Before revision:  Here we study SOC in an adaptive network considered first by Bornholdt and Rohlf PRL, 84(26), p.6114-6117, 2000 . We focus on the important convergence to criticality and discover time-scale and noise optimal behaviour as well as a noise-induced phase transition . Due to the complexity of adaptive networks dynamics we suggest to investigate each effect separately by developing simple models.
After revision:  Here we study SOC in an adaptive network considered first by Bornholdt and Rohlf PRL, 84(26), p.6114-6117, 2000 . We focus on the important convergence to criticality and discover time-scale optimality are reached at finite values. This is in sharp contrast to the previously believed optimal zero noise and infinite time scale separation case. Furthermore, we discover a noise induced phase transition for the breakdown of SOC. As a novel paradigm for adaptive networks SOC, we suggest to investigate each effect separately by developing simple models.