Police were called to Whitevale Street at around 9.55pm on Sunday, July 20.
A Police Scotland spokesman said: "A 16-year-old male was taken to the Southern General Hospital with a fractured skull and is currently in a stable condition.
"A 13-year-old boy was treated for a minor knee injury and was later released after treatment."
It is understood the injuries were caused by a blunt instrument.
Anyone with information is asked to to contact Police Scotland.