Two Ways To Understand Reachable Matrix

Last week we have learned how to use adjacency matrix to represent a social network. One question is why the power of matrix can indicate the reachability of social networks. Actually, the matrix mentioned in the class is named reachable matrix.

Let's see how it works. Here, I assume all of us have known the way to do matrix multiplication.

1. The First Way
One simple way to calculate reachable matrix is to regard it as logical operations,(And, Or, Not). We regard 0 and 1 represents false and true respectively.(you can also understand in this way: 0 represents two nodes are not connected; 1 represents two nodes are connected)

Another thing that should be declared is that in logic algebra, numeric operations of multiplication x*y, addition x + y are equal to the respective logical operations of conjunction x∧y, disjunction x∨y. The relevant logic operation rules are shown as below:

Let's review the example, here is a matrix:

Graph below is generated by matrix above: 

The two steps matrix within logic calculation:(which means the reachability going through two steps)

We can see that in X, n6 and n3 are not connected but in X^2 they are. So let's analyze the elements m'63(i=6, j=3).(using m for the elements in X matrix; using m' for the elements in X^2 matrix)

• m'63
  
  m'63=(m61∧m13)∨(m62∧m23)∨(m63∧m33)∨(m64∧m43)∨(m65∧m53)∨(m66∧m63)
  We have know that n6 is not connected with n3 within one step, So if n6 is connected with n3 within two steps, there must be at least one node nx which connects n3 and is connected by n6.
  
  Now look at the calculation process. It means if n6 n1 are connected and at the same time n1 n3 are connected; OR n6 n2 are connected and at the same time n2 n3 are connected;OR....., then n6 n3 are connected.
  
  In this process, only (m62∧m23) is equal to 1, others are all zero. So the path should be 6->2->3.

The thinking way is the same with the situation of more steps.

2. The Second Way
Another way to think this is to multiple matrix in normal way. With the foundation above, we can deepen our understanding.

The two steps matrix within normal way calculation:

With the knowledge in FIRST PART, we have understood the meaning of non-zero elements. Now let's analyze the elements m'16 and m'22

• m'16
  m'16=m11*m16+m12*m26+m13*m36+m14*m46+m15*m56+m16*m66
  We can see that only :
  
  m12*m26 =1
  m15*m56 =1
  Others are all equal to zero. So 
  m'16=m12*m26+m15*m56=2
  And you must find that 2 represents that there are two paths which make n1 n6 connected. Obviously, the paths can be 1->2->6 and 1->5->6
  
  
• m'22
  It is easy to find that two paths are 2->3->2 and 2->6->2

The thinking way is the same with the situation of more steps.

If you have any question, pls feel free to discuss with me.

Hope you enjoy this!

Difference Between Individual Work And Group Work

We've learned about communication and socail behavior on the Internet in last 2 lectures. And we had  activities which help us understand some theory deeply. After that, I thought more about what we learned in the class, and I'd like to share my view with others.

In this post, I will answer the questions mentioned in the class.

Answers of Activities

The answer of these two activities are shown as below (the answers are the same)

Q1. What is the definition of Social Cloud?

A1. A Social Cloud is a resource and service sharing framework utilizing relationships established between members of a social network.

Q2. What are the possible applications of a Social Cloud?

A2. A Social Computation Cloud

   A Social Storage Cloud

   A Social Collaborative Cloud

   A Social Cloud for Public Science

   An Enterprise Social Cloud

Difference In Epistemic Aims

Firstly, I want to mention about the JTB theory.

A subject S knows that a proposition P is true if and only if:
 1.P is true
 2.S believes that P is true, and
 3.S is justified in believing that P is true

In individual work, the epistemic aim is understanding the reading material,  finding out the answer of question and forming belief. In this process, our behavior reflect the second point of JTB theory: believe our answer is true.

In group work, I do think the epistemic aim changes. Since we have already understood the material and found the answer, understanding and finding out can't be regarded as our aim. In most case, we consider the answer which most people give as right. In this process, what we do is justifing whether our answer is true or not. So it reflects the third point of JTB theory: we are justified in believing our answer is true.

Difference In Epistemic Cognition

In individual work, people have no chance to communicate with others about their idea, and they might not be very sure about their answer, their idea, their view, etc. at the stage of certainty of knowing and criteria of knowing. And there are less things inspires people to generate new ideas.

However, in group work, by sharing their own ideas with others, people can find an answer more quick; by learning from others and discussing the issue with others, people can produce new knowledge more easily.

Our group changed our focus point after knowing our answers are the same. We discussed things we wanted to know and problems from the article etc..

 

Difference In Processes

I think the activities might not reflect the social behavior completely.The first reason is that in the second activity we have already find our own answer. The second reason is that the questions are very simple.

Firstly, you may notice a phenomenon that there are always few people doing nothing in a group work.(what we say "打酱油" in Chinese) So, If the article in Activity Two is different from Activity One, people mentioned above may show this character.

Secondly, as many students said, the questions are quite easy and almost everyone could affirm the answer. Students always check answers of advanced math homework during their university life. Why? Because the questions are very complicated and confirmation degree of such questions is lower than simple questions. In addition, with many controversial issue, different people hold different stand. So when questions given are complicated or controversial, the difference between  individual work and group work will be more obvious.

Reference:

[1]http://en.wikipedia.org/wiki/Justified_true_belief

[2]http://www.dic123.com/A/7/7F/7FD_25815.html

[3]http://www.docin.com/p-438217766.html