Oh, the book says: "Any decrease in cost travels fast, but increase travels slow"
That's actually nonsense. Both increase and decrease travel at exactly the same speed.
If an implementation does not do flash-updates, then there will be 30 seconds between updates. New routes will be advertised within 30 seconds (per hop) through the network. If routes need to be removed, then it can take 3 * 30 seconds per hop for that information to get everywhere. However, if a router uses "poison metric, it will advertise the deleted route with metric 15 for a while. And in that case, bad news travels exactly as fast as good news.
What the writer of the book probably means is: if a new route is added to the network, it will be advertised everywhere (almost) immediatelly. Especially when the routers do flash-updates. But if a route is removed from the network, there will be counting-to-infinity. And counting-to-infinity goes slow. It will take a while before an old route is completely removed from the network. (Several times 30 seconds. Worst case = 14 * 3 * 30 seconds = 21 minutes). It doesn't matter much that it takes that long, because the destination isn't reachable anyway. The only problem is that if someone keeps sending packets to that destination, traffic will loop for up to 21 minutes. Which can cause a burden on the network.
Suppose a network as many routers. And they are connected with many links. Suppose router X advertises prefix P. Suppose it advertises P to two other routers, Y and Z, both with cost 1. Y and Z advertise P to their neighbors, with cost 2.
Now suppose router X crashes. Y and Z don't receive their periodic updates. After 3x30=90 seconds, they declare that P is not reachable anymore. Suppose they don't make that decision at exactly the same time. Suppose router Y is a bit quicker. It will note P in its routing table as down. It will then advertise P with metric 15 to all its neighbors. Now if Z was a bit slow, it might advertise P one more time, with cost 2. If Y hears that advertisement, it will think: "my old route to P via X, metric 1 is not valid anymore. But here is a new route to P, via Z, with metric 2. I'll use that one". A few seconds later Z decides that its route to P via X is not valid anymore. But in the mean time, other routers have picked up the old route that was supposed to be deleted. This "old information" keeps being advertised around. And every 90 seconds, the router that started advertising that old information will stop doing that. But another router repeats the same mistake. The good thing is: every time that happens, the metric increases (depending on number of hops over which the old/outdate info traveled). After a while, the metric has been increased to 15. And 15 is unreachable. At that point, routers will start removing the route to prefix P from their routing table.
This is counting to infinity. (Infinity is 15, in RIP). It goes slow. It takes several times, times 90 seconds. Google it, I'm sure there are better explanations than mine.
As I tried to explain in my first post: all this behaviour depends on how the protocol has been implemented. A few smart trick can be done, to make the counting-to-infinity less bad.
Example tricks:
flash updates
poison reverse
split horizon
Google what those are, and you might get a better understanding for distance vector protocols, like RIP.
For completeness sake: the expression in routing is usually: "bad news travels fast, good news travels slow". What this means is: if new paths become available, routers become available, links are back working, etc, you can wait a little (a few seconds or more) before you announce the new paths. Because traffic was flowing over another path anyway. If it changes to the better path 1 second later, or 10 seconds later, doesn't matter much.
But if a router crashes, or a link goes down, then traffic will be lost. And that's bad. So when that happens, routers will *immediately* try to notify all the other routers of the problem, and try to find an alternate path. Just so that as little traffic as possible is lost.
The writer of your book mixes things up. It might not be a very good book, sorry.