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In this letter, we first define a cognitive network to be useful if at least one node can be scheduled to transmit without causing significant simultaneous interference to any primary user and then investigate the interaction between secondary network size and the probability of the secondary network being useful. First the size of the primary network is fixed, and we analyze how quickly the interference threshold limit of the primary network can be reduced as a function of secondary network size. Here there is a tradeoff between the rate of interference threshold reduction and the probability that the secondary network is useful which is completely characterized for Rician fading. We then allow both networks to grow simultaneously. Here the tradeoff is determined in the regime that the interference decreases sufficiently fast for Rayleigh fading. We also investigate the effect of primary channel correlation. Finally, we say that the secondary network is -useful provided at least one of any secondary nodes can be scheduled. We show that in the asymptotic regime, the probabilities of the secondary network being-useful are uniquely related and do not depend on the asymptotic behavior of the interference threshold, the rates at which the networks grow or even the distribution of the fading.