Degree Centrality

In these lecture notes we will go through the basic centrality metrics. Particularly, the “big three” according to Freeman (1979), namely, degree, closeness (in two flavors) and betweenness.

We first load our trusty Pulp Fiction data set from the networkdata package, which is an undirected graph of character scene co-appearances in the film:

   library(networkdata)
   library(igraph)
   g <- movie_559

Degree Centrality

Degree centrality is the simplest and most straightforward measure. In fact, we are already computed in the lecture notes on basic network statistics since it is the same as obtaining the graph’s degree sequence. So the igraph function degree would do it as we already saw.

Here we follow a different approach using the row (or column) sums of the graph’s adjacency matrix:

   A <- as_adjacency_matrix(g)
   A <- as.matrix(A)
   rowSums(A)

          BRETT           BUDDY           BUTCH      CAPT KOONS     ED SULLIVAN 
              7               2              17               5               2 
   ENGLISH DAVE       ESMARELDA        FABIENNE      FOURTH MAN       GAWKER #2 
              4               1               3               2               3 
    HONEY BUNNY          JIMMIE            JODY           JULES           LANCE 
              8               3               4              16               4 
        MANAGER       MARSELLUS          MARVIN         MAYNARD             MIA 
              5              10               6               3              11 
         MOTHER          PATRON      PEDESTRIAN        PREACHER         PUMPKIN 
              5               5               3               3               8 
         RAQUEL           ROGER SPORTSCASTER #1 SPORTSCASTER #2        THE GIMP 
              3               6               2               1               2 
       THE WOLF         VINCENT        WAITRESS         WINSTON           WOMAN 
              3              25               4               3               5 
      YOUNG MAN     YOUNG WOMAN             ZED 
              4               4               2 

The igraph function as_adjancency_matrix doesn’t quite return a regular R matrix object, so we have to further coerce the resulting object into a numerical matrix containing zeroes and ones using the as.matrix function in line 2. Then we can apply the native rowSums function to obtain each node’s degree. Note that this is same output we got using the degree function before.

Indegree and Outdegree

The movie network is based on the relationship of co-appearance in a scene which by nature lacks any natural directionality (it’s a symmetric relation) and can therefore be represented in an undirected graph. The concepts of in and outdegree, by contrast, are only applicable to directed relations. So to illustrate them, we need to switch to a different source of data.

We pick an advice network which is a classical directed kind of (asymmetric) relation. I can give advice to you, but that doesn’t necessarily mean you can give advice to me. The networkdata package contains one such data set collected in the late 80s early 1990s in a New England law firm (Lazega 2001) called law_advice:

   d.g <- law_advice
   V(d.g)

+ 71/71 vertices, from d1a9da7:
 [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
[26] 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
[51] 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71

   vertex_attr(d.g)

$status
 [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 1 1 1 1 1 1 1 1 1 2 2
[39] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

$gender
 [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 2 1 2 1 1 1 1 2 1 1 1 2
[39] 2 1 1 1 2 2 1 2 1 2 1 1 2 1 1 1 1 1 2 1 2 2 2 1 1 2 1 1 2 1 2 1 2

$office
 [1] 1 1 2 1 2 2 2 1 1 1 1 1 1 2 3 1 1 2 1 1 1 1 1 1 2 1 1 2 1 2 2 2 2 1 2 1 3 1
[39] 1 1 1 1 1 3 1 2 3 1 1 2 2 1 1 1 1 1 1 2 2 1 1 1 2 1 1 1 1 1 1 1 1

$seniority
 [1] 31 32 13 31 31 29 29 28 25 25 23 24 22  1 21 20 23 18 19 19 17  9 16 15 15
[26] 15 13 11 10  7  8  8  8  8  8  5  5  7  6  6  5  4  5  5  3  3  3  1  4  3
[51]  4  4 10  3  3  3  3  3  2  2  2  2  2  2  2  1  1  1  1  1  1

$age
 [1] 64 62 67 59 59 55 63 53 53 53 50 52 57 56 48 46 50 45 46 49 43 49 45 44 43
[26] 41 47 38 38 39 34 33 37 36 33 43 44 53 37 34 31 31 47 53 38 42 38 35 36 31
[51] 29 29 38 29 34 38 33 33 30 31 34 32 29 45 28 43 35 26 38 31 26

