2012-12-04

Authors in a Markov matrix: Which author do people find most inspiring? (10)

Last time there is only Alice. That was simple. If Alice likes herself, it is 1. If Alice doesn't like herself, it is 0. This is maybe too simple and not so interested in, so I will ask Cheshire Cat to join us now.


Figure 9: Graphs representing relationships between Alice and Cheshire Cat.
Figure 9 (a) shows that Alice likes herself, but she doesn't like Cheshire Cat. Cheshire Cat likes neither Alice nor himself. The adjacency matrix of this situation is the following. Note that the number of relationships is the square of the number of people represented by the matrix. There are two people in this matrix so there are \(2^2 = 4\) relationships.
\begin{eqnarray*} \begin{array}{ccc} & \mbox{Alice} & \mbox{Cheshire}\\ \begin{array}{c} \\ \mbox{Alice}\\ \mbox{Cheshire}\\ \end{array} & \left[ \begin{array}{c} 1 \\ 0 \\ \end{array} \right. & \left. \begin{array}{c} 0\\ 0\\ \end{array} \right] \end{array} \end{eqnarray*} Cheshire doesn't like himself in the graph in Figure 9 (b). But, I felt he was a bit kind to Alice in the book. Cheshire Cat might like Alice. Here I assume Alice doesn't like Cheshire Cat, but Cheshire Cat likes Alice. Such a situation is represented by the following adjacency matrix: \begin{eqnarray*} \begin{array}{ccc} & \mbox{Alice} & \mbox{Cheshire}\\ \begin{array}{c} \\ \mbox{Alice}\\ \mbox{Cheshire}\\ \end{array} & \left[ \begin{array}{c} 1 \\ 1 \\ \end{array} \right. & \left. \begin{array}{c} 0\\ 0\\ \end{array} \right] \end{array} \end{eqnarray*} Each element of the adjacency matrix means following: \begin{eqnarray*} \begin{array}{ccc} & \mbox{Alice} & \mbox{Cheshire}\\ \begin{array}{c} \\ \mbox{Alice}\\ \mbox{Cheshire}\\ \end{array} & \left[ \begin{array}{c} \mbox{A $\rightarrow$ A} \\ \mbox{C $\rightarrow$ A} \\ \end{array} \right. & \left. \begin{array}{c} \mbox{A $\rightarrow$ C}\\ \mbox{C $\rightarrow$ C}\\ \end{array} \right] \end{array} \end{eqnarray*} Where ``A'' is Alice and ``C'' is Cheshire Cat. We read ``A \(\rightarrow\) A''as ``Alice likes Alice,'' ``A \(\rightarrow\) C'' as ``Alice likes Cheshire Cat.'' Now you can make an adjacency matrix that represents any of the graphs in Figure 9. The graph of Figure 9 (c) means that Alice likes herself and Cheshire Cat, and that Cheshire Cat doesn't like himself, but that he likes Alice. The adjacency matrix of this scenario is the following: \begin{eqnarray*} \begin{array}{ccc} & \mbox{Alice} & \mbox{Cheshire}\\ \begin{array}{c} \\ \mbox{Alice}\\ \mbox{Cheshire}\\ \end{array} & \left[ \begin{array}{c} 1 \\ 1 \\ \end{array} \right. & \left. \begin{array}{c} 1\\ 0\\ \end{array} \right] \end{array} \end{eqnarray*} Please note here that a mutual like-relationship becomes an undirected edge (Figure 9 (d)) and the adjacency matrix has a special form --- it is called a symmetry matrix. In this case, if you switch the positions of Alice and Cheshire Cat, the matrix stays the same. The elements of the matrix are symmetrical along the diagonal --- the line formed by the elements from the upper left to the lower right --- as if they are reflected in a mirror.

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