The incidents all took place between 03:00 and 04:00 on Saturday.
A 38-year-old man was on Lothian Road when two men pushed him to the ground, leaving him with a minor head injury.  A 21-year-old man was also attacked and robbed on Princes Street.
Police are attempting to identify a second suspect who is believed to have been involved.
They have asked for any other potential victims or witnesses to contact them.
A 24-year-old man has been arrested after two other men were attacked in Edinburgh city centre at the weekend.