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You have to set up your router for port-forwarding. This setting allows the router to forward the data that arrives on a specific port to a local IP address previously set. This IP address would be the Edison's IP address. Please check the following links; they are about accessing an Arduino from an external network:
- Setting up Arduino Webserver for remote access
- Accessing Arduino over internet - All
Although the links above are for Arduino projects, the principles discussed there apply for the Edison too. You will find that information useful.