Social Network Analysis
Session 01
Data Structures
Meltem Odabaş
2021-10-26
(updated: 2021-10-25)
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Why Social Network Analysis?
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Why Social Network Analysis?
- Social relations and institutional connections shape our attitudes and behaviors
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Why Social Network Analysis?
Social relations and institutional connections shape our attitudes and behaviors
Allows researchers to do micro, meso and macro-level analyses:
- micro-level e.g. individuals and their dyadic relationships
- meso-level e.g. inter- and intra- organizational networks; social groups
- macro-level e.g. large-scale and complex networks; community structures
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What types of information do social network graphs contain?
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Persons are represented in circle, referred to as "nodes" or "vertexes";
Relations are represented with lines, referred to as "ties" or "edges".
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Relations can be directed (Mr. Collins is fond of Elizabeth may be, but she certainly not reciprocate the feeling!)
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These "feelings" may be strong, or weak:
Edges can take ordinal attributes.
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But does Lizzy love Mr. Darcy?!?
Not when Mr. Darcy thinks "she is tolerable"; but then for some reason turns full 180!
Edges can take cardinal attributes: such as love vs. hate.
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Not only edges but nodes/vertexes can take both ordinal and cardinal attributes:
Wealth is ordinal and Gender (in Jane Austen's 1813 Western world!) is cardinal.
(As you might know, Mr. Darcy earns 10,000 a year...)
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Network Graphs and Data Structures
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Network Graphs and Data Structures
So far, we have seen three groups of datasets:
Relational data (represented in edges/ties drawn across nodes)
Attributes of the edges
Attributes of the vertices
Graph objects in R will always request relational data; attribute data (either of edges or vertices) is rather optional.
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Network Graphs and Data Structures (cont'd)
If we are adding all three types of data, the R graph object will request:
Relational data and edge attributes as in one dataset (which I will shortly call as "Relational data"; but keep in mind that edge attributes go in this one as well)
Node attribute data in a separate dataset. (which I will shortly call as "Vertex/node attribute data")
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1- Relational Data Structure
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Relational Data structures
There are two ways to represent relational data:
Matrix notation,
Edgelist Notation.
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Relational data in the form of Matrix notation
Below is a Krackhardt's (1999) small hi-tech computer firm dataset. It looks at 36 employees and managers from the firm and their friendship ties.
bucket = "/Users/meltemodabas/github/meltemod/workshops/SNA/01-data-structure"options(width = 60)M = read.table(file.path(bucket, "data","Hi-tech_sociomatrix.txt"), header=TRUE,row.names=1)M## Abe Bob Carl Dale Ev Fred Gary Hal Ivo Jack Ken Len## Abe 0 0 0 0 0 1 0 0 0 0 0 0## Bob 0 0 0 1 0 0 0 0 0 1 0 0## Carl 0 0 0 0 0 1 0 1 0 0 0 1## Dale 0 1 0 0 0 0 0 0 0 0 0 0## Ev 0 0 0 0 0 0 0 0 0 0 0 0## Fred 0 0 0 0 0 0 0 0 0 0 0 0## Gary 0 0 0 1 0 0 0 0 0 0 0 0## Hal 0 0 1 0 0 1 0 0 0 0 0 0## Ivo 0 0 0 0 0 0 0 0 0 0 0 0## Jack 0 1 0 0 0 0 0 0 0 0 0 0## Ken 0 0 0 0 0 0 0 0 0 0 0 0## Len 0 0 1 0 0 0 0 0 0 0 0 0## Mel 0 1 0 1 0 0 0 0 0 0 0 0## Nan 0 1 0 0 0 0 1 0 1 0 1 0## Ovid 0 0 0 0 0 0 0 0 0 0 0 0## Pat 0 0 0 0 0 0 0 0 0 0 0 0## Quincy 0 0 0 0 0 0 0 0 0 0 0 0## Robin 0 0 0 0 0 0 0 0 0 0 1 0## Steve 0 0 0 0 1 0 0 0 0 0 0 0## Tom 0 0 0 1 0 0 0 0 0 0 1 0## Upton 0 0 0 0 0 1 0 0 1 0 0 0## Vic 0 0 0 0 0 0 0 0 0 0 1 0## Walt 0 0 0 0 0 0 0 0 0 0 0 0## Rick 0 0 0 1 1 0 1 0 0 0 1 0## York 0 0 0 0 0 0 0 0 0 0 0 0## Zoe 0 0 0 0 0 0 0 0 0 0 0 0## Alex 0 0 0 1 0 0 0 0 0 0 0 0## Ben 0 1 0 0 0 0 0 0 0 0 0 0## Chris 0 1 0 1 0 0 1 0 1 0 1 0## Dan 0 0 0 0 0 0 0 0 0 0 1 0## Earl 0 0 0 0 0 0 0 0 0 0 0 0## Fran 0 0 0 0 0 0 0 0 0 0 0 0## Gerry 0 0 0 0 0 0 0 0 0 0 1 0## Hugh 0 0 0 0 0 0 0 0 0 0 1 0## Irv 0 0 0 1 0 0 0 0 0 0 0 1## Jim 0 0 0 0 0 0 0 0 0 0 0 0## Mel Nan Ovid Pat Quincy Robin Steve Tom Upton Vic## Abe 0 0 0 0 0 0 0 0 0 0## Bob 0 0 1 0 0 0 0 0 0 0## Carl 0 0 0 0 0 0 0 0 0 0## Dale 1 0 0 0 0 0 0 1 0 0## Ev 0 0 0 0 0 0 1 0 0 0## Fred 0 0 0 0 0 0 1 0 1 0## Gary 0 0 0 0 0 0 0 0 0 0## Hal 0 0 0 0 0 0 0 0 0 0## Ivo 0 0 0 0 0 0 0 1 0 0## Jack 0 0 0 0 0 0 0 0 0 0## Ken 0 1 0 0 0 1 0 1 0 0## Len 0 0 0 0 0 0 0 0 0 0## Mel 0 0 0 1 0 1 0 1 0 0## Nan 0 0 1 0 0 0 0 0 0 0## Ovid 0 1 0 0 0 0 0 0 0 0## Pat 1 0 0 0 0 0 1 0 0 0## Quincy 0 0 0 0 0 0 0 0 0 0## Robin 1 0 0 0 0 0 0 0 0 0## Steve 0 0 0 1 0 0 0 0 0 0## Tom 1 0 0 0 0 0 0 0 1 1## Upton 0 0 0 0 0 0 1 1 0 0## Vic 0 0 0 0 0 0 0 1 0 0## Walt 0 0 0 0 0 0 0 0 0 0## Rick 0 0 0 0 0 0 1 1 1 0## York 0 0 0 0 0 0 0 0 0 0## Zoe 0 0 0 0 0 0 0 0 1 0## Alex 0 0 0 0 0 1 0 0 0 0## Ben 0 0 0 0 0 0 0 0 0 0## Chris 1 1 1 0 0 1 0 1 1 0## Dan 1 0 0 0 0 0 1 0 0 0## Earl 0 0 0 1 0 0 1 0 0 0## Fran 0 0 0 0 0 0 0 0 0 0## Gerry 0 1 0 0 0 0 0 1 0 0## Hugh 0 1 1 0 0 0 0 1 0 0## Irv 0 0 0 1 0 0 1 0 0 0## Jim 0 0 0 1 0 0 0 0 0 0## Walt Rick York Zoe Alex Ben Chris Dan Earl Fran## Abe 0 0 0 0 0 0 0 0 0 0## Bob 0 0 0 0 0 0 1 0 0 0## Carl 0 0 0 0 0 0 0 0 0 0## Dale 0 1 0 0 1 0 1 0 0 0## Ev 0 0 0 0 0 0 0 0 0 0## Fred 0 0 0 0 0 0 0 0 0 0## Gary 0 1 0 0 0 0 0 0 0 0## Hal 0 0 0 0 0 0 0 0 0 0## Ivo 0 0 0 0 0 0 1 0 0 0## Jack 0 0 0 0 0 0 0 0 0 0## Ken 0 1 0 0 0 0 0 1 0 0## Len 0 0 0 1 0 0 0 0 0 0## Mel 0 0 0 0 0 0 1 1 0 0## Nan 0 0 0 0 0 0 1 0 0 0## Ovid 0 0 0 0 0 0 1 0 0 0## Pat 0 0 0 0 0 0 0 0 0 0## Quincy 0 0 0 0 0 0 0 0 0 0## Robin 0 0 0 0 1 0 1 