How do adjacency bonuses for base facilities really work? [duplicate]

In the case you show, the second arrangement will grant you a greater bonus than the first. With links represented by lines, in the first arrangement:

X-X-X-X

There are three links. In the second arrangement:

X-X
| |
X-X

There are four links.

For every two adjacent facilities, you will get the adjacency bonus, and a building can be in multiple adjacency pairs.

Placing them in a square will net you one greater number of bonuses (whatever the bonuses are). Placing them in a line gives you three bonuses, as you have three adjacent links. Placing them in a square gives you four bonuses, as you have four adjacent links.