Appreciation of Sociomatrix

A matrix X of size g×g is defined to describe the connectivity inside the group, inside which an entry x_ij indicates the existence of a directional link from i to j. Such a matrix summarizes the whole status of the group and is sufficient to include all the information needed to describe a group. One example can be:

Here the power of X is 1, so each non-zero entry indicates a link of length 1. For example, x₁₂=x₁₅=1, the physical meaning is that node 1 has an out degree of 2, pointing to node 2 and node 5 once each.

By taking the n^th power (n < g) of X, say n=3, we obtain:

In this case, n=3 gives all the possible paths of length 3 among the existing nodes. The intensity of a certain entry gives the number of different paths.

Take =3 for example, this means from node 1 to node 3, there exist three paths of length 3, they each are:

Node 1 – Node 2 – Node 6 – Node 2	Node 1 – Node 2 – Node 3 – Node 2
Node 1 – Node 5 – Node 6 – Node 2

And this process exhausts all the possibilities.

This matrix representation method is concise and informative for either visual observation or statistics analysis. From my point of view, adopting matrices has advantages as follow:
1. Standardized computation.
By applying multiplication to the original matrix, we can easily obtain different level of information without introducing much computation. This way of information processing also allow computers to calculate answers in a much easier way, so that codes and programs can be as concise as possible.

2. Fixed structure of matrices
The format is still concise and matrix remains the same dimension and size as the original one. Such consistency can keep the systems function properly.

3. Easier reuse
This method is not the final goal of a topic. It's just a method used to simply other problems.
In the form of matrices, this calculation method can be easily implemented and encapsulated. In this way, black box reuse is possible, so that users don't necessarily need to know the exact calculation methodologies behind it, instead the numbers say for themselves, and should be enough to explain problems.

4. Can you guys think of more advantages? :)