What does "noise power is assumed to be the same at all nodes
Although noise is often called channel noise in various block diagrams of radio-frequency (RF) communications systems that communications analysts draw, the truth of the matter is that most actual RF channels are mostly noiseless, and what is referred to as the channel noise is primarily thermal noise in the resistive elements in the front end of the receiver.  The noise power in the receiver that communications engineers have to contend with is thus determined by the temperature of those resistors (and the value of those resistors).  Thus, assuming that the receivers in the various "nodes" are identical and also in the same geographical area so that all the receivers can be expected to be at the same temperature, it is reasonable to use a model in which the noise power is the same in all of the receivers.