IRN technology lvl affects how many research labs are taken in account. Here is a sample table:

IRN Technology Level / Research Labs Used
0 / 0
1 / 1
2 / 2
3 / 3
4 / 4
etc

This means that at IRN 1, you do not get any bonus in time.
The IRN network, always chooses your highest research labs to use.
The IRN network, only takes in account the labs that are eligible to research a technology.
The IRN network adds up eligible research labs.

Here is an example on these:
You got IRN lvl 2
You got 3 research labs. One at lvl 6, the other one at lvl 2 and the last at level 3.
The IPN will choose lvl 6 and lvl 3.
If you want to research Computer Technology, it will be like you have a lvl 9 laboratory (6 + 3)
However, if you want to research Weapons Technology, it will be like you have a lvl 6 laboratory, as the weapons technology have a minimum requirement of Level 4 Laboratory.

Another Example:

Lets say you got IPN 4 and 6 research labs as 12, 10, 6, 1, 5, 3
IRN will select in this order: 12, 10, 6, 5 (Total Level = 33)
But if you want to research Hyperspace Engine (Requirement is at least Level 7 laboratory) the total level will be 12 + 10 = 22, as lvl 6 and lvl 5 laboratories will not be taken in mind.

One more example:

Let say you have Espionage Technology lvl 2 and you want to research lvl 3.

With a lvl 13 research lab, you will need 1m 01s
With a lvl 13 research lab + IRN level 2 + a lvl 10 research lab, you will need 36s
With a lvl 13 research lab + IRN level 2 + a lvl 8 research lab, you will need 39s
With a lvl 13 research lab + IRN level 2 + a lvl 3 research lab, you will need 50s
With a lvl 13 research lab + IRN level 2 + a lvl 2 research lab, you will need 1m 01s (Same as without IRN because lvl2 research lab does not count for this tech as it has a requirement of at least lvl 3 lab)

This is very helpful. I have 1 additional question.
The research menu displays, "Construction Time". Does the Construction time reflect the IRN level?

Example:
Construction Time for Expedition Level 4 on my display from a planet with a level 12 research lab is 52m 21s.
Is this the Level 12 lab's construction time or the construction time for the IRN to which the Level 12 belongs?

Thanks in advance!

Assuming your research network have a level higher than 1 you could try looking at the same research from the planet with your secindmost developed lab. If the times shown are the same I would expect the research network is taken into consideration.

Is it supposed to be that if 1 lab is either reserching or building all labs elsewhere in the empire are closed to new research or building???

Seems v odd to me.

Ands if it's suposed to be, does that change after yu get IRN (I'm due to soon, he, he!)

?:-)

Yes, that is intentional. And no, that doesn't change. You are only allowed to research one tech at a time throughout your entire ZE career.

And either build a RL somewhere or research?

That seems a weird overlap.

?:-)