0 0 0## Steve 0 1 0 0 0 0 0 1 0 0## Tom 0 1 0 1 0 0 1 0 0 0## Upton 0 0 0 0 0 0 0 0 0 0## Vic 0 0 0 0 0 0 0 0 0 0## Walt 0 1 0 0 0 0 1 0 0 0## Rick 0 0 0 0 0 0 1 1 0 0## York 0 0 0 0 0 0 0 0 0 0## Zoe 0 0 0 0 0 0 0 0 0 0## Alex 0 1 0 0 0 0 0 0 0 0## Ben 0 0 0 0 0 0 0 0 0 0## Chris 1 1 0 0 1 0 0 0 0 0## Dan 0 1 0 0 0 0 0 0 0 0## Earl 0 0 0 0 0 0 0 0 0 0## Fran 0 0 0 0 0 0 0 0 0 0## Gerry 0 1 0 0 0 0 1 0 0 0## Hugh 0 1 0 0 0 0 1 0 0 0## Irv 0 0 0 1 0 0 0 1 0 0## Jim 0 0 0 0 0 0 0 0 0 0## Gerry Hugh Irv Jim## Abe 0 0 0 0## Bob 0 0 0 0## Carl 0 0 1 0## Dale 0 0 1 0## Ev 0 0 0 0## Fred 0 0 0 0## Gary 0 0 0 0## Hal 0 0 0 0## Ivo 0 0 0 0## Jack 0 0 0 0## Ken 1 1 0 0## Len 1 0 1 0## Mel 0 0 0 0## Nan 1 0 0 0## Ovid 0 1 0 0## Pat 0 0 1 1## Quincy 0 0 0 0## Robin 0 0 0 0## Steve 1 0 1 0## Tom 1 1 0 0## Upton 0 0 0 0## Vic 0 1 0 0## Walt 0 0 0 0## Rick 0 0 0 0## York 0 0 0 0## Zoe 0 0 1 0## Alex 0 0 0 0## Ben 0 0 0 0## Chris 1 1 0 0## Dan 0 0 1 0## Earl 0 0 0 0## Fran 0 0 0 0## Gerry 0 1 1 0## Hugh 1 0 0 0## Irv 1 0 0 0## Jim 0 0 0 016 / 75
You can also work with sparse matrices:
tmp = as.data.table(M)M_sp = sparsify(tmp)M_sp## 36 x 36 sparse Matrix of class "dgCMatrix"## [[ suppressing 36 column names 'Abe', 'Bob', 'Carl' ... ]]## ## [1,] . . . . . 1 . . . . . . . . . . . . . . . . . . . . .## [2,] . . . 1 . . . . . 1 . . . . 1 . . . . . . . . . . . .## [3,] . . . . . 1 . 1 . . . 1 . . . . . . . . . . . . . . .## [4,] . 1 . . . . . . . . . . 1 . . . . . . 1 . . . 1 . . 1## [5,] . . . . . . . . . . . . . . . . . . 1 . . . . . . . .## [6,] . . . . . . . . . . . . . . . . . . 1 . 1 . . . . . .## [7,] . . . 1 . . . . . . . . . . . . . . . . . . . 1 . . .## [8,] . . 1 . . 1 . . . . . . . . . . . . . . . . . . . . .## [9,] . . . . . . . . . . . . . . . . . . . 1 . . . . . . .## [10,] . 1 . . . . . . . . . . . . . . . . . . . . . . . . .## [11,] . . . . . . . . . . . . . 1 . . . 1 . 1 . . . 1 . . .## [12,] . . 1 . . . . . . . . . . . . . . . . . . . . . . 1 .## [13,] . 1 . 1 . . . . . . . . . . . 1 . 1 . 1 . . . . . . .## [14,] . 1 . . . . 1 . 1 . 1 . . . 1 . . . . . . . . . . . .## [15,] . . . . . . . . . . . . . 1 . . . . . . . . . . . . .## [16,] . . . . . . . . . . . . 1 . . . . . 1 . . . . . . . .## [17,] . . . . . . . . . . . . . . . . . . . . . . . . . . .## [18,] . . . . . . . . . . 1 . 1 . . . . . . . . . . . . . 1## [19,] . . . . 1 . . . . . . . . . . 1 . . . . . . . 1 . . .## [20,] . . . 1 . . . . . . 1 . 1 . . . . . . . 1 1 . 1 . 1 .## [21,] . . . . . 1 . . 1 . . . . . . . . . 1 1 . . . . . . .## [22,] . . . . . . . . . . 1 . . . . . . . . 1 . . . . . . .## [23,] . . . . . . . . . . . . . . . . . . . . . . . 1 . . .## [24,] . . . 1 1 . 1 . . . 1 . . . . . . . 1 1 1 . . . . . .## [25,] . . . . . . . . . . . . . . . . . . . . . . . . . . .## [26,] . . . . . . . . . . . . . . . . . . . . 1 . . . . . .## [27,] . . . 1 . . . . . . . . . . . . . 1 . . . . . 1 . . .## [28,] . 1 . . . . . . . . . . . . . . . . . . . . . . . . .## [29,] . 1 . 1 . . 1 . 1 . 1 . 1 1 1 . . 1 . 1 1 . 1 1 . . 1## [30,] . . . . . . . . . . 1 . 1 . . . . . 1 . . . . 1 . . .## [31,] . . . . . . . . . . . . . . . 1 . . 1 . . . . . . . .## [32,] . . . . . . . . . . . . . . . . . . . . . . . . . . .## [33,] . . . . . . . . . . 1 . . 1 . . . . . 1 . . . 1 . . .## [34,] . . . . . . . . . . 1 . . 1 1 . . . . 1 . . . 1 . . .## [35,] . . . 1 . . . . . . . 1 . . . 1 . . 1 . . . . . . 1 .## [36,] . . . . . . . . . . . . . . . 1 . . . . . . . . . . .## ## [1,] . . . . . . . . .## [2,] . 1 . . . . . . .## [3,] . . . . . . . 1 .## [4,] . 1 . . . . . 1 .## [5,] . . . . . . . . .## [6,] . . . . . . . . .## [7,] . . . . . . . . .## [8,] . . . . . . . . .## [9,] . 1 . . . . . . .## [10,] . . . . . . . . .## [11,] . . 1 . . 1 1 . .## [12,] . . . . . 1 . 1 .## [13,] . 1 1 . . . . . .## [14,] . 1 . . . 1 . . .## [15,] . 1 . . . . 1 . .## [16,] . . . . . . . 1 1## [17,] . . . . . . . . .## [18,] . 1 . . . . . . .## [19,] . . 1 . . 1 . 1 .## [20,] . 1 . . . 1 1 . .## [21,] . . . . . . . . .## [22,] . . . . . . 1 . .## [23,] . 1 . . . . . . .## [24,] . 1 1 . . . . . .## [25,] . . . . . . . . .## [26,] . . . . . . . 1 .## [27,] . . . . . . . . .## [28,] . . . . . . . . .## [29,] . . . . . 1 1 . .## [30,] . . . . . . . 1 .## [31,] . . . . . . . . .## [32,] . . . . . . . . .## [33,] . 1 . . . . 1 1 .## [34,] . 1 . . . 1 . . .## [35,] . . 1 . . 1 . . .## [36,] . . . . . . . . .17 / 75
dim(M) #36x36 matrix## [1] 36 36all.equal(rownames(M),colnames(M)) #rownames and colnames are equal## [1] TRUEisSymmetric(as.matrix(M)) #not symmetric; not all ties are reciprocated.## [1] FALSEWhat we have is called an adjacency matrix: a square matrix that represents a finite graph.
The employee names are the "nodes," and friendships are defined as connections from one employee to another.
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Using adjacency matrix to create a graph object
There are various packages to work with network graphs, such as:
igraphsnanetwork
I am more familiar with igraph, so this is what I will use for this tutorial. Needless to say, feel free to explore other packages on your spare time.
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Using adjacency matrix to create a graph object
Ok, once we have our matrix (or sparse matrix!) ready, we can simply create a graph object (or, "igraph object"!) using igraph package.
g_hitech = graph_from_adjacency_matrix(M_sp)g_hitech## IGRAPH fca88bf DN-- 36 147 -- ## + attr: name (v/c)## + edges from fca88bf (vertex names):## [1] Dale ->Bob Jack ->Bob Mel ->Bob Nan ->Bob ## [5] Ben ->Bob Chris->Bob Hal ->Carl Len ->Carl## [9] Bob ->Dale Gary ->Dale Mel ->Dale Tom ->Dale## [13] Rick ->Dale Alex ->Dale Chris->Dale Irv ->Dale## [17] Steve->Ev Rick ->Ev Abe ->Fred Carl ->Fred## [21] Hal ->Fred Upton->Fred Nan ->Gary Rick ->Gary## [25] Chris->Gary Carl ->Hal Nan ->Ivo Upton->Ivo ## [29] Chris->Ivo Bob ->Jack Nan ->Ken Robin->Ken ## + ... omitted several edgesThis is a directed network (DN) of 36 vertices and 147 edges. There is one vertex attribute in this graph called name (attr: name(v/c)) in character format. This attribute comes from the row and column names assigned.
