Exercise for Chapter 6: Centrality

Exercise 1 (a)  Calculate Degree centrality scores of each node in the network above, and complete the table below.

Degree centrality of a node refers to the number of edges attached to the node.  In order to know the standardized score, you need to divide each score by n-1 (n = the number of nodes).  Since the graph has 7 nodes, 6 (7-1) is the denominator for this question.

Degree Centrality

 Node Score Standardized  Score 1 1 1/6 2 1 1/6 3 3 3/6 = 1/2 4 2 2/6 = 1/3 5 3 3/6 = 1/2 6 2 2/6  = 1/3 7 2 2/6 = 1/3

(b)  Calculate Closeness centrality scores of each node in the network above, and complete the table below.  (Do not forget to take inverse.)

You need to calculate the inverted score after you count the total number of steps to a node.  In order to know the standardized score, you need to divide a score by (n-1), then take inverse.  Note that the most central node is node 4 while the most central node for degree centrality is node 3 and 5.

Closeness Centrality

 Node Score Standardized  Score 1 1/16 6/16 = 3/8 2 1/16 6/16 = 3/8 3 1/11 6/11 4 1/10 6/10 = 3/5 5 1/11 6/11 6 1/15 6/15 = 2/5 7 1/15 6/15 = 2/5

(c)  Calculate Betweenness centrality scores of each node in the network above, and complete the table below.

To calculate betweenness centrality, you take every pair of the network and count how many times a node can interrupt the shortest paths (geodesic distance) between the two nodes of the pair. For standardization, I note that the denominator is (n-1)(n-2)/2. For this network, (7-1)(7-2)/2 = 15.  Note that node 5 has a little smaller centrality score that node 3 and 4 because the connection between node 6 and 7 reduces the controllability of node 5.

Betweenness Centrality

 Node Score Standardized  Score 1 0 0 2 0 0 3 16/3 16/45 4 13/3 13/45 5 13/3 13/45 6 0 0 7 0 0

On the following questions (d), (e), and (f), you need to remember the lecture by
Motoki.  You should read the brief summary of each centrality on your notes. (I wrote it at the very end of lecture).  If you understand it well, you don't have any difficulties to answer the three questions.

(d) Suppose the above network refers to friendship network.  Each node represents a person, and each edge represents friendship between the persons at ends.  If you are interested in finding the most poplar person in the network, which centrality measure is the most appropriate?  Give the answer with reasons why it is the most appropriate.

The most popular person should have the highest number of friends.  Thus, degree centrality is the most appropriate measure.

(e) Suppose the above network refers to information flow network of an organization.  Each node represents a section in the organization, and each edge represents a possible information exchange between the sections at ends.  If you are interested in finding the section that can most efficiently obtain information from every other section, which centrality measure is the most appropriate?  Give the answer with reasons why it is the most appropriate.

To obtain information, one should be near from everyone.  In this sense, the node in the nearest position on average can most efficiently obtain information.  Thus, closeness centrality is the most appropriate.

(f) Again, suppose the above network refers to information flow network of an organization. If you are interested in finding the section that can most frequently control information flow in the network, which centrality measure is the most appropriate?  Give the answer with reasons why it is the most appropriate.

To control information flow, a node should be between other nodes because the node can interrupt infromation flow between them.  Thus, betweenness centrality is the most appropriate measure.

Exercise 2 (a) For each Network, calculate the three centrality indexes (degree, closeness, and betweenness) for each entire graph (Not for each node of a graph) .  Which graph is more centralized ?

Use equations 6.8, 6.9, and 6.10 on Page 171 in the textbook for this question.  The basic idea of centrality for entire graph is to calculate average deviance of each node from the most central node.

Degree centrality (Equation 6.8)

Network A

Node 1 - centrality score  3
Node 2 - centrality score 1
Node 3 - centrality score  1
Node 4 - centrality score  1

The maximum score is 3.

Use Eq.6.8

((3-3)+(3-1)+(3-1)+(3-1))/(16-12+2) = (0+2+2+2)/6 = 1

The degree centrality score of Network A is 1.

