OK, so, you'll need exactly one controller, and one or more amplifiers depending on how fancy you want to get. You don't show any of that on your drawing.
The Controller takes 12VDC power and outputs R,G,B and Common. This can be both the signal driving any number of amplifiers, or the power driving a limited number of LED strips.
The important component is the amplifier. It inputs two things: a power input of 12V in large amounts; and a signal input consisting of R,G,B and C (common) from a controller. The signal input is simply the RGBC output of the controller. The amplifier outputs LED drive output consisting of R, G, B and C (common). This can feed one or several LED strings, limited by the rating of the power supply and the amplifier.
The minimum configuration
Install the controller just past the power supply on the lower right. The controller will directly drive the bottom-edge LED string.
At the place you mark "connector", the botton LED strip will be totally severed from the other LED strips. However, a 4-wire (RGBC) cable will continue onward from the end of this LED strip, and it will go to the amplifier. These can be very small wires, they carry virtually no current.
The top center power supply will be replaced by a power supply and an amplifier. The power supply output goes to the amplifier power input. The signal input comes from the 4-wire cable from the bottom-edge strip. The amplifier's RGBC output will then drive both LED strips along the top, left and right.