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 Xij=p (Xij is the element in the ith
, row jth column), that implies the number of methods from ni to
nj walking length n is p. Therefore, in social networking, the
entries of the matrix Xp give the total number of walks of length p
from node ni to nj and the value of Xij in Xp
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.
CD(n*)
means the largest degree of actor, CD(ni) implies the
number of degree of actor ni. 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 CD(n*) become largest and CD(ni)
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 CD =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.
Such an in-depth study!... It's refreshing to read your concise yet clear blog Dorothy!
ReplyDeleteThe objective of this blog is so clear, and your description is worth reading. Good job. =P
I think your article focus more on mathematical because of your background. I think you may understand more about the power of matrix and the importance of centralization. Maybe you can show us more about centralization but not only degree centralization. By the way, PageRank is the most powerful algorithm nowadays but the shortcoming is it need too many calculate.
ReplyDeleteOh you write another good article! You pick up three elements of the SNA to share with us which are Powers of a Matrix, Group Degree Centralization as well as PageRank.
ReplyDeleteAfter reading these content, I know more about the SNA properties. Take the Powers of a Matrix for example, after calculating the matrix, we have an idea that the power means the two nodes' walking length. Yet your explanation is quiet good for me to understand more briefly.
The concept of PageRank also interests me and let me wanna know more about SNA. We can share more SNA methods later and exchange our ideas.
Your article has a complete description to degree centralization. These parts help me to have a deeper and clear understanding of SNA. The example about PageRank shown in the final several paragraphs would be pretty suitable to the article. And I can see your great mathematical background with such a brief and strict description. I have to say good article!
ReplyDeleteWow! So many mathematic in this blog.You must be very good at linear algebra and probability statistics. After reading your blog I have a clear understanding of the social matrix and the SNA method.
ReplyDeleteHello, your post can help me review the SNA. A question is that I don't understand the first equation "Pr(A) = Pr(B)". According to the socialmatrix, I think it should be "Pr(A) = Pr(C)". I am a little confusing.
ReplyDeleteAnd the question that "X[Σ] = X + X^2 + X^3 + … + X^(g-1) shows the reach ability of pairs of nodes" is ... First, the maximum step in a group is (g-1). Second, if you add them together that means Ni to Nj in every possible steps. Third, if the Ni to Nj in X[Σ] is non-zero, that means they are connected with each other. Fourth, the number X[i,j] in the X[Σ] shows the total kinds of path between Ni and Nj.
Initially, I suppose I have to mention that Pr(A) = Pr(B)is completely correct and you might make a mistake on p = X’p.Therefore, the matrix picture showed firstly is X, but we have to use X' to calculate instead of X. In this way, the answer would be unambiguous.
DeleteBesides that, thanks of your explanations in detail. In my opinion, you have done a lot research after class. Now, the equation is clear for me, and X[Σ] is just a sum of a series of powers of social matrix,right? "The following matrix is ……"in your article which implies the X[Σ], but it is not the following matrix of X.
I have misconstrued the meaning of "following matrix", and thank you for your hard work.
haha... the topic of this blog is really match with your background. Thank you for using the Mathematical Questions in social network analysis to explain the definiton of all the materials. It's realy a novel aspect to SNA
ReplyDeleteWow, that's amazing you girl with strong math background write such a blog!
ReplyDeleteAfter reading this post, I get a deep understanding of the degree centrality and page rank. It seems very interesting using math to analyze SNS questions. Looking forward to more sharing about math with SNA.
Ooops, Directed graph, I tried to do a SNA using the directed graph, but finally I found it was too complex, so I gave up and turned to a undirected graph to do SNA. There are so many elements we should concerned in the directed graph SNA and each centrality will spend much more time to calculate than the undirected one.
ReplyDelete