This page of the Astro's Playroom guide describes the Network Speed Run level.
Network Speed Run is located on the CPU Square, between the entrance to the GPU Jungle and the Cooling Springs. You can enter the race using the map as well.
There are 8 routes available in the speed run. Some routes will be unlocked only after completing levels from the Memory Meadow, SSD Highway, Cooling Springs, and GPU Jungle.
Network Speed Run is a mini-route, resembling one of the time-constrained levels in the game. You can repeat each route multiple times if you want to improve your result.
After completing all the routes, the game will calculate the total score. If the total time does not exceed 7 minutes, the Run, Astro, Run! trophy will be unlocked.