SNA for an Example

In lecture 6-8, we learnt a lot of things about social networking analysis, such as the graphical representation of social networks, terminologies for SNA, concepts on centrality and centralization, prestige, ranking algorithms and SNA examples. It is really a new word to me, I didn’t think that the social network can be analyzed using so many methods. To help our revision on SNA, we need analyze the following social network example.

Before analyzing the above social network, let me describe what SNA is first. Social network analysis (SNA) is the study of relationships and flows between individuals or entities such as people, groups, organizations, computers, URLs, and other connected information/knowledge entities. The nodes in the network represent the people or groups while the links or ties represent the relationships or flows between the nodes. SNA provides both visual and mathematical analysis of social relationships ^[1]. There are two kinds social network, one mode and two mode networks. One node networks only contain one type nodes, which means all nodes are of the same type; Two mode networks involve relations among two different types of nodes.

OK, it is time to analyze the above social network. First, let me describe this social network according what I learnt in lecture 6-8. It is a network contains five nodes, and 6 ties, which can be said 5 students, 6 relationships. This is a non-directional network. The relationships are:

(1)  For Alice, she has relationship with Bob, Carol and David;

(2)  For Bob, he has relationship with Alice and David;

(3)  For Carol, she has relationship with Alice and David;

(4)  For David, he has relationship with everyone in this network, i.e. Alice, Bob, Carol, and Eva;

(5)  For Eva, she only has relationship with David.

To finding patterns about the above social network more easily, we can use a simple matrix to represent it.

	Alice	Bob	Carol	David	Eva
Alice	-	1	1	1	0
Bob	1	-	0	1	0
Carol	1	0	-	1	0
David	1	1	1	-	1
Eva	0	0	0	1	-

From the above symmetrical matrix we can say it is undirectional network. May be we can treat the above social network as their friendships on facebook. We can calculate the density of this social network: 

{Alice, Bob, David} and {Alice, Carol, David} are cliques because they have maximum density(1). If we only consider {Alice, Bob, Carol, David}, it is a 2-plex. Because 4 nodes in which every nodes has a tie to at least 4-k=2 others in the set. In this 2-plex, every node is connected to at least two others in the set.

Secondly, let me calculate the three popular individual centrality measures(degree centrality, closeness centrality and betweeness centrality)  respectively to analyze different roles and grouping in this social network.

Degree Centrality

The concept of degrees - the number of direct connections a node has is used by social network researchers to measure network activity or popularity of a node. The following graph shows nodes (n_i) and their centrality C_D(n_i): Also

Also they can be normalized as C'_D= d(n_i)/(g-1) as:

Alice: 3/4=0.75 ; Bob: 2/4=0.5; Carol: 2/4=0.5; David: 4/4=1; Eva: 1/4=0.25. From this result, we can say David has the most direct connections in the network, making him the most active node in the network. He is most influential because he has relationships with all other in the network. By the way, we can calculate the freeman which used to measure the group degree centralization to analyze how large the sum of differences can actually be. Here the largest degrees of the network is C_D(n*) = 4

Closeness Centrality

This centrality measures the geodesic distances between some particular node and all other nodes connected with it. An actor is considered important is he/she is relatively close to all other actors. We can base on the closeness centrality formula to calculate the closeness centrality of each node.

P.S. 

C_C(Alice) =0.2; C_C(Bob) = C_C(Carol)≈0.17; C_C(David) =0.25; C_C(Eva) ≈0.14

To get the normalized closeness centrality C’_C(n_i)= C_C(n_i)(g-1), here g-1=4:

C'_C(Alice) =0.8; C'_C(Bob) = C'_C(Carol)≈0.68; C'_C(David) =1; C'_C(Eva) ≈0.56

The pattern of David's direct ties allow him to access all the nodes in the network more quickly than anyone else. He has the shortest paths to all others - he is close to everyone else. In another way, we can say David is most influential because he is close to everyone.By the way, we can also calculate the group closeness centralization to measure the overall level of closeness of this network.

C=(0.25-0.2)+2x(0.25-0.17)+(0.25-0.14)=0.5+0.16+0.11=0.77

Betweeness Centrality

Betweeness centrality is used to measure quantifying the control of a human on the communication between other humans in a social network. It counts the number shortest path between a node i and k that actor j resides on.



Then we can get,

C_B(Alice) = 0.5; C_B(Bob) = C_B(Carol) = C_B(Eva) = 0; C_B(David) = 3.5

To normalize the result:

C'_B(Alice) ≈ 0.083; C_B(Bob) = C_B(Carol) = C_B(Eva) = 0; C_B(David) ≈ 0.583

David has the most direct ties, he is able to act as a gatekeeper controlling the flow of resources between the alters that he or she connects. By the way, the following is the group betweeness centrality calculation:

Therefore,  C_B=[(0.583-0.083)+(0.583-0)x2+(0.583-0.583)] / 4≈0.5623=56.23%

Assumptions

At last, suppose I am conducting a research on the social network of these five students and the above results are obtained, the findings and their implications are discussed base on my data. David connects with everyone in this social network. He is the core person in this network. Also, David is in an excellent position to monitor the information flow in the network --he has the best visibility into what is happening in the network. Eva is the isolates of the network, she only connects with David. She is on the periphery. As the core person in the network, David may encourage Eva participate more in the network. 



When learning how to calculate the betweeness centrality, I could not get the same answer as the ring example on the lecture notes. After discussing with Dr. Rosanna and classmates, I found that I used the wrong formula (directional network) without over two in the denominator [ only (g-1)(g-2)]. I was more familiar with the calculation after discussing with others, participating in social network can help the learning process.