So can you tell me if the airport is your only wireless router in your home and how many Macs you have?
On one of your Macs can you go to System Preferences then select the Network tab.
From there can you click the Show pulldown menu and select AirPort.
You should then see a list of your available wireless networks arrainged alphabetically, If I am right the Linksys network will be listed above your home Airport network, So can you either delete the entry for Linksys or make sure your home network is above it in the list by dragging your Airport name to the top then saving your changes.
Once you have done that let me know if you now remain connected to your network or not.
Just take your time and let me know if you need any clarification.