Adjacency Matrix

Table 6.2: Undirected, Binary Network in Adjacency Matrix Format.
001 002 003 004 005 006 007 008 009
001 . 1 1 1 0 0 0 0 0
002 1 . 1 1 0 0 0 0 0
003 1 1 . 1 1 0 0 0 0
004 1 1 1 . 1 0 1 1 0
005 0 0 1 1 . 1 0 0 0
006 0 0 0 0 1 . 0 0 0
007 0 0 0 1 0 0 . 1 0
008 0 0 0 1 0 0 1 . 0
009 0 0 0 0 0 0 0 0 .

Table 6.2, Table 6.3 each represent the same network data, which is a symmetrized version of the network in Figure 1.1. To confirm the symmetry of the undirected network represented in Table 6.2, notice that any time there is a 1 in the {i, j} cell (e.g. {001,004}), there is also a 1 in the {j, i} cell (e.g. {004, 001}). In addition to the presence/absence of relationships between each pair of nodes, the table also contains information about “self ties” along the main diagonal. Here, those are defined as missing, because in many social network applications ties from a person to themself is not meaningful. If the case under study meaningfully allows for self ties, these entries could be replaced with values.

Table 6.3: Non-Redundant Undirected Adjacency Matrix, Equivalent to Table B.2.
001 002 003 004 005 006 007 008 009
001 . 1 1 1 0 0 0 0 0
002 . 1 1 0 0 0 0 0
003 . 1 1 0 0 0 0
004 . 1 0 1 1 0
005 . 1 0 0 0
006 . 0 0 0
007 . 1 0
008 . 0
009 .

If the network data being represented are directed, the adjacency matrix will be asymmetrical. Table 6.4 represents the adjacency matrix that corresponds with the untransformed version of Figure 1.1. Note, for example that while node 1 sends a tie to person 2 ({1, 2} = 1), there is no corresponding tie from 2 to 1 ({2, 1} = 0). Similarly, note that while node 5 receives three ties, they send none, and node 6 sends one tie, while receiving none.

Table 6.4: Directed, Binary Network in Adjacency Matrix Format.
001 002 003 004 005 006 007 008 009
001 . 1 1 0 0 0 0 0 0
002 0 . 0 1 0 0 0 0 0
003 0 1 . 1 1 0 0 0 0
004 1 0 0 . 1 0 1 1 0
005 0 0 0 0 . 0 0 0 0
006 0 0 0 0 1 . 0 0 0
007 0 0 0 1 0 0 . 1 0
008 0 0 0 1 0 0 1 . 0
009 0 0 0 0 0 0 0 0 .

Adjacency matrices are convenient for many types of social network analysis in that matrix manipulations provide a simple basis for computing several common measures. However, as social networks get larger, matrices are a decreasingly efficient way to store data. When the field was dominated by relatively small networks, this wasn’t a problem, but as the field has increasingly incorporated larger networks for study, adjacency matrices have become a much less common way to store network data. This leads to an alternative data format, known as an edge list.