Suppose all people are incapacitated or disappear: how long will the global internet connectivity remain working?
Will re-routing via still functional paths occur automatically, or does it require human intervention (especially in light of undersea cable wrecks that require many hours to mitigate)?
You're asking a very simple question that has a remarkably complex answer
The internet has never seen a world war. In fact, the 40 years in which the internet has existed—and the mere 20 in which it has been widely used—have been anomalously peaceful in the long view of human history. What conflicts have arisen have been relatively local in scope: skirmishes, cross-border conflicts and internal civil unrest. And these conflicts have reliably resulted in local disruptions to the internet. If local conflicts cause local internet disruptions, what sorts of disruptions would global conflicts cause? (Source)
The "global Internet" is, as a phrase, a fantasy. The Internet is highly complex in countries and areas like Europe, the U.S., and Finland. And it's nothing more than smartphones on a cell network in many third-world countries. Even in a country like the U.S., "the Internet" is something quite a bit different between rural and urban areas, in the densely populated East vs. the Midwest. And let's not even mention the difference between military/government connectivity and citizen-grade connectivity. It's often a temptation to believe that globally everyone can do what I do when in reality the "global Internet" isn't nearly as global as one might think.
To understand the internet during a world war, let’s begin with a more popular topic: internet fragmentation. Broadly, this term refers to the sense that the internet is growing increasingly different in different countries. As our measurements show, internet fragmentation is a real phenomenon, though its form is considerably more complex than popular narratives of illiberal internets rising to threaten a supposedly global one.
What fragmentation we observe is not solely the maneuverings of an ascendant illiberal order, but also a global response to an internet whose management by policymakers no longer (and perhaps never did) align perfectly with the strategic interests of all the world’s nations. Popular narratives of “cyber conflict” focus on the “domain” reading, framing the internet as a medium through which conflict occurs; but it is equally an effect of conflict. The internet is, in other words, not only the domain but also the object of international competition and co-operation. (ibid.)
By the way, you should read the entirety of the article I'm quoting. It won't directly answer your question, but it will give you an idea of why your question is both important and remarkably difficult to answer with a single objective number.
In other words, the global Internet is already hugely at risk simply because it's not uniformly the same level of technology, the same level of maintenance, the same level of security or the same level of reliability everywhere in the world. But it's worse than that. Just s few days ago Yemen's Houthi rebels attacked a ship. As it sank, it destroyed three of fourteen international telecommunications cables. Did it bring down the global Internet? No. It did cause some grief, but the remaining eleven cables (and other routes) were used to reroute the data.
This suggests that the "global Internet" is reasonably resilient to some problems. Unfortunately, the ability to power the Internet isn't one of them. The internet in areas served by nuclear power would survive longer than areas served by hydroelectric power — and that would survive longer than areas served by coal or propane power. Some localized sections of the Internet (notably rural or more sparsely populated) are shifting to solar power.
What this means is that if your measure of success is "given every operating Internet node today and ignoring all outages currently in progress, how many would be operating after everybody disappeared?" Answer: immediately, all of them. After a day or two, some of them. After a week or more only a small portion of them — some of which will be international in nature. Some will collapse faster than others due to the weather in the area at the time (colder areas depend on more energy and thus will deplete faster). Others still will depend on whether or not major storms that depend on human maintenance to keep everything running have occurred.
You ask about automatic re-routing. Yes, that ability exists to a degree, but explaining all the ins-and-outs of how the Internet is managed top-to-bottom is well beyond the scope of this site.
From a worldbuilding perspective, what does this mean?
It means that between technologically advanced countries, you can depend on a day, maybe a week, and could believably rationalize months. From a storybuilding perspective, you simply need to ensure the right kind of conditions (hydroelectric, dams, moderate weather, etc.) exist to justify your choice of time.
The problem is that between two arbitrary points it's next to impossible to argue how long the Internet would remain running. E.G., between my house and the house of someone in Annecy, France. Between government/military centers it is easy it would remain active for a very long time. Between major cities, not as long. Between rural Texas and rural Egypt, maybe an hour. The Internet is too fractured for better estimates than that and the idea of a "global Internet" is actually quite meaningless.
Out of curiosity, why do you care?
Suppose all people are incapacitated or disappear.
Nobody is around to care if the Internet is running or not. Why, then, does it matter how long it stays running?
You ask about automatic re-routing. Yes, that ability exists to a degree, but explaining all the ins-and-outs of how the Internet is managed top-to-bottom is well beyond the scope of this site. Some of the routing needed human intervention, some was automatic. You have no idea the volume of information that had to be re-routed. The spillway gates on many dams are in many cases automated... but they take minutes to hours to cycle depending on the dam and how much water is involved. Time vs. volume.