Network B

Node 1 - centrality score  3
Node 2 - centrality score  3
Node 3 - centrality score  3
Node 4 - centrality score  3

The maximum score is 3.

Use Eq.6.8

((3-3)+(3-3)+(3-3)+(3-3))/(16-12+2) = (0+0+0+0)/6 = 0

The degree centrality score of Network B is 0.

Thus, Network A is more centralized than Network B for degree centrality.

Closeness centrality (Equation 6.9)
* You need to use standardized score.

Network A

Node 1 - centrality score  3/3 = 1
Node 2 - centrality score  3/5
Node 3 - centrality score  3/5
Node 4 - centrality score  3/5

The maximum score is 3/3 = 1.

Use Eq.6.9.

((1-1)+(1-3/5)+(1-3/5)+(1-3/5))/((16-12+2)/(8-3))
= (0+2/5+2/5+2/5)/(6/5)
= (6/5)/(6/5) = 1 .

The closeness centrality score of Network A is 1.

Network B

Node 1 - centrality score  3/3 = 1
Node 2 - centrality score  3/3 = 1
Node 3 - centrality score  3/3 = 1
Node 4 - centrality score  3/3 = 1

The maximum score is 1.

Use Eq.6.9.

((1-1)+(1-1)+(1-1)+(1-1))/((16-12+2)/(8-3)) = (0+0+0+0)/(6/5) = 0

The closeness centrality score of Network B is 0.

Thus, Network A is more centralized than Network B for closeness centrality.

Betweenness centrality (Equation 6.10)
* You need to use standardized score.

Network A

Node 1 - centrality score  3/3 = 1
Node 2 - centrality score  0
Node 3 - centrality score  0
Node 4 - centrality score  0

The maximum score is 3/3 = 1.

Use Eq.6.10.

((1-1)+(1-0)+(1-0)+(1-0))/(4-1) = (0+1+1+1)/3 = 1.

The closeness centrality score of Network A is 1.

Network B

Node 1 - centrality score  0
Node 2 - centrality score  0
Node 3 - centrality score  0
Node 4 - centrality score  0

The maximum score is 0.

Use Eq.6.10.

((0-0)+(0-0)+(0-0)+(0-0))/(4-1) = (0+0+0+0)/3 = 0

The closeness centrality score of Network B is 0.

Thus, Network A is more centralized than Network B for betweenness centrality.

I note that centrality score of Network A is 1 for any centrality measures while centrality score of Network B is 0 for any centrality measures because Network A is the maximally centralized network with four nodes and Network B is the non-centralized network with four nodes.

(b) Comparing the centralized and non-centralized graph above, discuss advantages and disadvantages of centralized network.

This is an open-ended question. The aim of the question is to let you imagine some centralized and non-centralized networks in the real world and compare them.  Since this course is a sociology course, it is important that you apply the mathematical approach to actual social phenomenon.  I give you a sample answer.

If the network represents information flow, Node 1 in Network A has huge controllability of information flow.  If node 1 works very well, the network is very efficient because there is no redundant edges in the network.  However, if node 1 does not work, node 2, 3, and 4 are isolated. In this sense, Network A is risky in that efficiency of information flow totally depends on node 1's performance.  On the other hand, Network B is more flexible.  Even if node 1 does not work, all other nodes can still communicate one another.  However, in another sense, this network has many redundant edges.  If it takes costs to make edges, this network is very costly.

Exercise 3

Centrality can be used as an index of power in networks, because the most central node is more likely to occupy the most powerful position. Do you think it is true ? Answer with reasons why you think so.

The answer is  actually "no".  An experimental study by Cook et. al. (1983) demonstrates that centrality is not necessarily the index of power. You do not need to know about this study, but you need to think whether the powerful position is always central in a network.  I do not explain details of this study.  If you are interested in it, contact me. I will explain it.  Since the main purpose of this question is let you think about the meaning of centrality, you can write your own idea.

If you have questions, ask Motoki.  mwatabe@ucla.edu