Why is it that when we lose power, we also lose Verizon 4G internet?

I live in a suburb north of Boston, and we lose power fairly frequently due to storms. Last year we lost power for almost 5 days. We lost it last week for 36 hours, and we’re expecting another storm in a few hours with possible power outages.

Every time we lose power, our Verizon internet service on our phones no longer works. How is this possible? This happens on every phone in my house and others in my neighborhood. We also have spoken with others in towns near us with the same experience. Verizon is of no help.

Note: We actually have the same level of 4G signal – it’s just that the internet is not working. It will occasionally (rarely) load a website or refresh email, but then it won’t for the next hour or more.

Does anyone know how 4G signals and internet works and how it might be affected by power outages?