POST 3:Social Network Analysis(SNA)

What is SNA?
In lecture 6 and 7, I have learned a lot of things about Social Network Analysis (SNA), for example, graphical representation of social networks, some terminologies, centrality and etc. I am so amazed that social network can be calculated and analyzed by using so many mathematical methods. 

So what is SNA? Social network analysis (SNA) is the mapping and measuring of relationships and flows between people, groups, organizations, computers, URLs, and Social network analysis views social relationships in terms of network theory consisting of nodes and ties. Nodes are the individual actors within the networks, and ties are the relationships between the actors. SNA provides both a visual and a mathematical analysis of human relationships. 

A SNA example

In order to have a better understanding of SNA, I'd like to introduce some concepts of it by analyzing the following example.

From the above picture, we can easily see it is a non-directional network which involves 5 nodes and 6 ties. In other words, in the sociograph there are 5 actors and 6 relationships. We can represent the relationships among the members of this social network by using a matrix as follows. And it is possible to find patterns about the communities that the social network represent.

Cutpoint: A node which, if delected, will make the network disconnected. So in this case, we can easily know David is the cutpoint.

Bridge: A tie which, if delected, will make network disconnected. In the example, the tie connected David and Eva is the bridge.

Degree: The degree of a node is the number of links that are incident with it. Equvalently, the degree of a node is the number of nodes adjacent to it. With regard to this instance, the degree of each nodes is shown as follows. 

Density: The proportion of ties that exist out of all possible ties, which is equal to the number of links divided by the number of vertices in a complete graph with the same number of nides. We can calculate the density of this social network like this:

 

Geodesic Distances: The shortest of all the paths between two nodes is called the geodesic path, and the distance of the geodesic path is the geodesic distance. In this case, the distances between each two nodes are shown below.

Clique: Maximum set of nodes in which every node is connected to every other is a Clique. In this network, {Alice, Bob, David} and {Alice, Carol, David} are cliques.

K-Plex: A set of n nodes in which every node has a tie to at least n-k others in the set. In the case, {Alice, Bob, Carol, David} is a 2-plex, because every node has a tie to at least 2 other nodes.

Calculations & Measurements

When analyzing different roles in the social network, many methods can be used, such as Centrality and Influence Range.

 Centrality: Centrality identifies which nodes are in the "center" of the network, in other words, the "key player". So we can use this method to find who is the most influencial person in our example. There are 3 standard centrality measures capture a wide range of "importance" in a network: Degree Centrality, Closeness Centrality and Betweenness Centrality.

Degree Centrality: Degree Centrality is the sum of all other actors who are directly connected to the actor in concern. It signifies activity or popular. It can be normalized as :

 

Closeness Centrality: An actor is considered important if he/she is relatively close to all other actors. Closeness represents the mean of the geodesic distances between some particular node and all other nodes connected with it. It can be understood as how long does it take for a message to spread inside the network from a particular node. Closeness is based on the inverse of the geodesic distance of each actor to every other actor in the network. Closeness Centrality can be expressed as:

Normalized Closeness Centrality can be expressed as:

 Betweenness Centrality: The number of times a node connects pairs of other nodes, who otherwise would not be able to reach one another. It is a measure of the potencial for control as an actor who is high in "betweenness"is able to act as a gatekeeper controlling the flow of resources between the alters that he/she connects. Betweenness Centrality counts the number of shortest paths between i and j that actor j resides on. Betweenness Centrality can be expressed as :

 g_jk = the number of geodesics connecting jk, g_jk(n_i) = the number that actor i is on.
The formula can also be nomalized as: 

As introduced above, we can calculate the centrality of each node and analyze as follows:

 From the matrix we can see David is the most influental actor from 3 aspects:
1. David has relationships with all the other actors, and he has the most direct connections in this social network, which makes him the most active node in the social network. He also has the largest value of each attribute.
2.The distances between David and other actors are the shortest, that is to say, he has the shortest way anyone else, which makes him communicate with others more quikly than anyone in the same network.
3. David has the most direct connection with others, and for Alice, Bob and Carol, if they want to communicate with Eva, they must via David, vice versa. It means that David acts as a gatekeeper controlling the flow of resources between the alters he connects. It is also made him the most influencial one.

Influence Range  
Define influence range of n_i as the set of actors who are reachable from n_i. Define J_i as the number of actors in the influence range of actor i (excluding i itself). It is an "improved" actor-level centrality closeness index considers how proximate n_i is to the actors in its influence range. It can be expressed as follows:

This index is a ratio of the fraction of the actors in the group who are reachable, to the average distance that these actors are from the actor n_i. The calculation result shows as follows:

 From the result, we can easily see, by using this measure we can also decide that David is the most influential actor.

Conclusion  
 From our analysis, we can draw a conclusion that David is the "key player"of this social network, and absolutely he is the most influential actor.

Findings 
After analyzing the this social network, I obtained much.
1. SNA plays an important role in analyzing social network. By using different measures of SNA, we can have a deep understanding of a particular social network, such as knowing its characteristics and finding the "key player" of it. So I found SNA is an useful way to measuring social networks.
2. When dealing with some cases, we can have different measures, for example, in this case we can use influence range and centrality to decide who is the most influential actor in our social network. And no doubt, this two measurs help us get the same answer. That is to say, despite different measures are used for analyzing the same social network, they can acheive the same goal and even the same answer, and in turn, they can also be used for mutual autentication, verifing whether the answer is correct.
3. SNA can be also used in our daily life or in the work to help us dealing with different situations and problems. Government can use it, companies can use it , everyone can use. It can help us have a better understanding of our social network, our business and our life.