$practice
 [1] 1 2 1 2 1 1 2 1 2 2 1 2 1 2 2 2 2 1 2 1 1 1 1 1 2 1 1 2 2 1 1 1 1 2 2 1 2 1
[39] 1 1 1 2 1 2 2 2 1 2 1 2 1 1 2 1 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1 2 1

$law_school
 [1] 1 1 1 3 2 1 3 3 1 3 1 2 2 1 3 1 1 2 1 1 2 3 2 2 2 3 1 2 3 3 2 3 3 2 3 3 3 2
[39] 1 1 2 2 2 1 3 2 3 3 2 2 3 3 3 3 3 2 2 3 2 2 3 2 2 2 3 3 2 3 3 2 2

We can see that the graph has 71 vertices, and that there are various attributes associated with each vertex, like gender, age, seniority, status in the law firm, etc. We can query those attributes using the igraph function vertex_attr, which takes the graph object as input.

To keep things manageable, we will restrict our analysis to partners. To do that we need to select the subgraph that only includes the vertices with value of 1 in the “status” vertex attribute. From the data description, we know the first 36 nodes (with value of 1 in the status attribute) are the law firm’s partners (the rest are associates). In igraph we can do this as using the subgraph function:

   d.g <- subgraph(d.g, 1:36)
   V(d.g)

+ 36/36 vertices, from 756dfbd:
 [1]  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
[26] 26 27 28 29 30 31 32 33 34 35 36

   V(d.g)$status

 [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 1 1 1 1 1 1 1 1 1

The first line just tells igraph to generate the subgraph containing the first 36 nodes (the partners). The subgraph function thus takes two main inputs: The graph object, and then a vector of node ids (or node labels) telling the function which nodes to select to create the node-induced subgraph.

Computing in and outdegree

OK, going back to the partners subgraph, we can now create our (asymmetric) adjacency matrix and compute the row and column sums:

   d.A <- as_adjacency_matrix(d.g)
   d.A <- as.matrix(d.A)
   rowSums(d.A)

 [1]  3  7  7 17  4  0  4  2  3  7  5 18 11 13 10 19 17  5 21 10  9 12  9 16  8
[26] 22 18 22 13 15 16  9 15  6 15  7

   colSums(d.A)

 [1] 18 17  8 14 10 17  4  8 10  6 11 14 14 12 15 13 21  7  7 17 11 10  2 14  7
[26] 20  2 14 12 11  8 13  2 16  8  2

Note that in contrast to the undirected case the row and column sums give you two different sets of numbers. The row sums provide the directed graph’s outdegree set (number of outgoing links incident to each node), and the column sums provide the graph’s indegree set (number of incoming links incident to each node). So if you are high in the first vector, you are an advice giver (perhaps indicating informal status or experience) and if you are high in the second you are advice taker.

Of course igraph has a dedicated function for this, which is just our old friend degree with an extra option mode, indicating whether you want the in or outdegrees:

   d.o <- degree(d.g, mode = "out")
   d.i <- degree(d.g, mode = "in")
   d.o

 [1]  3  7  7 17  4  0  4  2  3  7  5 18 11 13 10 19 17  5 21 10  9 12  9 16  8
[26] 22 18 22 13 15 16  9 15  6 15  7

   d.i

 [1] 18 17  8 14 10 17  4  8 10  6 11 14 14 12 15 13 21  7  7 17 11 10  2 14  7
[26] 20  2 14 12 11  8 13  2 16  8  2

Note that the graph attributes are just vectors of values, and can be accessed from the graph object using the $ operator attached to the V() function as we did above.

So if we wanted to figure out the correlation between some vertex attribute and in or out degree centrality, all we need to do is correlate the two vectors:

   r <- cor(d.o, V(d.g)$age)
   round(r, 2)

[1] -0.43

Which tells us that at least in this case, younger partners are more sought after as sources of advice than older partners.

References

Freeman, Linton C. 1979. “Centrality in Social Networks Conceptual Clarification.” Social Networks 1: 215–39.

Lazega, Emmanuel. 2001. The Collegial Phenomenon: The Social Mechanisms of Cooperation Among Peers in a Corporate Law Partnership. Oxford University Press.