Understanding the relation between the structure of brain networks and their functions is a fundamental open question. Simple models of neural activity based on real anatomical networks have proven to be effective in describing features of whole-brain spontaneous activity when tuned at their critical point. In this work, we show that structural networks are indeed a crucial ingredient in the emergence of collective oscillations in a whole-brain stochastic model at criticality. We study analytically a stochastic Greenberg–Hastings cellular automaton in the mean-field limit, showing that it undergoes an abrupt phase transition with a bistable region. In particular, no global oscillations emerge in this limit. Then, we show that by introducing a network structure in the homeostatic normalization regime, the bistability may be disrupted, and the transition may become smooth. Concomitantly, through an interplay between network topology and weights, a large peak in the power spectrum appears around the transition point, signaling the emergence of collective oscillations. Hence, both the structure of brain networks and criticality are fundamental in driving the collective responses of whole-brain stochastic models.