*Katherine Ognyanova, www.kateto.net*

*NetSciX 2016 School of Code Workshop, Wroclaw, Poland*

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 (maintained 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")`

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 # Assignmentx # Evaluate the expression and print resulty <- 4 # Assignmenty + 5 # Evaluation, y remains 4z <- x + 17*y # Assignmentz # Evaluation`

`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 # Equality2!=2 # Inequalityx <= 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 data5 + NA # When used in an expression, the result is generally NAis.na(5+NA) # Check if missing# NULL - an empty object, e.g. a null/empty list10 + 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:

`5/0is.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.

`0/0is.nan(0/0)`

*1.4 Vectors*

Vectors can be constructed by combining their elements with the important R function `c()`

.

`v1 <- c(1, 5, 11, 33) # Numeric vector, length 4v2 <- 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 zeroesv <- 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 # 1,2,3,4,5v2 <- 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 additionv1 + 1 # Add 1 to each elementv1 * 2 # Multiply each element by 2v1 + c(1,7) # This doesn't work: (1,7) is a vector of different length`

Mathematical operations:

`sum(v1) # The sum of all elementsmean(v1) # The average of all elementssd(v1) # The standard deviationcor(v1,v1*5) # Correlation between v1 and v1*5 `

Logical operations:

`v1 > 2 # Each element is compared to 2, returns logical vectorv1==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`

Vector elements:

`v1[3] # third element of v1v1[2:4] # elements 2, 3, 4 of v1v1[c(1,3)] # elements 1 and 3 - note that your indexes are a vectorv1[c(T,T,F,F,F)] # elements 1 and 2 - only the ones that are TRUEv1[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") #vectoreye.col.f <- factor(c("brown", "green", "brown", "blue", "blue", "blue")) #factoreye.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`

`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 1dim(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 1sm <- matrix(1,5,4) # same matrix as abovedim(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 matrixm <- 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 cellm[2,] # The whole second row of m as a vectorm[,2] # The whole second column of m as a vectorm[1:2,4:6] # submatrix: rows 1 and 2, columns 4, 5 and 6m[-1,] # all rows *except* the first one`

Other operations with matrices:

`# 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 mm %*% t(m) # %*% does matrix multiplicationm * 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 3x3x2a <- 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 componentsl2 <- 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 vectorl1[[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 l3l4[[3]] <- c(22, 23) # add a vector as element 3 in the empty list l4. `

Since we added element 3 to the list `l4`

above, elements 1 and 2 will be generated and empty (NULL).

`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 list`l1`

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 columnsdfr1[,1] # First column, all rowsdfr1$Age # Age column, all rowsdfr1[1:2,3:4] # Rows 1 and 2, columns 3 and 4 - the gender and age of John & Jimdfr1[c(1,3),] # Rows 1 and 3, all columns`

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 expr2x <- 5; y <- 10if (x==0) y <- 0 else y <- y/x # y`

`## [1] 2`

`# for (variable in sequence) exprASum <- 0; AProd <- 1for (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) exprwhile (x > 0) {print(x); x <- x-1;}# repeat expr, use break to exit the looprepeat { print(x); x <- x+1; if (x>10) break}`

*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)) `

If you plan on using the built-in color names, here’s how to list all of them:

`colors() # List all named colorsgrep("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, opaquepal2 <- rainbow(5, alpha=.5) # 5 colors from the heat palette, transparentplot(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.

`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:

*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`

.*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.*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`

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()`

.

`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`

`class(g1)`

`## [1] "igraph"`

`g1`

`## 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) `

`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 neededplot(g3)`

`g3`

`## IGRAPH DN-- 3 3 -- ## + attr: name (v/c)## + edges (vertex names):## [1] John->Jim Jim ->Jill Jill->John`

`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) `

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`

`plot(graph_from_literal(a--+b, b+--c))`

`plot(graph_from_literal(a+-+b, b+-+c)) `

`plot(graph_from_literal(a:b:c---c:d:e))`

`gl <- graph_from_literal(a-b-c-d-e-f, a-g-h-b, h-e:f:i, j)plot(gl)`

*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`

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 edgesE(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"`

`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")] ) `

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)`

`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:

- D or U, for a directed or undirected graph
- N for a named graph (where nodes have a
`name`

attribute) - W for a weighted graph (where edges have a
`weight`

attribute) - 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)`

**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**

`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) `

**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)`

**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) `

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) `

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:

`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)`

———————————–

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`

We also have easy access to nodes, edges, and their attributes with:

`E(net) # The edges of the "net" objectV(net) # The vertices of the "net" objectE(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")`

*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).

`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)])`

`plot(net2.bp$proj2, vertex.label.color="black", vertex.label.dist=1, vertex.size=7, vertex.label=nodes2$media[ is.na(nodes2$media.type)])`

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 |

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)`

`# 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) `

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.

`# 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)$arrow.size <- .2E(net)$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") `

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)`

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")`

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).

`net.bg <- sample_pa(80) V(net.bg)$size <- 8V(net.bg)$frame.color <- "white"V(net.bg)$color <- "orange"V(net.bg)$label <- "" E(net.bg)$arrow.mode <- 0plot(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)`

`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 verticesl <- layout_randomly(net.bg)plot(net.bg, layout=l)`

`# Circle layoutl <- layout_in_circle(net.bg)plot(net.bg, layout=l)`

`# 3D sphere layoutl <- layout_on_sphere(net.bg)plot(net.bg, layout=l)`

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 columnsplot(net.bg, layout=layout_with_fr)plot(net.bg, layout=layout_with_fr)plot(net.bg, layout=l)plot(net.bg, layout=l)`

`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)`

`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.

`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) }`

*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)`

:

`cut.off <- mean(links$weight) net.sp <- delete_edges(net, E(net)[weight<cut.off])plot(net.sp) `

Another way to think about this is to plot the two tie types (hyperlink & mention) separately.

`E(net)$width <- 1.5plot(net, edge.color=c("dark red", "slategrey")[(E(net)$type=="hyperlink")+1], vertex.color="gray40", layout=layout.circle)`

`net.m <- net - E(net)[E(net)$type=="hyperlink"] # another way to delete edgesnet.h <- net - E(net)[E(net)$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")`

`# 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")`

`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 openl <- tkplot.getcoords(tkid) # grab the coordinates from tkplottk_close(tkid, window.close = T)plot(net, layout=l)`

*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)$mediarownames(netm) <- V(net)$mediapalf <- colorRampPalette(c("gold", "dark orange")) heatmap(netm[,17:1], Rowv = NA, Colv = NA, col = palf(100), scale="none", margins=c(10,10) )`

*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=.4V(net2)$label.font=2plot(net2, vertex.label.color="white", vertex.size=(2-V(net2)$type)*8) `

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)`