Paths in a two-mode network

Question

I'm reading through this blog, and having trouble with some really basic graph theory. Could someone please explain this to me as if I'm five?

There is a claim:

"In the first panel (A), there are five 4-paths, three of which are closed."

This is the two-mode graph to which the author is referring:

I have two questions:

(1) Why does the author claim there are only five 4-paths?

(2) Given the definition "a 4-path is closed if it is part of a loop of 6 ties with five nodes each", which paths are closed and why?

Here are the 4-paths I've found:

1-A-2-C (closed, because it's part of the cycle 1-A-2-C-3-B-1)

1-B-3-C (closed, because it's part of the cycle ...)

A-2-C-4 (not closed, because 4 is not in the cycle...)

A-2-C-3 (closed ...)

A-1-B-3 (closed...)

2-C-3-B (closed...)

2-A-1-B (closed...)

Answer 1

A $k$-path in this terminology is a path made up of $k$ "ties" (edges). In a two-mode network, we add a further requirement: the $k$-path must start and end at one of the "primary nodes" (in this case, $1$, $2$, $3$, and $4$ are primary).

In the example graph, a closed $4$-path is part of the only $6$-cycle in this network: the cycle $1 - A - 2 - C - 3 - B - 1$. Its endpoints are two nodes from the set $\{1,2,3\}$, so there are three possibilities:

1. $1 - B - 3 - C - 2$.
2. $1 - A - 2 - C - 3$.
3. $2 - A - 1 - B - 3$.

An open $4$-path isn't part of this $6$-cycle, so it uses the node $4$, so it must start or end at $4$. There are two more possibilities:

4. $1 - B - 3 - C - 4$.
5. $1 - A - 2 - C - 4$.