As we learned in How to make a pom-pom? they are fairly easy to make. The problem is that they are just as easy to take apart.
So you could get your pom pom trimmed to make the perfect sphere and it takes little to no effort to pull individual strings from the ball. Do this enough and the integrity of the ball starts to fail and exacerbates the issue and they can start to fall out on there own.
I noticed that all of the commercial products that use pom poms e.g. clothing, toys, etc. do nothing to prevent this (Wife hates it when I test them). That tells me it is not a huge issue that industry, as a whole, does not attempt to mitigate it. It's not that can't. I'm sure its a matter of cost justification and its not worth it.
However, when I make something I would like it to last as long as possible.
What can I do to reduce the likelihood of a pom pom falling apart? Or at least slow it down. Glue comes to mind.....