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In each igraph object, you can see the list of edges by using E(graph) function, and list of vertices by typing V(graph) function:
E(g_hitech)## + 147/147 edges from fca88bf (vertex names):## [1] Dale ->Bob Jack ->Bob Mel ->Bob Nan ->Bob ## [5] Ben ->Bob Chris->Bob Hal ->Carl Len ->Carl## [9] Bob ->Dale Gary ->Dale Mel ->Dale Tom ->Dale## [13] Rick ->Dale Alex ->Dale Chris->Dale Irv ->Dale## [17] Steve->Ev Rick ->Ev Abe ->Fred Carl ->Fred## [21] Hal ->Fred Upton->Fred Nan ->Gary Rick ->Gary## [25] Chris->Gary Carl ->Hal Nan ->Ivo Upton->Ivo ## [29] Chris->Ivo Bob ->Jack Nan ->Ken Robin->Ken ## [33] Tom ->Ken Vic ->Ken Rick ->Ken Chris->Ken ## [37] Dan ->Ken Gerry->Ken Hugh ->Ken Carl ->Len ## + ... omitted several edges21 / 75
In each igraph object, you can see the list of edges by using E(graph) function, and list of vertices by typing V(graph) function:
V(g_hitech)## + 36/36 vertices, named, from fca88bf:## [1] Abe Bob Carl Dale Ev Fred Gary Hal ## [9] Ivo Jack Ken Len Mel Nan Ovid Pat ## [17] Quincy Robin Steve Tom Upton Vic Walt Rick ## [25] York Zoe Alex Ben Chris Dan Earl Fran ## [33] Gerry Hugh Irv JimTo list an attribute by itself, such as the vertex attribute name(v/c), use the dollar sign (in base R):
V(g_hitech)$name## [1] "Abe" "Bob" "Carl" "Dale" "Ev" "Fred" ## [7] "Gary" "Hal" "Ivo" "Jack" "Ken" "Len" ## [13] "Mel" "Nan" "Ovid" "Pat" "Quincy" "Robin" ## [19] "Steve" "Tom" "Upton" "Vic" "Walt" "Rick" ## [25] "York" "Zoe" "Alex" "Ben" "Chris" "Dan" ## [31] "Earl" "Fran" "Gerry" "Hugh" "Irv" "Jim"22 / 75
Let's plot the network graph:
l = layout.fruchterman.reingold(g_hitech)plot(g_hitech, vertex.size = 10, layout = l)
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Although not all ties are reciprocated in this graph,, we can rather set the igraph as "undirected"
g_hitech_und = graph_from_adjacency_matrix(M_sp, mode = "undirected")g_hitech_und## IGRAPH 836d002 UN-- 36 91 -- ## + attr: name (v/c)## + edges from 836d002 (vertex names):## [1] Abe --Fred Bob --Dale Bob --Jack Bob --Mel ## [5] Bob --Nan Bob --Ovid Bob --Ben Bob --Chris## [9] Carl--Fred Carl--Hal Carl--Len Carl--Irv ## [13] Dale--Gary Dale--Mel Dale--Tom Dale--Rick ## [17] Dale--Alex Dale--Chris Dale--Irv Ev --Steve## [21] Ev --Rick Fred--Hal Fred--Steve Fred--Upton## [25] Gary--Nan Gary--Rick Gary--Chris Ivo --Nan ## [29] Ivo --Tom Ivo --Upton Ivo --Chris Ken --Nan ## + ... omitted several edgesLet's read the information: g_medici_und is an undirected network (UN) of 36 vertices and 91 edges (not 147!).
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Plotting the graph, we see:
plot(g_hitech_und, vertex.size = 10, layout = l)
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It is possible to assign tie weights to matrices: rather than using 0-1 dichotomy, you may assign values between [0-Z], where Z is a reel number.
Although the original dataset does not have weights for friendships, we can make up our own values (just for the sake of practicing!) I will replace the cells that are equal to 1 to a randomly selected list of values ranging across 1 and 5
M_sp[M_sp == 1] #prints all the values in flo_sp that equal to 1## <sparse>[ <logic> ] : .M.sub.i.logical() maybe inefficient## [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1## [28] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1## [55] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1## [82] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1## [109] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1## [136] 1 1 1 1 1 1 1 1 1 1 1 1length(M_sp[M_sp == 1]) #147 ties## <sparse>[ <logic> ] : .M.sub.i.logical() maybe inefficient## [1] 14726 / 75
random_values = sample(c(1:5), length(M_sp[M_sp == 1]), replace = T) #get 147 randomly seleccted constant numbers ranging from 1 to 5## <sparse>[ <logic> ] : .M.sub.i.logical() maybe inefficientrandom_values## [1] 2 4 2 4 5 5 1 1 1 4 3 3 2 3 4 5 2 5 2 1 2 2 1 1 5 2 3## [28] 5 5 5 2 4 4 2 2 1 3 4 4 4 2 5 3 5 3 5 1 3 5 2 2 1 4 2## [55] 2 1 2 5 5 1 1 5 4 3 4 5 2 5 5 4 1 5 4 4 3 3 1 1 4 1 1## [82] 3 3 3 5 5 2 2 4 3 4 1 1 3 5 2 2 2 4 2 2 2 2 1 5 3 3 4## [109] 5 1 1 2 2 1 5 2 5 2 1 3 2 1 5 4 2 2 5 5 5 1 1 5 5 2 2## [136] 1 5 2 1 2 2 5 3 4 3 3 3M_sp_weighted = M_sp #create a new sparse matrixM_sp_weighted[M_sp_weighted == 1] <- random_values #assign these random values to existing ties27 / 75
M_sp_weighted## 36 x 36 sparse Matrix of class "dgCMatrix"## [[ suppressing 36 column names 'Abe', 'Bob', 'Carl' ... ]]## ## [1,] . . . . . 2 . . . . . . . . . . . . . . . . . . . . .## [2,] . . . 1 . . . . . 5 . . . . 4 . . . . . . . . . . . .## [3,] . . . . . 1 . 2 . . . 4 . . . . . . . . . . . . . . .## [4,] . 2 . . . . . . . . . . 5 . . . . . . 4 . . . 4 . . 5## [5,] . . . . . . . . . . . . . . . . . . 5 . . . . . . . .## [6,] . . . . . . . . . . . . . . . . . . 2 . 3 . . . . . .## [7,] . . . 4 . . . . . . . . . . . . . . . . . . . 1 . . .## [8,] . . 1 . . 2 . . . . . . . . . . . . . . . . . . . . .## [9,] . . . . . . . . . . . . . . . . . . . 3 . . . . . . .## [10,] . 4 . . . . . . . . . . . . . . . . . . . . . . . . .## [11,] . . . . . . . . . . . . . 3 . . . 5 . 3 . . . 1 . . .## [12,] . . 1 . . . . . . . . . . . . . . . . . . . . . . 2 .## [13,] . 2 . 3 . . . . . . . . . . . 2 . 4 . 1 . . . . . . .## [14,] . 4 . . . . 1 . 3 . 2 . . . 2 . . . . . . . . . . . .## [15,] . . . . . . . . . . . . . 5 . . . . . . . . . . . . .## [16,] . . . . . . . . . . . . 3 . . . . . 5 . . . . . . . .## [17,] . . . . . . . . . . . . . . . . . . . . . . . . . . .## [18,] . . . . . . . . . . 4 . 5 . . . . . . . . . . . . . 3## [19,] . . . . 2 . . . . . . . . . . 5 . . . . . . . 3 . . .## [20,] . . . 3 . . . . . . 4 . 3 . . . . . . . 5 4 . 5 . 2 .## [21,] . . . . . 2 . . 5 . . . . . . . . . 5 1 . . . . . . .## [22,] . . . . . . . . . . 2 . . . . . . . . 4 . . . . . . .## [23,] . . . . . . . . . . . . . . . . . . . . . . . 2 . . .## [24,] . . . 2 5 . 1 . . . 2 . . . . . . . 4 1 5 . . . . . .## [25,] . . . . . . . . . . . . . . . . . . . . . . . . . . .## [26,] . . . . . . . . . . . . . . . . . . . . 2 . . . . . .## [27,] . . . 3 . . . . . . . . . . . . . 3 . . . . . 2 . . .## [28,] . 5 . . . . . . . . . . . . . . . . . . . . . . . . .## [29,] . 5 . 4 . . 5 . 5 . 1 . 5 2 2 . . 4 . 1 2 . 3 2 . . 3## [30,] . . . . . . . . . . 3 . 1 . . . . . 1 . . . . 4 . . .## [31,] . . . . . . . . . . . . . . . 5 . . 5 . . . . . . . .## [32,] . . . . . . . . . . . . . . . . . . . . . . . . . . .## [33,] . . . . . . . . . . 4 . . 2 . . . . . 3 . . . 2 . . .## [34,] . . . . . . . . . . 4 . . 1 1 . . . . 3 . . . 2 . . .## [35,] . . . 5 . . . . . . . 2 . . . 1 . . 4 . . . . . . 1 .## [36,] . . . . . . . . . . . . . . . 1 . . . . . . . . . . .## ## [1,] . . . . . . . . .## [2,] . 4 . . . . . . .## [3,] . . . . . . . 1 .## [4,] . 5 . . . . . 2 .## [5,] . . . . . . . . .## [6,] . . . . . . . . .## [7,] . . . . . . . . .## [8,] . . . . . . . . .## [9,] . 1 . . . . . . .## [10,] . . . . . . . . .## [11,] . . 3 . . 2 5 . .## [12,] . . . . . 2 . 2 .## [13,] . 1 2 . . . . . .## [14,] . 2 . . . 5 . . .## [15,] . 2 . . . . 2 . .## [16,] . . . . . . . 5 3## [17,] . . . . . . . . .## [18,] . 1 . . . . . . .## [19,] . . 1 . . 5 . 3 .## [20,] . 5 . . . 5 2 . .## [21,] . . . . . . . . .## [22,] . . . . . . 1 . .## [23,] . 2 . . . . . . .## [24,] . 5 5 . . . . . .## [25,] . . . . . . . . .## [26,] . . . . . . . 4 .## [27,] . . . . . . . . .## [28,] . . . . . . . . .## [29,] . . . . . 1 5 . .## [30,] . . . . . . . 3 .## [31,] . . . . . . . . .## [32,] . . . . . . . . .## [33,] . 2 . . . . 2 3 .## [34,] . 1 . . . 1 . . .## [35,] . . 4 . . 5 . . .## [36,] . . . . . . . . .28 / 75
