Mathematical Questions in Social Network Analysis

In last three lectures, the SNA which is the most useful and accurate tool to explain the pattern of interaction between actors has been introduced in detail by Prof. Rosanna. Through the SNA, abstract social relationship could be transformed into math model to measure and calculate which might be more convenient to study and make decisions for researchers in further. More specifically, there are numerous methods in SNA, from graph theory to socio matrix, from centrality to prestige, and all the way to ranking. However, in my opinion, there are some trifles need to be decomposed, although the lecture notes has involved almost of all aspects in SNA.

1.      Powers of a Matrix

From the lecture, I am confirmed that every classmate could write the zero-one matrix to express any social relationship. However, in terms of powers of a matrix, maybe we have something confused. Now, let us review the knowledge of linear algebra learnt before. To matrix X, if X_ij=p (X_ij is the element in the i^th , row j^th column), that implies the number of methods from n_i to n_j walking length n is p. Therefore, in social networking, the entries of the matrix X^p give the total number of walks of length p from node n_i to n_j and the value of X_ij in X^p gives the quantity of probability. Here, we might surprisingly find why there is relationship between A and B in matrix X, the value being 0 in X² , but in X³ they have relationship as well. Maybe we can draw a picture like this.

From the picture, length 1, A to B, length 2, we cannot find any ways, and length 3, there is a way A to C to D to B. Obviously, these properties would be same for both directed and nondirected graph.

2.      Group Degree Centralization

In terms of the calculation formula

Where, g is the number of actors.

C_D(n^*) means the largest degree of actor, C_D(n_i) implies the number of degree of actor n_i. So we could calculate the numerator easily. For the denominator, there might be something confused with the meaning of “max”. In fact, it could be understood as the max value of numerator in entire probable topology patterns including star, circle and line etc.. To estimate the denominator, we should let C_D(n^*) become largest and C_D(n_i) become smallest. Undoubtedly, maximum value of the denominator occurs when the network is in star shape, which equals to numerator in star shape. Therefore, we could deduce it is (g-1)(g-2) and C_D =1 in star shape definitively.

3.      PageRank

Before discussion of the PageRank, there is need to have a review of Rank prestige. To the topology

The sociomatrix X is

p = X’p, Which corresponds to the system of equations

On the other hand, the calculation of PageRank is still ambiguous after reading the content of Slide 25 in Week 9. What does the formula  mean? And how to use it?

We could analysis every element step by step first. To Actor A, the direction to it is just C, but the Actor C directing to other actors is just A as well. So

 To Actor B, the direction to B is just A, and A directing to B and C which make up of 1/2. Hence,

To Actor C, the direction to C is A and B, A directing to B and C which make up of 1/2, and B only direct to C. Therefore, 

Having a detailed understanding and recognition of these mathematical questions, there are no obstacles to analysis the social model and psychology. Actually, there are some other interesting parts of SNA not included in the article, and you might dig them personally. 

Social Behaviors on the Internet

To learn social behaviors on the Internet, we studied a case about social cloud computing last class. The main idea of the article is that each community could achieve resource sharing through cloud computing and cloud storage.

In terms of the definition of social cloud, my understanding is that a social cloud is a resource and service sharing framework utilizing relationships established between members of a social network. In addition to that, the second problem Prof. Rosanna proposed “what are the possible applications of a social cloud?” In my opinion, it might conclude social computation cloud, social storage cloud, social collaborative cloud, social cloud for public science and enterprise social cloud, which could be resolved easily for the unambiguous structure of the article. What we should admit is that there are several disadvantages and imprecision in the article, among these, the biggest drawback might be the privacy. As the former part said in the article, “Users are more likely to trust information from a “friend” if the digital relationship between the two is based on a real world relationship (friend, family, and colleague) rather than a purely online relationship (second life, online games, etc).” Therefore, we would public our private information and life status, but we cannot guarantee that all the online friends could know each other and like to share our stories. What’s more, some people just look like our “friends”, and we maybe have a chip on their shoulders. Nowadays, there is a status between cognizance and incognizance named “I have his Facebook”. It might be a kidding, but a common sight in our life. Hence, the problem of individual privacy in social networking has emerged and extends further. On the other hand, socially motivated resource sharing indeed provides us so much convenience in study, life and work. Thanks to wireless application, more and more people use Dropbox instead of USB flash disk. Then the issue has emerged, we could guarantee the truth of materials in USB flash disk in the reason that we could hold it everywhere and every time. However, the resources on social cloud are easily shared by everyone. Whether the materials have been cut or modified we might be not sure. As a result, we face the truth of them with a grain of salt.

During the period of class activity two, our group discussed deeply the former questions as well. As a consequence, we hold the same opinion about the definition of social cloud and possible applications, because these problems are so simple for us to find the answer in article fast. However, in term of drawbacks, participants in our group had numerous arguments, such as private issues, and offer resolutions, like setting the authority. Meanwhile, we also provide some questions about confused parts. A simple example would be like this, Sabrina said “I wonder why people will trust a real world friend more than a pure online relationship? I know it may true but who know the basis.” I am confirmed that it is the networking security bringing about fear for users. Belonging with more and more Internet fraud emerging, users feel less the trust on friends who know online, especially facing the personal privacy. When come to a stranger who meet right now online and want to know your phone number, you might hesitate, which means you do not trust in him completely. But the person would be your friend in real life, and you might not feel that hesitation. Besides that, this question can be explanted in epistemic cognition theory, which is so simple and do not be discussed here.

  

Thus it can be seen that there exist some differences between individual and group epistemic cognition. When I learnt the case, first of all, I scanned total content, understand and explain. Step by step, to acquire true, justify beliefs and to avoid false beliefs, I could promote cognition into metacognition and epistemic level. However, during this period, there is no any new knowledge to be come into being. On the contrary, in the process of group discussion, participants could exchange their ideas and build knowledge together in a social environment. In term of given questions, participants tried to find the answers, form the new knowledge system which help to collaborative knowledge construction. All participants set forth their ideas and negotiate a fit between personal ideas and ideas of others. Through a sustained improvement of ideas and understanding, participants create higher level concepts. For the limited time, participants were also encouraged to contribute new information to central resources, to reference and build on authoritative sources.

Personally, the way to approach to the problem individually and in group, respectively, is various as well. On one hand, individual behavior could not be immune by curiosity or anger over being criticized. On the other hand, in group works, total participants can interact continuously to improve the quality, coherence, and utility of ideas. At the same time, a good social environment that facilitates knowledge building should provide support for theory construction and refinement and for view ideas in the context of related but different ideas. Further, team members produce ideas of value to others and share responsibility for the overall advancement of knowledge in the community. In addition, participants can embed authoritative sources (e.g. news, videos), along with other information sources, as data for their own knowledge building and idea-improving processes.

To conclude, we cannot deny that community with more possibilities than individuals to improve and emerge new knowledge construction. Moreover, it is an indispensable part in the process of cognition.