2.4 Adjacency matrix

The adjacency matrix of a network that has \(N\) nodes has \(N\) rows and \(N\) columns. If there is a link from node \(i\) to node \(j\), then \(A_ij = 1\). If node \(i\) and node \(j\) are not connected, then \(A_ij = 0\).

For an undirected network, the link \((i, j)\) has two representations: \(A_ij\) = \(A_ji\). That’s why the adjacency matrix of an undirected network is always symmetric.

Lets have a look at the adjacency matrix of g1 and g2.

For g1:

par(mar = c(1, 1, 1, 1))
plot(g1)

as_adjacency_matrix(g1)

## 4 x 4 sparse Matrix of class "dgCMatrix"
##             
## [1,] . 1 1 .
## [2,] 1 . 1 .
## [3,] 1 1 . 1
## [4,] . . 1 .

For g2:

par(mar = c(1, 1, 1, 1))
plot(g2)

as_adjacency_matrix(g2)

## 4 x 4 sparse Matrix of class "dgCMatrix"
##             
## [1,] . 1 1 .
## [2,] . . 1 .
## [3,] . . . 1
## [4,] . . . .

Each dot in the matrix means \(0\). The adjacency matrices here did not label the column name. To better understand it, you can imagine that the columns are also labeled as [1, ], [2, ], [3, ] and [4, ] as the rows are.