generate node color variable to plot the path:
vcol <- rep("gray40", vcount(net))
vcol[unlist(news.path$vpath)] <- "gold"
plot(net, vertex.color=vcol, edge.color=ecol,
edge.width=ew, edge.arrow.mode=0)
Identify the edges going into or out of a vertex, for instance the WSJ. For a single node, use
incident(), for multiple nodes use incident_edges()
inc.edges <- incident(net, V(net)[media=="Wall Street Journal"], mode="all")
# Set colors to plot the selected edges.
ecol <- rep("gray80", ecount(net))
ecol[inc.edges] <- "orange"
vcol <- rep("grey40", vcount(net))
vcol[V(net)$media=="Wall Street Journal"] <- "gold"
plot(net, vertex.color=vcol, edge.color=ecol)
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We can also easily identify the immediate neighbors of a vertex, say WSJ. The neighbors function
finds all nodes one step out from the focal actor.To find the neighbors for multiple nodes, use
adjacent_vertices() instead of neighbors(). To find node neighborhoods going more than one
step out, use function ego() with parameter order set to the number of steps out to go from the
focal node(s).
neigh.nodes <- neighbors(net, V(net)[media=="Wall Street Journal"], mode="out")
# Set colors to plot the neighbors:
vcol[neigh.nodes] <- "#ff9d00"
plot(net, vertex.color=vcol)
Special operators for the indexing of edge sequences: %–%, %->%, %<-%
E(network)[X %–% Y] selects edges between vertex sets X and Y, ignoring direction
E(network)[X %->% Y] selects edges from vertex sets X to vertex set Y
E(network)[X %->% Y] selects edges from vertex sets Y to vertex set X
For example, select edges from newspapers to online sources:
58
E(net)[ V(net)[type.label=="Newspaper"] %->% V(net)[type.label=="Online"] ]
## + 7/48 edges (vertex names):
## [1] s01->s15 s03->s12 s04->s12 s04->s17 s05->s15 s06->s16 s06->s17
Co-citation (for a couple of nodes, how many shared nominations they have):
cocitation(net)
## s01 s02 s03 s04 s05 s06 s07 s08 s09 s10 s11 s12 s13 s14 s15 s16 s17
## s01 0 1 1 2 1 1 0 1 2 2 1 1 0 0 1 0 0
## s02 1 0 1 1 0 0 0 0 1 0 0 0 0 0 2 0 0
## s03 1 1 0 1 0 1 1 1 2 2 1 1 0 1 1 0 1
## s04 2 1 1 0 1 1 0 1 0 1 1 1 0 0 1 0 0
## s05 1 0 0 1 0 0 0 1 0 1 1 1 0 0 0 0 0
## s06 1 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 2
## s07 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0
## s08 1 0 1 1 1 0 0 0 0 2 1 1 0 1 0 0 0
## s09 2 1 2 0 0 0 1 0 0 1 0 0 0 0 1 0 0
## s10 2 0 2 1 1 0 0 2 1 0 1 1 0 1 0 0 0
## s11 1 0 1 1 1 1 0 1 0 1 0 2 1 0 0 0 1
## s12 1 0 1 1 1 1 0 1 0 1 2 0 0 0 0 0 2
## s13 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0
## s14 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 0 0
## s15 1 2 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0
## s16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
## s17 0 0 1 0 0 2 0 0 0 0 1 2 0 0 0 1 0
8. Subgroups and communities
Before we start, we will make our network undirected. There are several ways to do that conversion:
• We can create an undirected link between any pair of connected nodes (mode="collapse" )
• Create undirected link for each directed one in the network, potentially ending up with a
multiplex graph (mode="each" )
• Create undirected link for each symmetric link in the graph (mode="mutual" ).
In cases when we may have ties A -> B and B -> A ties collapsed into a single undirected link, we
need to specify what to do with their edge attributes using the parameter ‘edge.attr.comb’ as we
did earlier with simplify(). Here we have said that the ‘weight’ of the links should be summed,
and all other edge attributes ignored and dropped.