Let's create a new graph with this weighted matrix:
g_hitech_weighted = graph_from_adjacency_matrix(M_sp_weighted, mode = "directed", weighted = TRUE)g_hitech_weighted## IGRAPH 6f5749e DNW- 36 147 -- ## + attr: name (v/c), weight (e/n)## + edges from 6f5749e (vertex names):## [1] Dale ->Bob Jack ->Bob Mel ->Bob Nan ->Bob ## [5] Ben ->Bob Chris->Bob Hal ->Carl Len ->Carl## [9] Bob ->Dale Gary ->Dale Mel ->Dale Tom ->Dale## [13] Rick ->Dale Alex ->Dale Chris->Dale Irv ->Dale## [17] Steve->Ev Rick ->Ev Abe ->Fred Carl ->Fred## [21] Hal ->Fred Upton->Fred Nan ->Gary Rick ->Gary## [25] Chris->Gary Carl ->Hal Nan ->Ivo Upton->Ivo ## [29] Chris->Ivo Bob ->Jack Nan ->Ken Robin->Ken ## + ... omitted several edgesg_hitech_weighted is a directed and weighted graph with 36 vertices and 147 ties.
The graph has now two attributes: An edge attribute called weight (weight (e/n)) in numeric format and a vertex attribute called name(name (v/c)) in character format.
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Let's plot this new graph:
plot(g_hitech_weighted, edge.width = E(g_hitech_weighted)$weight, #use edge weight attr to edge width edge.curved = .5, #curve the edges vertex.size = 5, layout = l)
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Ok, this got a bit cluttered.
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We could instead make up "friend" vs. "enemy" ties; assigning 1 foor friend and -1 for enemy:
We could also create a love-hate matrix: by only using values of +1 or -1 for existing ties.
random_values_v2 = sample(c(1,-1), length(M_sp[M_sp == 1]), replace = T) #randomly select -1 or 1## <sparse>[ <logic> ] : .M.sub.i.logical() maybe inefficientrandom_values_v2## [1] -1 -1 -1 -1 1 -1 1 1 -1 1 1 -1 1 -1 1 1 1 1## [19] -1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 1 -1 -1 -1 1## [37] -1 -1 1 -1 1 1 1 1 1 -1 -1 1 -1 1 1 1 1 1## [55] 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1## [73] 1 1 -1 -1 -1 1 1 -1 1 1 1 -1 1 -1 -1 1 -1 -1## [91] -1 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 1 -1 1 -1## [109] 1 1 -1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1## [127] -1 1 1 -1 1 -1 1 1 -1 1 -1 -1 -1 -1 1 -1 1 -1## [145] -1 -1 1M_sp_hatred = M_sp #create a new sparse matrixM_sp_hatred[M_sp_hatred == 1] <- random_values_v2 #assign these random values to existing ties in flo_sp32 / 75
M_sp_hatred## 36 x 36 sparse Matrix of class "dgCMatrix"## [[ suppressing 36 column names 'Abe', 'Bob', 'Carl' ... ]]## ## [1,] . . . . . -1 . . . . . . . . . . . . . .## [2,] . . . -1 . . . . . -1 . . . . 1 . . . . .## [3,] . . . . . -1 . 1 . . . -1 . . . . . . . .## [4,] . -1 . . . . . . . . . . 1 . . . . . . 1## [5,] . . . . . . . . . . . . . . . . . . -1 .## [6,] . . . . . . . . . . . . . . . . . . -1 .## [7,] . . . 1 . . . . . . . . . . . . . . . .## [8,] . . 1 . . -1 . . . . . . . . . . . . . .## [9,] . . . . . . . . . . . . . . . . . . . -1## [10,] . -1 . . . . . . . . . . . . . . . . . .## [11,] . . . . . . . . . . . . . 1 . . . 1 . -1## [12,] . . 1 . . . . . . . . . . . . . . . . .## [13,] . -1 . 1 . . . . . . . . . . . 1 . -1 . -1## [14,] . -1 . . . . -1 . 1 . -1 . . . 1 . . . . .## [15,] . . . . . . . . . . . . . -1 . . . . . .## [16,] . . . . . . . . . . . . 1 . . . . . 1 .## [17,] . . . . . . . . . . . . . . . . . . . .## [18,] . . . . . . . . . . 1 . 1 . . . . . . .## [19,] . . . . 1 . . . . . . . . . . -1 . . . .## [20,] . . . -1 . . . . . . -1 . 1 . . . . . . .## [21,] . . . . . 1 . . 1 . . . . . . . . . 1 1## [22,] . . . . . . . . . . -1 . . . . . . . . 1## [23,] . . . . . . . . . . . . . . . . . . . .## [24,] . . . 1 1 . 1 . . . -1 . . . . . . . -1 -1## [25,] . . . . . . . . . . . . . . . . . . . .## [26,] . . . . . . . . . . . . . . . . . . . .## [27,] . . . -1 . . . . . . . . . . . . . 1 . .## [28,] . 1 . . . . . . . . . . . . . . . . . .## [29,] . -1 . 1 . . -1 . 1 . 1 . -1 1 1 . . 1 . 1## [30,] . . . . . . . . . . -1 . -1 . . . . . -1 .## [31,] . . . . . . . . . . . . . . . -1 . . 1 .## [32,] . . . . . . . . . . . . . . . . . . . .## [33,] . . . . . . . . . . -1 . . 1 . . . . . 1## [34,] . . . . . . . . . . 1 . . 1 -1 . . . . 1## [35,] . . . 1 . . . . . . . 1 . . . 1 . . 1 .## [36,] . . . . . . . . . . . . . . . -1 . . . .## ## [1,] . . . . . . . . . . . . . . . .## [2,] . . . . . . . . -1 . . . . . . .## [3,] . . . . . . . . . . . . . . -1 .## [4,] . . . -1 . . 1 . 1 . . . . . -1 .## [5,] . . . . . . . . . . . . . . . .## [6,] -1 . . . . . . . . . . . . . . .## [7,] . . . 1 . . . . . . . . . . . .## [8,] . . . . . . . . . . . . . . . .## [9,] . . . . . . . . 1 . . . . . . .## [10,] . . . . . . . . . . . . . . . .## [11,] . . . -1 . . . . . -1 . . -1 1 . .## [12,] . . . . . -1 . . . . . . -1 . 1 .## [13,] . . . . . . . . -1 -1 . . . . . .## [14,] . . . . . . . . -1 . . . -1 . . .## [15,] . . . . . . . . 1 . . . . 1 . .## [16,] . . . . . . . . . . . . . . -1 1## [17,] . . . . . . . . . . . . . . . .## [18,] . . . . . . -1 . -1 . . . . . . .## [19,] . . . 1 . . . . . 1 . . 1 . 1 .## [20,] 1 -1 . 1 . -1 . . -1 . . . 1 -1 . .## [21,] . . . . . . . . . . . . . . . .## [22,] . . . . . . . . . . . . . 1 . .## [23,] . . . -1 . . . . 1 . . . . . . .## [24,] -1 . . . . . . . 1 -1 . . . . . .## [25,] . . . . . . . . . . . . . . . .## [26,] -1 . . . . . . . . . . . . . -1 .## [27,] . . . 1 . . . . . . . . . . . .## [28,] . . . . . . . . . . . . . . . .## [29,] 1 . -1 1 . . 1 . . . . . -1 -1 . .## [30,] . . . -1 . . . . . . . . . . -1 .## [31,] . . . . . . . . . . . . . . . .## [32,] . . . . . . . . . . . . . . . .## [33,] . . . -1 . . . . -1 . . . . -1 -1 .## [34,] . . . 1 . . . . 1 . . . 1 . . .## [35,] . . . . . 1 . . . -1 . . -1 . . .## [36,] . . . . . . . . . . . . . . . .33 / 75
rather than using the edge weights, let's use colors to define "love" and "hate" this time:
#for all ties, first assign the color red, which is for hate.edge_colors = rep("red", length(M_sp[M_sp == 1]))## <sparse>[ <logic> ] : .M.sub.i.logical() maybe inefficientedge_colors[1:10]## [1] "red" "red" "red" "red" "red" "red" "red" "red" "red"## [10] "red"#change the value to "green" where random_values_v2 equals 1edge_colors[random_values_v2 == 1] = "green" edge_colors[1:10]## [1] "red" "red" "red" "red" "green" "red" "green"## [8] "green" "red" "green"random_values_v2[1:10]## [1] -1 -1 -1 -1 1 -1 1 1 -1 1g_hitech_hatred = graph_from_adjacency_matrix(M_sp_hatred, mode = "directed", weighted = TRUE)34 / 75
plot(g_hitech_hatred, edge.color = edge_colors, edge.curved = .5, vertex.size = 5, layout = l)
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Relational Data structures
There are two ways to represent relational data:
Matrix notation,
Edgelist Notation.
