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$\frac{23550\text{ stations}}{(\frac{5000ly}{1000ly})^23}$ $=\frac{23550\text{ stations}}{25}$ $=942\text{ stations}$

$\frac{23550\text{ stations}}{(\frac{5000ly}{1000ly})^2}$ $=\frac{23550\text{ stations}}{25}$ $=942\text{ stations}$

What you're trying to do doesn't make sense unless you send a few thousand (or more, depending on sensitivity) mapping drones into the galaxy to keep up-to-date mapping data. The drones would be connected in an FTL sneakernet so you can just ping the nearest drones for data from time to time and always be within a few hours of current data.

Mapping within explored solar systems would be done routinely so there will be no issues there.

Without a lot of drones, or very good resolution on them, there will always be uncertainty about the exact positions of small bodies in the unexplored / infrequently visited solar systems. As such, you'll need to send a scout drone ahead to be absolutely certain there's nothing where you're going in you're in unexplored areas.

I didn't do any math, but the entire network could potentially be setup within days if your mapping drones can jump hundreds of light years at a time.

How do we update that in real-time?

What you're trying to do doesn't make sense unless you send a few thousand (or more, depending on sensitivity) mapping drones into the galaxy to keep up-to-date mapping data. The drones would be connected in an FTL sneakernet so you can just ping the nearest drones for data from time to time and have pretty accurate data.

Note that if you're in the thousands of probes range, the data will necessarily be predictions from light thousands of years old. The simulations should be very accurate with advanced math, but things like "the Death Star exploded your planet" won't show up.

Additionally, without extremely good resolution on the drones (not likely due to diffraction limits), there will always be uncertainty about the exact positions of small bodies in the unexplored / infrequently visited solar systems. As such, you'll need to send a scout drone ahead to be absolutely certain there's nothing where you're going in you're in unexplored areas.

However, mapping within explored solar systems would be done routinely so there will be no issues there.

If you want one-year "real-time" accuracy, you'll need hundreds of millions to hundreds of billions of probes. Same thing if you want to see all the planets in the system without probes jumping ahead.

I didn't do any math, but the entire network could potentially be setup within days if your mapping drones can jump hundreds of light years at a time. However, it will take some time to calculate precise velocity information to make your models accurate. A few years should be fine, but an advanced civilization could probably do it faster. From there, updates every few decades should be sufficient.

How do we update that in "real-time"?

What is "real-time"?

As kingledion notes in a comment, "real-time" isn't really real-time. Each node still has a pretty big lag between the events happening and the light hitting the node. My assumption was that we just needed to see the stars closely enough that we could keep the calculations accurate.

If my assumption is correct (the question seems to indicate that's all we need), then the "real-time" aspect isn't really important. You just need to update once every few decades or centuries to keep your simulations reasonably accurate. This is a good thing, because it means you can get by with far fewer drones and energy requirements.

However, if you want more accurate data, you'd need to either:

A) Have a lot of probes. To keep everything within 1 year accuracy, you'd need to bring the probe distance to 1 light year. That's around $4\cdot10^{13}$, or $40\text{ trillion}$ stations. A lot more than you likely want.

To be fair, there are only 200-400 billion stars in the galaxy, and there's not a really good reason to use more than one station per star except highly traveled FTL routes. So 200-400 billion is a more "reasonable" cap.

B) Have probes that do a lot of hopping around. Depending on your FTL energy requirements, this might be quite difficult. But you can just have (relatively) a few probes that pop from star to star. From this site about cloaking (and how you can't do it in space), they calculate about 4 hours to scan the entire sky for things the size of spaceships.

Our futuristic space probes could likely do it in 30 minutes or less, though they'll need to do three or four scans from different positions, so let's call it 2 hours (probably a lot less). You say these guys can jump hundreds of light years at a time, so a probe should easily be able to hop the five to ten light years between nearby stars in a single jump.

Add the jump time to the scan time (the probes can absorb most of this by running calculations and charging the jump drives while scanning). Let's say jump time is pretty minimal, so the total is 2.5 hours per system.

