Explain what each entry in this matrix means and construct a hexagonal diagram r

Question

Explain what each entry in this matrix means and construct a hexagonal diagram representing this communication network. Explain the implications for the network of the fact that this matrix is not symmetric and that its diagonal entries arc all 0. What could be done to the communication system in order to have a symmetric adjacency matrix? Compute A^2 (you can use a calculator) and explain: the meaning of its entries in relation to the communication system how such meaning is justified by the method we use to multiply matrices. Compute A^2 and. b y using what you learned from question 3, describe how the information it contains tells you that a minimum of three steps are needed to communicate from a to J.

Explanation / Answer

The Matrix A represents the adjacency matrix of the following directed graph with vertices 1,2,3,4,5 and 6.

There are no loops (connection from a node to itself). Non-symmetric means there might be an edge from i to j but may not be from j to i. (For example there is an edge from 1 to 2 but not the other way).

In view of the above remarks, the matrix could be made symmetric by providing two way communication between a pair of nodes which share an edge .

A2 =

The (i,j)th entry counts the number of paths from i to j of length at most 2.

The matrix multiplication is justified as it picks up 1's wherever there are edges.

Same interpretation as before, except that paths upto lengths 3 are accounted for.

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