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Relational data in the form of Edgelist notation
Below is a Krackhardt's (1999) small hi-tech computer firm dataset. It looks at 36 employees and managers from the firm and their friendship ties.
df_edgelist = fread( # read data as data.table file.path(bucket, "data","Hi-tech_edgelist.csv"))df_edgelist## from to## 1: Nan Bob## 2: Mel Bob## 3: Jack Bob## 4: Chris Bob## 5: Dale Bob## --- ## 143: Carl Irv## 144: Dale Irv## 145: Dan Irv## 146: Pat Irv## 147: Pat Jim37 / 75
We can also add our artificially-made "friendship weight" and "friend vs. enemy" edge attributes as separate columns to the edgelist:
df_edgelist[, weight:= random_values]df_edgelist[, frenemy:= random_values_v2]df_edgelist## from to weight frenemy## 1: Nan Bob 2 -1## 2: Mel Bob 4 -1## 3: Jack Bob 2 -1## 4: Chris Bob 4 -1## 5: Dale Bob 5 1## --- ## 143: Carl Irv 3 1## 144: Dale Irv 4 -1## 145: Dan Irv 3 -1## 146: Pat Irv 3 -1## 147: Pat Jim 3 138 / 75
Next is to use the edgelist to generate an igraph object:
g_hitech_from_edgelist = graph_from_data_frame(df_edgelist, directed = TRUE)g_hitech_from_edgelist## IGRAPH 6fa5d82 DNW- 33 147 -- ## + attr: name (v/c), weight (e/n), frenemy (e/n)## + edges from 6fa5d82 (vertex names):## [1] Nan ->Bob Mel ->Bob Jack ->Bob Chris->Bob ## [5] Dale ->Bob Ben ->Bob Len ->Carl Hal ->Carl## [9] Tom ->Dale Rick ->Dale Gary ->Dale Irv ->Dale## [13] Bob ->Dale Chris->Dale Mel ->Dale Alex ->Dale## [17] Steve->Ev Rick ->Ev Abe ->Fred Carl ->Fred## [21] Upton->Fred Hal ->Fred Chris->Gary Nan ->Gary## [25] Rick ->Gary Carl ->Hal Nan ->Ivo Upton->Ivo ## [29] Chris->Ivo Bob ->Jack Robin->Ken Tom ->Ken ## + ... omitted several edgesThis is a directed and weighted graph (DNW) with 33 (not 36!) vertices and 147 ties. Why do you think we missed 3 out of 36 vertices?
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Next is to use the edgelist to generate an igraph object:
g_hitech_from_edgelist = graph_from_data_frame(df_edgelist, directed = TRUE)g_hitech_from_edgelist## IGRAPH 1dbb475 DNW- 33 147 -- ## + attr: name (v/c), weight (e/n), frenemy (e/n)## + edges from 1dbb475 (vertex names):## [1] Nan ->Bob Mel ->Bob Jack ->Bob Chris->Bob ## [5] Dale ->Bob Ben ->Bob Len ->Carl Hal ->Carl## [9] Tom ->Dale Rick ->Dale Gary ->Dale Irv ->Dale## [13] Bob ->Dale Chris->Dale Mel ->Dale Alex ->Dale## [17] Steve->Ev Rick ->Ev Abe ->Fred Carl ->Fred## [21] Upton->Fred Hal ->Fred Chris->Gary Nan ->Gary## [25] Rick ->Gary Carl ->Hal Nan ->Ivo Upton->Ivo ## [29] Chris->Ivo Bob ->Jack Robin->Ken Tom ->Ken ## + ... omitted several edgesThis is a directed and weighted graph (DNW) with 33 (not 36!) vertices and 147 ties. Why do you think we missed 3 out of 36 vertices? Because we had 3 isolate vertices (i.e., with no ties). This does not mean we cannot register isolate nodes with edgelist datasets, however. I will come to that.
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plot(g_hitech_hatred, edge.color = edge_colors, edge.curved = .5, vertex.size = 5, layout = l)
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Next is to use the edgelist to generate an igraph object:
g_hitech_from_edgelist## IGRAPH 1dbb475 DNW- 33 147 -- ## + attr: name (v/c), weight (e/n), frenemy (e/n)## + edges from 1dbb475 (vertex names):## [1] Nan ->Bob Mel ->Bob Jack ->Bob Chris->Bob ## [5] Dale ->Bob Ben ->Bob Len ->Carl Hal ->Carl## [9] Tom ->Dale Rick ->Dale Gary ->Dale Irv ->Dale## [13] Bob ->Dale Chris->Dale Mel ->Dale Alex ->Dale## [17] Steve->Ev Rick ->Ev Abe ->Fred Carl ->Fred## [21] Upton->Fred Hal ->Fred Chris->Gary Nan ->Gary## [25] Rick ->Gary Carl ->Hal Nan ->Ivo Upton->Ivo ## [29] Chris->Ivo Bob ->Jack Robin->Ken Tom ->Ken ## + ... omitted several edgesThis graph has three network attributes:
- names at vertex level in character format (although 3 missing at the moment),
- weight at edge level, numeric
- frenemy at edge level, numeric.
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plot(g_hitech_from_edgelist, edge.color = edge_colors, edge.width = E(g_hitech_from_edgelist)$weight, edge.curved = .5, vertex.size = 5, layout = l)
Note that we were able to register two edge attributes using the edgelist notation quite easily; and now using both in our visualization!
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2- Vertex Attribute Data Structure
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Network Graphs and Data Structures (cont'd)
If we are adding all three types of data, the R graph object will request:
Relational data and edge attributes as in one dataset (which I will shortly call as "Relational data"; but keep in mind that edge attributes go in this one as well)
Node attribute data in a separate dataset. (which I will shortly call as "Vertex/node attribute data")
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Vertex Attribute data
the vertex attribute data always comes in the edgelist format, and has to include the exact same labels used in the:
Relational data matrix notation row and column names, or
Relational data edgelist notation entries in the first two columns: "from" and "to" (the column names do not matter -- could be named as "ego" and "alter", and so forth. Mainly informative.)