Now, let's say you want data less than 1 year old. Each probe can jump through $\frac{8760 \frac{h}{yr}}{2.5 \frac{h}{\text{system}}}=3504\text{ systems}$ per year. This means you need around a hundred million probes to cover the galaxy.

Both of these options also have the advantage that you can see all the planets and so forth inside every system. At high cost, of course.

The blue dot is a star at the edge of the galaxy. It's maximal distance to a station given a single layer of cubes is given by the green vectors, which are equivalent to summing the maroon, red, and orange vectors. The distance is $\sqrt{m^2+r^2+o^2}$ The maroon and red vectors are both half the distance between stations, or $m=r=\frac{a}{\sqrt{3}}$, and we need the total distance to be less than $a$ (where $a$ is the maximum distance we need). Plug that all in to get:

In the other two cases, where $433ly<866ly$ and $a>866ly$, station count is proportional to $a^2$. Same thing, but square the difference.

You'd need a pretty big dataset to hold all this information, but a society this advanced shouldn't have much trouble with that. Each node has a complete copy of the data at all times. Far away nodes will be slightly out of date, but the FTL nature of the network means they'll be accurate within about $JumpTime\cdot NumberOfNodes$. This is a linear node count; worst case scenario is going to be about $JumpTime\cdot\frac{2\cdot GalaxyDiameter}{NodeDistance}$.

For a 2 minute jump time (time to jump, transfer data, jump back, transfer new data) and 2900 ly node distance (corresponding to a 5000 ly view distance), that's $2min\cdot\frac{200000ly}{2900ly}$ $\approx139min$, or about 2 hours. The number is linear, so if you double the jump time, you'll double the lag time. Also, it depends on how many drones there are per station. The 2 hour number assumes six active drones per station; if you only have one drone popping to 6 stations, that's six times the delay, or about 12 hours.

The blue dot is a star at the edge of the galaxy. It's maximal distance to a station given a single layer of cubes is given by the green vectors, which are equivalent to summing the maroon, red, and orange vectors. The distance is $\sqrt{m^2+r^2+o^2}$. The maroon and red vectors are both half the distance between stations, or $m=r=\frac{a}{\sqrt{3}}$, and we need the total distance to be less than $a$ (where $a$ is the maximum distance we need). Plug that all in to get:

In the other two cases, where $433ly<a<866ly$ and $a>866ly$, station count is proportional to $a^2$. Same thing, but square the difference.

You'd need a pretty big dataset to hold all this information, but a society this advanced shouldn't have much trouble with that. Each node has a complete copy of the data at all times. Far away nodes will be slightly out of date, but the FTL nature of the network means they'll be accurate within about $JumpTime\cdot NumberOfNodes$. This is a linear node count; worst case scenario is going to be about $JumpTime\cdot\frac{2\cdot GalaxyDiameter}{NodeDistance}$ (data is coming diagonally across the grid, so it has to jump south, east, south, east, etc., for example, meaning the delay is about twice as long as going straight along the nodes).

For a 2 minute jump time (time to jump, transfer data, jump back, transfer new data) and 5774 ly node distance (corresponding to a 5000 ly view distance), that's $2min\cdot\frac{200000ly}{5774ly}$ $\approx69min$, or about 1 hour. The number is linear, so if you double the jump time, you'll double the lag time. 

Also, it depends on how many drones there are per station. The 2 hour number assumes six active drones per station; if you only have one drone popping to 6 stations, that's six times the delay, or about 12 hours. For the 2D grid of cubes, you only need 5 drones per station, and for the 2D square grid, you only need 4 drones per station.

In a highly 3D grid (in a spherical galaxy or similar), worst case is $\frac{3}{2}$ worse, because it's doing south, east, down, south, east, down, etc., for example. Note that a disc-shaped galaxy (like ours) will still use the above numbers, even if you have a lot of nodes. In this case, the number of down jumps will be small compared to the number of south and east jumps, so you can ignore them.