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A day or so.
Without anyone balancing the grid, power plants will start to shut down after a day or so, and without power the internet will start to break down. Localized power plants e.g. dams will last longer, but the global internet isn't designed to survive with no power.
IT Professional here.
So - I want to expand on JBH's excellent answer - with some more food for thought:
The Biggest driver of Human interventions to fix problems is... Humans!
IIRC - something like 90% of all IT outages are caused by Human Error. Sure, there are hardware failures - but again, we've got to put that into perspective - what's one of the main driving forces for Hardware failures? It's usage! Consider a physical HDD - each time it has to do a Read or a Write - that is a small bit of wear and tear - eventually, it will fail. But if it's sat there idling, with very little load - only doing say a daily automated backup - then the load is much lower (someone may argue that regular use vs intermittent use is worse - but we can argue semantics later).
So let's consider what happens to the Internet if everyone just stops - For a period of time, there's still be a bunch of traffic - things like Torrents would still be seeding and downloading, other automated programs like Folding@home would still run - Bitcoin Miners would still be running etc. However human generated traffic (think of Facebook, Youtube, Netflix etc.) would all stop - that will drastically reduce the load on all systems involved in the Internet.
No Humans means no one making changes (major driver of incidents)
No Humans means no one reporting bugs or things that need to be fixed (major driver of incidents)
No Humans means no one generating fringe-scenarios that no one thought of (major driver of incidents)
No Humans means no one attempting to hack or break systems (major driver of Incidents)
You get the picture.
In short - so long as all the associated infrastructure still get power - then the 'Internet' as you know it would be up for a long time, add in JBH's points about it being a very decentralized setup (both for historical reasons and by subsequent design) - then it's really just a matter of how long the Servers last, how long the Cabling lasts etc.
I would say that so long as the Building remain weather-tight (and Datacentres are pretty robust), most of the Telephony/Networking Infrastructure is very resilient - you could probably say in 50-100 years, you could plug in a Generator to all the required components and getting working networking.
Ask instead how long the electricity will be on. I don't know of any fully automated coal fired plants. Some hydro and some nuclear may remain operating for days or months.
See S. M. Stirlings "Dies the Fire" first in the Emberverse series.
He had other things happening as well, so that internal combustion engines didn't work either, but that was icing on the cake for later. One electricity stops, data centers stop. backbone routers stop.
Many of these places have emergency generators that kick in, but few of them have days or weeks of fuel on hand. One major relay point goes out, then perhaps automated software routes around it. How well is this tested?
I'd estimate 3-5 days for a service failure to the point it's easier to count what still is connected than to count the failure points.
Locally people may be able to establish regional networks and keep them up.
But the connecting points where fibre is switched from one regional net to another are the weak point.
Starlink can work 5-7 years, GPS up to 12 years(base 10 yeras), some other satelites up to 50 years.
Switching/routing infrastructure - as long get power - max is around 50 years for some devices with passive cooling and around 25 years if need active cooling.
Undersea cables can hold up to 100 years if not destroyed by outside forces (ships, fisches, earthquakes etc.) Ground cables can work up to 50 Years if not destroyed by outside forces(animals, earthquakes etc.). Cables in buildings can work up to 50 Years if have conditions but if they are on weather conditions then 5 to 10 years max.
as long get power is, of course, the key.
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That is because their passive safety design concept. The "natural working" of the reactors is that they safely shut down. All the devices and humans are working on to keep them working. If any of them falls out, this time the humans, the result is automatic shutdown. Its timeline must be much more short a shift change in the reactor operators' room (which is mostly 8 hours) and depends probably a lot on the details. I suspect some minutes to some hours.
This is not something what they could handle automatically. If such an outage would happen, humans would be needed to say should be done. Thing are also here tend to follow a passive safety, thus large parts of the network will be likely shut down. They also have a pre-determined "ruleset", in which orders should the network (cities) turned off, and the system will try to follow probably this.
In this ruleset, there is a priority list of the consumers:
Actually, (4) is more important as (3), the cause of the reverse priority is electoral (power outages might alter your voting behavior on the next elections).
...and most of the things needed for ordinary home internet, belong to (3).
Thus, if you are not in some privileged area, like in a hospital, then you won't have power. Even if you are in some privileged area, your mobil network provider likely won't be. Beside that, most important servers, network centers have a redundant, safety power supply, but also these are depleted in some hours. Beside that, for example, the closest base tower of your mobile phone network company, that has probably nothing, or at most some hours.
