meaning of plural of plural v.s. singular of plural

I am confused of the followings:

The locations of nodes are given.

The location of nodes is given.

Node locations are given.

Nodes' location is given.

Nodes' locations are given.

Is there any difference among these?

Let's go through them one at a time:

The locations of nodes are given.

This is the different positions of multiple nodes.

The location of nodes is given.

This is the singular place where there are multiple nodes.

Node locations are given.

This is the multiple locations of plural nodes (because I assume one node cannot be in multiple places, but if it could, that would be included here).

Nodes' location is given.

This is the singular location of multiple nodes, but isn't very idiomatic IMO.

Nodes' locations are given.

These are multiple locations of multiple nodes.

SovereignSun's explanations are great except this one:

The "Nodes' locations" is wrong. The locations don't belong to the nodes. It should be "Nodes locations".).

The locations do belong to the nodes. The locations are where they are located. The location(s) of the nodes, making them possessive. Both of these are correct: * Nodes' location is given. * Nodes' locations are given.

• When I say a set X={(x1,y1), (x2,y2), ..., (xn,yn)}, where (xi,yi) is the location of the node i, do I have to write like "a set X of the locations of all the nodes"? Commented Dec 20, 2017 at 2:55

You need to understand that you can have one or many nodes and one or many locations.

• The location of a node. (one node has one location)
• The location of nodes. (one location for many nodes that are probably in one place)
• The locations of nodes. (many nodes and many locations, but this may note imply that each location can have only one node)
• The locations of a node. (this is doubtly possible in any context except if we are tracing a node)

The "Nodes' locations" is wrong. The locations don't belong to the nodes. It should be "Nodes locations".).

• Then, both node locations and nodes location are correct?? Commented Nov 13, 2017 at 9:11
• No, "nodes location" is wrong. It can be used as in "the location of the nodes", or the "the nodes' (belonging to the plural) location". This means there are multiple nodes in the same place. Commented May 27, 2021 at 7:22