59
net.sym <- as.undirected(net, mode= "collapse",
edge.attr.comb=list(weight="sum", "ignore"))
8.1 Cliques
Find cliques (complete subgraphs of an undirected graph)
cliques(net.sym) # list of cliques
sapply(cliques(net.sym), length) # clique sizes
largest_cliques(net.sym) # cliques with max number of nodes
vcol <- rep("grey80", vcount(net.sym))
vcol[unlist(largest_cliques(net.sym))] <- "gold"
plot(as.undirected(net.sym), vertex.label=V(net.sym)$name, vertex.color=vcol)
s01
s02
s03
s04
s05
s06
s07
s08 s09
s10
s11
s12
s13
s14
s15
s16
s17
8.2 Community detection
A number of algorithms aim to detect groups that consist of densely connected nodes with fewer
connections across groups.
Community detection based on edge betweenness (Newman-Girvan)
High-betweenness edges are removed sequentially (recalculating at each step) and the best parti-
tioning of the network is selected.
ceb <- cluster_edge_betweenness(net)
dendPlot(ceb, mode="hclust")
60
s0
4
s1
3
s1
6
s1
7
s0
6
s1
2
s1
4
s1
1
s0
8
s0
7
s0
3
s0
2
s0
9
s1
0
s0
5
s1
5
s0
1
plot(ceb, net)
Let’s examine the community detection igraph object:
class(ceb)
## [1] "communities"
length(ceb) # number of communities
## [1] 5
membership(ceb) # community membership for each node
## s01 s02 s03 s04 s05 s06 s07 s08 s09 s10 s11 s12 s13 s14 s15 s16 s17
## 1 2 3 4 1 4 3 3 5 5 4 4 4 4 1 4 4
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modularity(ceb) # how modular the graph partitioning is
## [1] 0.292476
crossing(ceb, net) # boolean vector: TRUE for edges across communities
High modularity for a partitioning reflects dense connections within communities and sparse
connections across communities.
Community detection based on based on propagating labels
Assigns node labels, randomizes, than replaces each vertex’s label with the label that appears most
frequently among neighbors. Those steps are repeated until each vertex has the most common label
of its neighbors.
clp <- cluster_label_prop(net)
plot(clp, net)
Community detection based on greedy optimization of modularity
cfg <- cluster_fast_greedy(as.undirected(net))
plot(cfg, as.undirected(net))
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We can also plot the communities without relying on their built-in plot:
V(net)$community <- cfg$membership
colrs <- adjustcolor( c("gray50", "tomato", "gold", "yellowgreen"), alpha=.6)
plot(net, vertex.color=colrs[V(net)$community])
8.3 K-core decomposition
The k-core is the maximal subgraph in which every node has degree of at least k. This also means
that the (k+1)-core will be a subgraph of the k-core.
The result here gives the coreness of each vertex in the network. A node has coreness D if it belongs
to a D-core but not to (D+1)-core.
kc <- coreness(net, mode="all")
plot(net, vertex.size=kc*6, vertex.label=kc, vertex.color=colrs[kc])
63
44
4
4
4
3
4
4 44
3
3
3
3
4
3
3
9. Assortativity and Homophily
Homophily: the tendency of nodes to connect to others who are similar on some variable.
• assortativity_nominal() is for categorical variables (labels)
• assortativity() is for ordinal and above variables
• assortativity_degree() checks assortativity in node degrees
assortativity_nominal(net, V(net)$media.type, directed=F)
## [1] 0.1715568
# Matching of attributes across connected nodes more than expected by chance
assortativity(net, V(net)$audience.size, directed=F)
## [1] -0.1102857
# Correlation of attributes across connected nodes
assortativity_degree(net, directed=F)
## [1] -0.009551146
# As above, with the focal attribute being the node degree D-1
64