An automation testing company always recommends to execute the tests on a stable environment. Otherwise, the automation engineer have to put extra effort in reviewing the failure logs and scrips.
This becomes a time consuming task and increases the fixing cost as well.
If your network is slow, then you should implement appropriate Wait in your automated tests. This would allow the application to respond even when the network is slow or the page is taking time to load.
So, you can use Explicit and Implicit Waits in your Selenium tests to avoid any failure due to network issue.