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Vertex Attribute data (cont'd)
Below is the vertex attributes Krackhardt's (1999) small hi-tech computer firm dataset: whether the employees (including managers) support unionization in the firm or not:
df_vattr = fread( # read data as data.table file.path(bucket, "data","Hi-tech_vertex_attributes.csv"))df_vattr## labels union_support union_support_text## 1: Abe 0 no support## 2: Bob 0 no support## 3: Carl 0 no support## 4: Dale 2 oppose## 5: Ev 0 no support## 6: Fred 0 no support## 7: Gary 0 no support## 8: Hal 1 support## 9: Ivo 1 support## 10: Jack 1 support## 11: Ken 0 no support## 12: Len 0 no support## 13: Mel 2 oppose## 14: Nan 0 no support## 15: Ovid 1 support## 16: Pat 3 oppose + top manager## 17: Quincy 0 no support## 18: Robin 2 oppose## 19: Steve 3 oppose + top manager## 20: Tom 0 no support## 21: Upton 0 no support## 22: Vic 0 no support## 23: Walt 0 no support## 24: Rick 0 no support## 25: York 0 no support## 26: Zoe 0 no support## 27: Alex 0 no support## 28: Ben 0 no support## 29: Chris 1 support## 30: Dan 0 no support## 31: Earl 0 no support## 32: Fran 0 no support## 33: Gerry 0 no support## 34: Hugh 0 no support## 35: Irv 0 no support## 36: Jim 3 oppose + top manager## labels union_support union_support_text47 / 75
Now, let's add our df_vattr data to the graph objects. First, using the matrix notation.
Honestly, I couldn't find any straight-forward way of adding the attribute data directly to the graph usinfg the graph_from_adjacency_matrix notation. So this is what I will do:
#g_hitech = graph_from_adjacency_matrix(M_sp)g_hitech = set_vertex_attr(g_hitech, "union_support", value = df_vattr$union_support)g_hitech = set_vertex_attr(g_hitech, "union_support_text", value = df_vattr$union_support_text)g_hitech## IGRAPH fca88bf DN-- 36 147 -- ## + attr: name (v/c), union_support (v/n),## | union_support_text (v/c)## + edges from fca88bf (vertex names):## [1] Dale ->Bob Jack ->Bob Mel ->Bob Nan ->Bob ## [5] Ben ->Bob Chris->Bob Hal ->Carl Len ->Carl## [9] Bob ->Dale Gary ->Dale Mel ->Dale Tom ->Dale## [13] Rick ->Dale Alex ->Dale Chris->Dale Irv ->Dale## [17] Steve->Ev Rick ->Ev Abe ->Fred Carl ->Fred## [21] Hal ->Fred Upton->Fred Nan ->Gary Rick ->Gary## [25] Chris->Gary Carl ->Hal Nan ->Ivo Upton->Ivo ## + ... omitted several edges48 / 75
Now using the edgelist notation:
#g_hitech_from_edgelist = graph_from_data_frame(df_edgelist, directed = TRUE)g_hitech_from_edgelist_and_vattr = graph_from_data_frame( df_edgelist, directed = TRUE, vertices = df_vattr)g_hitech_from_edgelist_and_vattr## IGRAPH 2053aed DNW- 36 147 -- ## + attr: name (v/c), union_support (v/n),## | union_support_text (v/c), weight (e/n), frenemy## | (e/n)## + edges from 2053aed (vertex names):## [1] Nan ->Bob Mel ->Bob Jack ->Bob Chris->Bob ## [5] Dale ->Bob Ben ->Bob Len ->Carl Hal ->Carl## [9] Tom ->Dale Rick ->Dale Gary ->Dale Irv ->Dale## [13] Bob ->Dale Chris->Dale Mel ->Dale Alex ->Dale## [17] Steve->Ev Rick ->Ev Abe ->Fred Carl ->Fred## [21] Upton->Fred Hal ->Fred Chris->Gary Nan ->Gary## + ... omitted several edgesWe got our 3 vertices back!!!!!
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Data Structures for Affiliation Networks
a.k.a "two-mode" or "bipartite" networks
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Data Structures for Affiliation Networks
So far, we worked with "one-mode" networks: all nodes were people, and they were connected to one another.
But in some cases, we are interested in looking at connections between, for example,
- people and institutions (which NGOs are people affiliated with),
- which countries are affiliated with what unions? (e.g., countries affiliated with EU, NATO, etc.)
Which means we have more than one class of entities
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Matrix notation for Affiliation Networks
Below is Joe Galaskiewicz's (1985) "CEOs and Clubs" dataset. This data gives the affiliation network of 26 CEO's and their spouses of major corporations and banks in the Minneapolis area to 15 clubs, corporate and cultural boards.
M_ceo = read.table(file.path(bucket, "data","galaskie_ceos_sociomatrix.txt"), header=TRUE,row.names=1)M_ceo## Club.1 Club.2 Club.3 Club.4 Club.5 Club.6 Club.7## CEO1 0 0 1 1 0 0 0## CEO2 0 0 1 0 1 0 1## CEO3 0 0 1 0 0 0 0## CEO4 0 1 1 0 0 0 0## CEO5 0 0 1 0 0 0 0## CEO6 0 1 1 0 0 0 0## CEO7 0 0 1 1 0 0 0## CEO8 0 0 0 1 0 0 1## CEO9 1 0 0 1 0 0 0## CEO10 0 0 1 0 0 0 0## CEO11 0 1 1 0 0 0 0## CEO12 0 0 0 1 0 0 1## CEO13 0 0 1 1 1 0 0## CEO14 0 1 1 1 0 0 0## CEO15 0 1 1 0 0 1 0## CEO16 0 1 1 0 0 1 0## CEO17 0 1 1 0 1 0 0## CEO18 0 0 0 1 0 0 0## CEO19 1 0 1 1 0 0 1## CEO20 0 1 1 1 0 0 0## CEO21 0 0 1 1 0 0 0## CEO22 0 0 1 0 0 0 0## CEO23 0 1 1 0 0 1 0## CEO24 1 0 1 1 0 1 0## CEO25 0 1 1 0 0 0 0## CEO26 0 1 1 0 0 0 0## Club.8 Club.9 Club.10 Club.11 Club.12 Club.13 Club.14## CEO1 0 1 0 0 0 0 0## CEO2 0 0 0 0 0 0 0## CEO3 0 0 0 0 1 0 0## CEO4 0 0 0 0 0 0 0## CEO5 0 0 0 0 0 1 1## CEO6 0 0 0 0 0 0 1## CEO7 0 0 1 1 0 0 0## CEO8 0 0 1 0 0 0 0## CEO9 1 0 1 0 0 0 0## CEO10 0 1 0 0 0 0 0## CEO11 0 1 0 0 0 0 0## CEO12 0 0 0 0 0 0 0## CEO13 0 1 0 0 0 0 0## CEO14 0 0 0 1 1 1 0## CEO15 0 0 0 0 0 1 0## CEO16 1 0 0 0 0 0 1## CEO17 0 0 0 1 1 0 0## CEO18 0 1 0 0 1 1 0## CEO19 0 1 0 0 0 0 0## CEO20 0 0 0 1 0 0 0## CEO21 1 0 0 0 0 0 0## CEO22 1 0 0 0 0 0 0## CEO23 0 0 0 0 0 0 0## CEO24 0 0 0 0 0 0 0## CEO25 0 0 0 0 0 1 0## CEO26 0 0 0 0 1 0 0## Club.15## CEO1 0## CEO2 0## CEO3 0## CEO4 1## CEO5 0## CEO6 0## CEO7 0## CEO8 0## CEO9 0## CEO10 0## CEO11 0## CEO12 0## CEO13 0## CEO14 1## CEO15 1## CEO16 0## CEO17 1## CEO18 1## CEO19 0## CEO20 1## CEO21 0## CEO22 1## CEO23 1## CEO24 1## CEO25 0## CEO26 052 / 75
Matrix notation for Affiliation Networks
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Matrix notation for Affiliation Networks
Note that the rownames and column names are different.
This also means that our incidence matrix (not adjacency matrix!) is not necessarily square:
(An incidence matrix shows the relationship between two classes of objects.)
dim(M_ceo)## [1] 26 15We have 26 CEOs affiliated with 16 social clubs.
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Graph from indicence matrix
The logic is pretty much the same, but we will use a different function to register the affiliation matrix to the graph:
g_ceo = graph_from_incidence_matrix(M_ceo, directed = FALSE)g_ceo## IGRAPH f55a7d8 UN-B 41 98 -- ## + attr: type (v/l), name (v/c)## + edges from f55a7d8 (vertex names):## [1] CEO1 --Club.3 CEO1 --Club.4 CEO1 --Club.9 ## [4] CEO2 --Club.3 CEO2 --Club.5 CEO2 --Club.7 ## [7] CEO3 --Club.3 CEO3 --Club.12 CEO4 --Club.2 ## [10] CEO4 --Club.3 CEO4 --Club.15 CEO5 --Club.3 ## [13] CEO5 --Club.13 CEO5 --Club.14 CEO6 --Club.2 ## [16] CEO6 --Club.3 CEO6 --Club.14 CEO7 --Club.3 ## [19] CEO7 --Club.4 CEO7 --Club.10 CEO7 --Club.11## [22] CEO8 --Club.4 CEO8 --Club.7 CEO8 --Club.10## + ... omitted several edgesPlease note that affiliation networks are generally not directed -- if a person is affiliated to a club, the club has that person as a member, so the relationship is most frequently mutual.
