### c - transitive relation in an adjacency matrix - Stack Overflow

This MATLAB function returns the sparse adjacency matrix for graph G. To see this, suppose that σ is a permutation on the vertices. Let P be the corresponding permutation matrix. Then the adjacency matrices are. Let A be the n×n adjacency matrix of the graph G. Element aij = 1 if and Powers of an Adjacency Matrix . Suppose that there is a linear relationship between.

You may take the average of the values in the block if the data are binary, taking the average is the same thing as calculating the densitysum the values in the block, select the highest value or the lowest value, or select a measure of the amount of variation among the scores in the block -- either the sums of squares or the standard deviation.

The command outputs two new matrices. The "PreImage" data set contains the original scores, but permuted; the "Reduced image dataset" contains a new block matrix containing the block densities. We might select, for example, to combine columns 1, 2, and 5, and rows 1, 2, and 5 by taking the average of the values we could also select the maximum, minimum, or sum.

The command creates a new matrix that has collapsed the desired rows or columns using the summary operation you selected. The data menu also gives you some tools for this kind of work: You simply specify the new order with a list. If I wanted to group rows 1, 2, and 5 to be new rows 1, 2, and 3; and rows 3 and 4 to be new rows 4 and 5, I would enter 1 2 4 5 3.

If you data are valued i.

If you want a more complicated sort say "all the 3's first, then all the 1's, then all the 2's you can use an external UCINET data file to specify this as a vector i. Social network analysts use a number of other mathematical operations that can be performed on matrices for a variety of purposes matrix addition and subtraction, transposes, inverses, matrix multiplication, and some other more exotic stuff like determinants and eigenvalues.

Without trying to teach you matrix algebra, it is useful to know at least a little bit about some of these mathematical operations, and what they are used for in social network analysis. If you do know some matrix algebra, you will find that this tool lets you do almost anything to matrix data that you may desire.

But, you do need to know what you are doing. The help screen for this command shows how to identify the matrix or matrices that are to be manipulated, and the algorithms that can be applied. Transposing a matrix This simply means to exchange the rows and columns so that i becomes j, and vice versa. If we take the transpose of a directed adjacency matrix and examine its row vectors you should know all this jargon by now!

The degree of similarity between an adjacency matrix and the transpose of that matrix is one way of summarizing the degree of symmetry in the pattern of relations among actors. That is, the correlation between an adjacency matrix and the transpose of that matrix is a measure of the degree of reciprocity of ties think about that assertion a bit.

Reciprocity of ties can be a very important property of a social structure because it relates to both the balance and to the degree and form of hierarchy in a network. Taking the inverse of a matrix This is a mathematical operation that finds a matrix which, when multiplied by the original matrix, yields a new matrix with ones in the main diagonal and zeros elsewhere which is called an identity matrix. Without going any further into this, you can think of the inverse of a matrix as being sort of the "opposite of" the original matrix.

Matrix inverses are used mostly in calculating other things in social network analysis.

They are sometimes interesting to study in themselves, however. It is sort of like looking at black lettering on white paper versus white lettering on black paper: For Example 2, the square of the adjacency matrix is This means that there is a path from vertex 4 to vertex 2, because the entry on fourth row and second column is 1.

Similarly there is a path from 3 to 1, as one can easily see from Example 1.

## Journal of Applied Mathematics

Consider a directed graph and a positive integer k. Then the number of directed walks from node i to node j of length k is the entry on row i and column j of the matrix Ak, where A is the adjacency matrix. In general, a matrix is called primitive if there is a positive integer k such that Ak is a positive matrix. A graph is called connected if for any two different nodes i and j there is a directed path either from i to j or from j to i.

On the other hand, a graph is called strongly connected if starting at any node i we can reach any other different node j by walking on its edges. We add the identity matrix I in order to deal with edges from a vertex to itself.

In other words, if there is at least one path from node i to node j of length at most k, then we can travel from node i to j.

Thus if matrix B has a positive entry on row i and column j then it is possible to reach node j starting from i. If this happens for all nodes, then the graph is strongly connected. One can easily see that the graph in Example 1 is connected, but not strongly connected because there is no edge from vertex 1 to vertex 3. For the matrix in Example 2, we notice that A4 is a matrix having only zeros, and so for all k greater than 4, Ak will be a matrix filled with zeros.

Since the matrix B is not positive, the graph in Example 1 is not strongly connected as we already saw. Construct the adjacency matrix of the graph below. Is this graph connected or strongly connected? How many paths of length 3 from node 3 to node 2 are there?

In the examples above we noticed that for every vertex i there is a number of edges that enter that vertex i is a head and a number of edges that exit that vertex i is a tail. Thus we define the indegree of i as the number of edges for which i is a head.

## Powers of the adjacency matrix and the walk matrix

Similarly, the outdegree of i as the number of edges for which i is a tail. For example, for the graph in the Problem 1, the indegree of node 2 is 2 and the outdegree of node 1 is 1.

The transition matrix A associated to a directed graph is defined as follows. If there is an edge from i to j and the outdegree of vertex i is di, then on column i and row j we put.

Otherwise we mark column i, row j with zero. Notice that we first look at the column, then at the row. We usually write on the edge going from vertex i to an adjacent vertex j, thus obtaining a weighted graph.

This will become clear through the following example. Consider the graph from Example 1 with weights on its edges as described above. Then the transition matrix associated to it is: Notice that the sum of the entries on the first column is 1. The same holds for the third and fourth column.