This paper investigates how a pairwise stable network changes upon the arrival of a new agent. Comparing the new networks obtained for the society after the arrival of a new agent with the initially pairwise stable network, we propose four different population invariance properties of a network: link invariance, distance invariance, connectedness invariance, and network invariance. We show that pairwise stability is incompatible with link invariance under certain assumptions on the allocation rule. However, if we consider specific models, positive results can be obtained. For instance, in the symmetric connections model, pairwise stability implies connectedness invariance.