The "network connection" provision of AGPL doesn't require you to open source anything you use to talk to it, just modifications of it. To make it clearer:
Client <--> Server.
The server is AGPL licenced.
I can write a client to talk to the server and I don't have any licence restrictions. For example - if I write a web client to talk to an AGPL web server, I don't have to open source the client.
If I modify the AGPL server, and then put that modified server on the internet then the AGPL terms require me to make available the modified source code.
Client <--> Server. The server is AGPL licenced.
I can write a client to talk to the server and I don't have any licence restrictions. For example - if I write a web client to talk to an AGPL web server, I don't have to open source the client.
If I modify the AGPL server, and then put that modified server on the internet then the AGPL terms require me to make available the modified source code.
I am not a lawyer, but this is my understanding of this: https://choosealicense.com/licenses/agpl-3.0/