It is unknown at this time what exactly is causing the seemingly random logouts, though The Verge reports that app makers have narrowed down the culprit to a mechanism in Background App Refresh.
Of the more popular apps affected, Mailbox was "in crisis mode" to find a solution, but was unable to rectify the situation, saying Apple's Background App Refresh is the root cause of the issue. Consequently, an updated version of the app disabling access to the feature was submitted to the App Store on Friday.
AppleInsider can confirm that certain apps do require repeated log-ins when Background App Refresh is activated, including the aforementioned Mailbox, PayPal and Snapchat. Curiously, other titles like Facebook seem unaffected by the purported bug.
One of the highlights of Apple's iOS 7, Background App Refresh allows apps to receive dynamic updates without running in the foreground. The feature is extremely useful when using time-sensitive apps like Mailbox, or those that do not support push notifications.
Users experiencing continuous logouts can simply turn off global Background App Refresh in Settings or, if certain apps appear to be working, toggle the feature on a per-app basis.