Social Network Analysis

Social Network, a highlight in social networking, is a social structure made up of actors(which could be individuals and organizations) and ties between them(representing relationships between actors). 

Two Sociograms (graphical representation of social networks)

By studying the pattern of interaction between those actors in social networks, we can identify and locate influential entities and observe network dynamics. This is also known as Social Network Analysis which would be our topic today.

To better illustrate some metrics and basic concepts in Social Network Analysis, let's consider a simple Social Network as follows:

A sociogram consists of 4 students: A, B, C and D,
which represents their relationships between others

Fundamental Concepts:

1. Actor
Consider Student D alone. He is an actor and a social entity in this social network. In one mode networks, all actors are found having the same type inside. Other examples of actors are a crew in a ship, a region in a country and so on.

2. Relational tie
In the Network above, student A and D are linked by a line which we call a tie in Social Network Analysis. As this is an undirected graph, we can consider student A and D knowing each other. Ties can also be directional in other circumstances such as representing the subscriptions in Twitter, which is in one direction with the arrow pointing to the node of person being subscribed from the node of the subscriber.

3. Dyad
Dyad comprises a pair of actors and the tie between them. Student A and D is a Dyad in the previous graph.

4. Triad
It is a similar concept as Dyad but with 3 actors involved. It is composed of a subset of 3 actors and the ties between them in a social network. Student A, B and C forms a Triad as you may notice in the graph.

5. Cutpoint
Cutpoint is a node causing the network disconnected if it is deleted. Obviously, Student A is the cutpoint here because when it is deleted, Student D would be disconnected from others.

6. Bridge
It represents a tie that will make the network disconnected when it is deleted. In this sociogram, the tie among Student A and D is the bridge since Student D will be disconnected if the tie is broken.

Metrics:

1. Nodal Degree

It is the number of lines incident with it and can be regarded as a measure of the activity of an actor. A node with degree equal to 0 is called an isolate node.

Taking the previous social network as an example, we can find that the degrees of every node are as below.

Node	Degree
A	3
B	2
C	2
D	1

2. Density

Density can be treated as a measure of existing ties in a network out of the possible number. To calculate, we first assume L is the number of links and g is the number of nodes in a network. As the maximum possible ties between nodes are g(g-1)/2, the formula for calculating Density of a social network would be L/[g(g-1)/2], which gives 2L/g(g-1).

Therefore, the density of the above graph is: 2*4/4(4-1) = 2/3.

3. Sociomatrix

The relationship between each node in a social network can be represented by a sociomatrix, which has a size of g x g (g rows and g colums, g is the number of nodes). The entries in a sociomatrix record which pairs of nodes are adjacent and which are not. Each entry is called a cell in a matrix. ‘1’ will be input in the cell when there is a tie among two nodes in the network, while ‘0’ will be inserted when there are no ties between a pair of nodes, representing they have no relations.

Student	A	B	C	D
A	-	1	1	1
B	1	-	1	0
C	1	1	-	0
D	1	0	0	-

From the matrix, we can differentiate between directed and undirected graphs of social networks since for undirected graphs, the cell for 2 nodes in the row A for A and B(i.e. AB) must have equal value of the cell for the 2 nodes in the row B for B and A(i.e. BA). Matrix operations are important for Social Network Analysis, especially for directed graphs as patterns in the network can be highlighted by then.

4. Centrality

It determines the relative importance of a node in a social network, and tries to identify the most important nodes in the network. Here demonstrates 2 centrality measures:

a) Degree Centrality

As mentioned previously, degree of a node is the number of adjacent nodes the node has. Degree centrality is the sum of other nodes connected to that node in concern, which also represents activity and popularity of the node.

Provided that C_D(n_i) is defined as a degree centrality index in actor-level, and the normalized actor degree centrality is: 

where C_D(n_i) = d(n_i) and g is the number of nodes

From our example, a following result table can be obtained as follows:

Student	CD(ni)	C’D(ni)
A	3	1
B	2	2/3
C	2	2/3
D	1	1/3

b) Closeness Centrality

Closeness is based on the inverse of the geodesic distance of a node to other nodes in a network. It describes the average distances between one node and others node connected to it and identifies nodes that are relatively close to all other nodes. Closeness is taken on the inverse of geodesic distance to make sure that the larger the value is, the more important the node is in the network, which aligns with other metrics used in social network analysis.

Actor Closeness centrality is represented by:  

and normalized Closeness centrality is: C’_c(n_i) = (C_c(n_i))(g-1) 

where d is the shortest distance between 2 nodes and g is the number of nodes in the network

The following result table can be obtained from the previous social network:

Student	CC(ni)	C’C(ni)
A	1/3	1
B	1/4	3/4
C	1/4	3/4
D	1/5	3/5

We can see that Student A got the highest Degree centrality and Closeness centrality, which implies that he is the most popular among students and is relatively close to other students.

These metrics mentioned above facilitates the analysis of social network. Apart from collecting data in social networks and perform analysis to formulate marketing strategies by companies, organizations and even sports teams could even make use of it to examine the relationships between staff and members. With the rapid development of the mobile network and the increasing number of the users, it is undoubtedly that Social Network Analysis will have a more important role in the near future.

References:
[1] Wikipedia, Social Network Analysis
http://en.wikipedia.org/wiki/Cascade_(Social Network_Theory)

[2] Wikipedia, Social Network
http://en.wikipedia.org/wiki/Social_Network

[3] www.analytictech.com
http://www.analytictech.com/networks/whatis/htm