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Graph from indicence matrix
g_ceo## IGRAPH f55a7d8 UN-B 41 98 -- ## + attr: type (v/l), name (v/c)## + edges from f55a7d8 (vertex names):## [1] CEO1 --Club.3 CEO1 --Club.4 CEO1 --Club.9 ## [4] CEO2 --Club.3 CEO2 --Club.5 CEO2 --Club.7 ## [7] CEO3 --Club.3 CEO3 --Club.12 CEO4 --Club.2 ## [10] CEO4 --Club.3 CEO4 --Club.15 CEO5 --Club.3 ## [13] CEO5 --Club.13 CEO5 --Club.14 CEO6 --Club.2 ## [16] CEO6 --Club.3 CEO6 --Club.14 CEO7 --Club.3 ## [19] CEO7 --Club.4 CEO7 --Club.10 CEO7 --Club.11## [22] CEO8 --Club.4 CEO8 --Club.7 CEO8 --Club.10## + ... omitted several edgesDifferent from graph_from_adjacency_matrix() function, graph_from_incidence_matrix() automatically generates a vertex attribute called "type": in our example, this helps us distinguish CEOs from clubs.
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V(g_ceo)$type## [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE## [10] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE## [19] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE## [28] TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE## [37] TRUE TRUE TRUE TRUE TRUE57 / 75
Edgelist notation for Affiliation Networks
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Edgelist notation for Affiliation Networks
We can also use edgelists for creating two-mode network graphs. But keep in mind that the graph_from_data_frame() function will not automatically detect the graph as two-mode network. Therefore you will need to make sure that your vertex attribute data has that information.
Note: there is a separate package that is designed for two-mode networks called tnet (https://toreopsahl.com/tnet/), but I believe this is beyond the scope of this tutorial, but feel free to check it out.
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Edgelist notation for Affiliation Networks
df_ceo_edgelist = fread( # read data as data.table file.path(bucket, "data","galaskie_ceos_edgelist.csv"))df_ceo_edgelist## from to## 1: CEO24 Club 1## 2: CEO9 Club 1## 3: CEO19 Club 1## 4: CEO15 Club 2## 5: CEO25 Club 2## 6: CEO17 Club 2## 7: CEO6 Club 2## 8: CEO11 Club 2## 9: CEO4 Club 2## 10: CEO20 Club 2## 11: CEO14 Club 2## 12: CEO16 Club 2## 13: CEO26 Club 2## 14: CEO23 Club 2## 15: CEO1 Club 3## 16: CEO14 Club 3## 17: CEO6 Club 3## 18: CEO2 Club 3## 19: CEO7 Club 3## 20: CEO11 Club 3## 21: CEO3 Club 3## 22: CEO5 Club 3## 23: CEO15 Club 3## 24: CEO4 Club 3## 25: CEO24 Club 3## 26: CEO10 Club 3## 27: CEO21 Club 3## 28: CEO16 Club 3## 29: CEO17 Club 3## 30: CEO23 Club 3## 31: CEO19 Club 3## 32: CEO20 Club 3## 33: CEO13 Club 3## 34: CEO26 Club 3## 35: CEO22 Club 3## 36: CEO25 Club 3## 37: CEO1 Club 4## 38: CEO12 Club 4## 39: CEO24 Club 4## 40: CEO19 Club 4## 41: CEO21 Club 4## 42: CEO13 Club 4## 43: CEO18 Club 4## 44: CEO9 Club 4## 45: CEO8 Club 4## 46: CEO20 Club 4## 47: CEO7 Club 4## 48: CEO14 Club 4## 49: CEO2 Club 5## 50: CEO13 Club 5## 51: CEO17 Club 5## 52: CEO24 Club 6## 53: CEO15 Club 6## 54: CEO16 Club 6## 55: CEO23 Club 6## 56: CEO12 Club 7## 57: CEO8 Club 7## 58: CEO2 Club 7## 59: CEO19 Club 7## 60: CEO21 Club 8## 61: CEO22 Club 8## 62: CEO9 Club 8## 63: CEO16 Club 8## 64: CEO10 Club 9## 65: CEO1 Club 9## 66: CEO11 Club 9## 67: CEO18 Club 9## 68: CEO13 Club 9## 69: CEO19 Club 9## 70: CEO8 Club 10## 71: CEO7 Club 10## 72: CEO9 Club 10## 73: CEO7 Club 11## 74: CEO17 Club 11## 75: CEO14 Club 11## 76: CEO20 Club 11## 77: CEO14 Club 12## 78: CEO3 Club 12## 79: CEO18 Club 12## 80: CEO17 Club 12## 81: CEO26 Club 12## 82: CEO25 Club 13## 83: CEO18 Club 13## 84: CEO15 Club 13## 85: CEO14 Club 13## 86: CEO5 Club 13## 87: CEO6 Club 14## 88: CEO5 Club 14## 89: CEO16 Club 14## 90: CEO4 Club 15## 91: CEO24 Club 15## 92: CEO18 Club 15## 93: CEO23 Club 15## 94: CEO15 Club 15## 95: CEO20 Club 15## 96: CEO17 Club 15## 97: CEO22 Club 15## 98: CEO14 Club 15## from to60 / 75
Edgelist notation for Affiliation Networks
df_ceo_vattr = fread( # read data as data.table file.path(bucket, "data","galaskie_ceos_vertex_attributes.csv"))df_ceo_vattr## labels type## 1: CEO1 ceo## 2: CEO2 ceo## 3: CEO3 ceo## 4: CEO4 ceo## 5: CEO5 ceo## 6: CEO6 ceo## 7: CEO7 ceo## 8: CEO8 ceo## 9: CEO9 ceo## 10: CEO10 ceo## 11: CEO11 ceo## 12: CEO12 ceo## 13: CEO13 ceo## 14: CEO14 ceo## 15: CEO15 ceo## 16: CEO16 ceo## 17: CEO17 ceo## 18: CEO18 ceo## 19: CEO19 ceo## 20: CEO20 ceo## 21: CEO21 ceo## 22: CEO22 ceo## 23: CEO23 ceo## 24: CEO24 ceo## 25: CEO25 ceo## 26: CEO26 ceo## 27: Club 1 club## 28: Club 2 club## 29: Club 3 club## 30: Club 4 club## 31: Club 5 club## 32: Club 6 club## 33: Club 7 club## 34: Club 8 club## 35: Club 9 club## 36: Club 10 club## 37: Club 11 club## 38: Club 12 club## 39: Club 13 club## 40: Club 14 club## 41: Club 15 club## labels type61 / 75
Edgelist notation for Affiliation Networks
g_ceo_from_edgelist = graph_from_data_frame( df_ceo_edgelist, directed = FALSE, vertices = df_ceo_vattr)g_ceo_from_edgelist## IGRAPH 7c8144c UN-B 41 98 -- ## + attr: name (v/c), type (v/c)## + edges from 7c8144c (vertex names):## [1] CEO24--Club 1 CEO9 --Club 1 CEO19--Club 1 CEO15--Club 2## [5] CEO25--Club 2 CEO17--Club 2 CEO6 --Club 2 CEO11--Club 2## [9] CEO4 --Club 2 CEO20--Club 2 CEO14--Club 2 CEO16--Club 2## [13] CEO26--Club 2 CEO23--Club 2 CEO1 --Club 3 CEO14--Club 3## [17] CEO6 --Club 3 CEO2 --Club 3 CEO7 --Club 3 CEO11--Club 3## [21] CEO3 --Club 3 CEO5 --Club 3 CEO15--Club 3 CEO4 --Club 3## [25] CEO24--Club 3 CEO10--Club 3 CEO21--Club 3 CEO16--Club 3## [29] CEO17--Club 3 CEO23--Club 3 CEO19--Club 3 CEO20--Club 3## + ... omitted several edges62 / 75
One more thing
before we conclude...
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One-mode projections of
affiliation networks
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One-mode projections of affiliation networks
In some cases, we are not particularly interested in which CEOs go to which sociall clubs, but rather we want to know:
- which CEOs attend the same clubs?
or,
- Which social clubs share CEO members?
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One-mode projections of affiliation networks (cont'd)
This means, if we could find a way to convert the CEO-club connections to CEO-CEO connections or club-club connections, what would be more than enough. We would then use those "one-mode projection" adjacency matrices for our analyses.
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One-mode projections of affiliation networks (cont'd)
In matrix notation, this can simply be achieved by matrix multiplication.
