Markov matrix
The train station example shows that applying an adjacency matrix to a stations vector can tell us which station can we reach from any starting station. Let's think one step further, we can look into the quantity side of the movement in stead of possible side of the movement.There is a city called Potsdam 40 minutes away from Berlin by a train (S-Bahn). These two cities are close and a lot f movement of people between them. Some move Potsdam to Berlin and others move other way around. The adjacency matrix of the two cities is the following.
\begin{eqnarray*} \begin{array}{|c|c|} \hline \mbox{Station} & \mbox{1 step} \\ \hline \mbox{Weinmeisterstr} & 1 \\ \mbox{Alexanderplatz} & 0 \\ \mbox{Hackescher Markt} & 0 \\ \mbox{Jannowitzbruecke} & 0 \\ \hline \end{array} \end{eqnarray*}
Let me show you what is the meaning of columns and rows of the matrix.
\begin{eqnarray*} \begin{array}{ccc} & \mbox{Berlin} & \mbox{Potsdam} \\ \begin{array}{c} \\ \mbox{Berlin} \\ \mbox{Potsdam} \\ \end{array} & \left[ \begin{array}{c} 1 \\ 1 \\ \end{array} \right. & \left. \begin{array}{c} 1\\ 1\\ \end{array} \right] \end{array} \end{eqnarray*}
This matrix only represents the connectivity. This means we can stay both cities and we can also move one to another. Though, this is not so interesting. An interest is a feeling, this can not be explained by mathematics. Therefore, some might think this is a strange as a mathematics discussion. Mathematician has some passion about a problem: interesting or not. In this case, there are two cities connected, and what the adjacency matrix can tell is that you can be any city. It's trivial, isn't it? At least I am not interested in this case. But it became more interesting when we introduce how many people is moving in the matrix.
Assume a following vector that represents population of the cities. Say this vector a population vector.
\begin{eqnarray*} \left[ \begin{array}{c} p_{b} \\ p_{p} \\ \end{array} \right] &=& \left[ \begin{array}{l} \mbox{Population of Berlin} \\ \mbox{Population of Potsdam} \\ \end{array} \right] \end{eqnarray*}
Let's think about the change of population. Assume the year 2000 is the starting point, time \(t=0\). We write this as:
\begin{eqnarray*} \left[ \begin{array}{c} p_{b} \\ p_{p} \\ \end{array} \right]_{t=0} \end{eqnarray*}
The population change is observed every year. Assume how this year's change happens based on the last year and let it a matrix \(M\).
\begin{eqnarray*} \left[ \begin{array}{c} p_{b} \\ p_{p} \\ \end{array} \right]_{t=k+1} &=& M \left[ \begin{array}{c} p_{b} \\ p_{p} \\ \end{array} \right]_{t=k} \end{eqnarray*}
To make the example simple, the change of population only defined by people's moving ratio. This means no baby is born, no one die. Also, the people only move between these two cities and the ratio of moving people are fixed.
This assumption is a big limitation. It's easy to imagine that the people's moving ratio may change every year, babies will be born, and some will die. Some people surely move to other cities. None of the assumptions is realistic. We can make the model more complex to be more realistic, i.e., number of cities can be arranged by other matrix size, birth and death ratio also can be considered. My motivation here is not accurately predicting the population. My motivation here is to show how the matrix analysis works. Therefore, I would like to stick the simplest model. This is a common problem when we try to explain a mathematician idea. A simpler example is easy to understand, but it is usually less realistic. I can imagine some people think the class mathematics is not useful because the assumption is usually not realistic and can not be applied to the real problems. But this is a learning problem. If the example is too complex, we might not be able to understand it, if the example is too simple, we might not be able to apply it to the real problem. I show a simple but unrealistic example for understanding. Then, I would like to apply the method to more complex problem later.
Today's story became a bit longer, yet just the introduction. Next article, I would like to talk about this population matrix.
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