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1.? If by network you mean WiFi, there's an option for that inside the advanced settings for wifi.

2.? If it's a gprs network, it should be a problem with the lock screen software. We suggest you to remove the lock screen software that comes with your system and install other lock screen software after rooting.

For example, the Dim Sum lock screen works very well, and I've been using it for a long time without any problems.