CEO to CEO connections:
# multiply M_ceo with its transpose # [26x15] %*% [15x26] --> [26,26] (CEO to CEO)M1_ceo = as.matrix(M_ceo) %*% t(as.matrix(M_ceo))M1_ceo## CEO1 CEO2 CEO3 CEO4 CEO5 CEO6 CEO7 CEO8 CEO9 CEO10## CEO1 3 1 1 1 1 1 2 1 1 2## CEO2 1 3 1 1 1 1 1 1 0 1## CEO3 1 1 2 1 1 1 1 0 0 1## CEO4 1 1 1 3 1 2 1 0 0 1## CEO5 1 1 1 1 3 2 1 0 0 1## CEO6 1 1 1 2 2 3 1 0 0 1## CEO7 2 1 1 1 1 1 4 2 2 1## CEO8 1 1 0 0 0 0 2 3 2 0## CEO9 1 0 0 0 0 0 2 2 4 0## CEO10 2 1 1 1 1 1 1 0 0 2## CEO11 2 1 1 2 1 2 1 0 0 2## CEO12 1 1 0 0 0 0 1 2 1 0## CEO13 3 2 1 1 1 1 2 1 1 2## CEO14 2 1 2 3 2 2 3 1 1 1## CEO15 1 1 1 3 2 2 1 0 0 1## CEO16 1 1 1 2 2 3 1 0 1 1## CEO17 1 2 2 3 1 2 2 0 0 1## CEO18 2 0 1 1 1 0 1 1 1 1## CEO19 3 2 1 1 1 1 2 2 2 2## CEO20 2 1 1 3 1 2 3 1 1 1## CEO21 2 1 1 1 1 1 2 1 2 1## CEO22 1 1 1 2 1 1 1 0 1 1## CEO23 1 1 1 3 1 2 1 0 0 1## CEO24 2 1 1 2 1 1 2 1 2 1## CEO25 1 1 1 2 2 2 1 0 0 1## CEO26 1 1 2 2 1 2 1 0 0 1## CEO11 CEO12 CEO13 CEO14 CEO15 CEO16 CEO17 CEO18 CEO19## CEO1 2 1 3 2 1 1 1 2 3## CEO2 1 1 2 1 1 1 2 0 2## CEO3 1 0 1 2 1 1 2 1 1## CEO4 2 0 1 3 3 2 3 1 1## CEO5 1 0 1 2 2 2 1 1 1## CEO6 2 0 1 2 2 3 2 0 1## CEO7 1 1 2 3 1 1 2 1 2## CEO8 0 2 1 1 0 0 0 1 2## CEO9 0 1 1 1 0 1 0 1 2## CEO10 2 0 2 1 1 1 1 1 2## CEO11 3 0 2 2 2 2 2 1 2## CEO12 0 2 1 1 0 0 0 1 2## CEO13 2 1 4 2 1 1 2 2 3## CEO14 2 1 2 7 4 2 5 4 2## CEO15 2 0 1 4 5 3 3 2 1## CEO16 2 0 1 2 3 5 2 0 1## CEO17 2 0 2 5 3 2 6 2 1## CEO18 1 1 2 4 2 0 2 5 2## CEO19 2 2 3 2 1 1 1 2 5## CEO20 2 1 2 5 3 2 4 2 2## CEO21 1 1 2 2 1 2 1 1 2## CEO22 1 0 1 2 2 2 2 1 1## CEO23 2 0 1 3 4 3 3 1 1## CEO24 1 1 2 3 3 2 2 2 3## CEO25 2 0 1 3 3 2 2 1 1## CEO26 2 0 1 3 2 2 3 1 1## CEO20 CEO21 CEO22 CEO23 CEO24 CEO25 CEO26## CEO1 2 2 1 1 2 1 1## CEO2 1 1 1 1 1 1 1## CEO3 1 1 1 1 1 1 2## CEO4 3 1 2 3 2 2 2## CEO5 1 1 1 1 1 2 1## CEO6 2 1 1 2 1 2 2## CEO7 3 2 1 1 2 1 1## CEO8 1 1 0 0 1 0 0## CEO9 1 2 1 0 2 0 0## CEO10 1 1 1 1 1 1 1## CEO11 2 1 1 2 1 2 2## CEO12 1 1 0 0 1 0 0## CEO13 2 2 1 1 2 1 1## CEO14 5 2 2 3 3 3 3## CEO15 3 1 2 4 3 3 2## CEO16 2 2 2 3 2 2 2## CEO17 4 1 2 3 2 2 3## CEO18 2 1 1 1 2 1 1## CEO19 2 2 1 1 3 1 1## CEO20 5 2 2 3 3 2 2## CEO21 2 3 2 1 2 1 1## CEO22 2 2 3 2 2 1 1## CEO23 3 1 2 4 3 2 2## CEO24 3 2 2 3 5 1 1## CEO25 2 1 1 2 1 3 2## CEO26 2 1 1 2 1 2 367 / 75
One-mode projections of affiliation networks (cont'd)
Club to club connections:
# multiply transpose of M_ceo with itself (M_ceo)# [15x26] %*% [26x15] --> [26,26] (CEO to CEO)M2_ceo = t(as.matrix(M_ceo)) %*% as.matrix(M_ceo)M2_ceo## Club.1 Club.2 Club.3 Club.4 Club.5 Club.6 Club.7## Club.1 3 0 2 3 0 1 1## Club.2 0 11 11 2 1 3 0## Club.3 2 11 22 8 3 4 2## Club.4 3 2 8 12 1 1 3## Club.5 0 1 3 1 3 0 1## Club.6 1 3 4 1 0 4 0## Club.7 1 0 2 3 1 0 4## Club.8 1 1 3 2 0 1 0## Club.9 1 1 5 4 1 0 1## Club.10 1 0 1 3 0 0 1## Club.11 0 3 4 3 1 0 0## Club.12 0 3 4 2 1 0 0## Club.13 0 3 4 2 0 1 0## Club.14 0 2 3 0 0 1 0## Club.15 1 6 8 4 1 3 0## Club.8 Club.9 Club.10 Club.11 Club.12 Club.13## Club.1 1 1 1 0 0 0## Club.2 1 1 0 3 3 3## Club.3 3 5 1 4 4 4## Club.4 2 4 3 3 2 2## Club.5 0 1 0 1 1 0## Club.6 1 0 0 0 0 1## Club.7 0 1 1 0 0 0## Club.8 4 0 1 0 0 0## Club.9 0 6 0 0 1 1## Club.10 1 0 3 1 0 0## Club.11 0 0 1 4 2 1## Club.12 0 1 0 2 5 2## Club.13 0 1 0 1 2 5## Club.14 1 0 0 0 0 1## Club.15 1 1 0 3 3 3## Club.14 Club.15## Club.1 0 1## Club.2 2 6## Club.3 3 8## Club.4 0 4## Club.5 0 1## Club.6 1 3## Club.7 0 0## Club.8 1 1## Club.9 0 1## Club.10 0 0## Club.11 0 3## Club.12 0 3## Club.13 1 3## Club.14 3 0## Club.15 0 968 / 75
One-mode projections of affiliation networks (cont'd)
In edgelist notation, this can simply be achieved by merging the dataset with itself.
CEO to CEO connections:
#from column is CEOs, to column is clubs df_ceo_edgelist1 = merge(df_ceo_edgelist,df_ceo_edgelist, by= "to", allow.cartesian=TRUE)df_ceo_edgelist1 = df_ceo_edgelist1[, .(from.x,from.y)]df_ceo_edgelist1 = df_ceo_edgelist1[, .N, by = c("from.x", "from.y")]df_ceo_edgelist1## from.x from.y N## 1: CEO24 CEO24 5## 2: CEO24 CEO9 2## 3: CEO24 CEO19 3## 4: CEO9 CEO24 2## 5: CEO9 CEO9 4## --- ## 590: CEO16 CEO9 1## 591: CEO10 CEO18 1## 592: CEO11 CEO18 1## 593: CEO18 CEO10 1## 594: CEO18 CEO11 169 / 75
One-mode projections of affiliation networks (cont'd)
Club to club connections:
#from column is CEOs, to column is clubs df_ceo_edgelist2 = merge(df_ceo_edgelist,df_ceo_edgelist, by= "from", allow.cartesian=TRUE)df_ceo_edgelist2 = df_ceo_edgelist2[, .(to.x,to.y)]df_ceo_edgelist2 = df_ceo_edgelist2[, .N, by = c("to.x", "to.y")]df_ceo_edgelist2## to.x to.y N## 1: Club 3 Club 3 22## 2: Club 3 Club 4 8## 3: Club 3 Club 9 5## 4: Club 4 Club 3 8## 5: Club 4 Club 4 12## --- ## 143: Club 1 Club 10 1## 144: Club 8 Club 1 1## 145: Club 8 Club 10 1## 146: Club 10 Club 1 1## 147: Club 10 Club 8 170 / 75
COMING UP NEXT...
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Network Centralities
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Communities
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Thanks!
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