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We consider a cognitive network consisting of n cognitive users uniformly distributed with constant density among primary users. Each user has a single transmitter and a single receiver, and the primary and cognitive users transmit concurrently. The cognitive users use single-hop transmission in two scenarios: (i) with constant transmit power, and (ii) with transmit power scaled according to the distance to a designated primary transmitter. We show that, in both cases, the cognitive users can achieve a throughput scaled linearly with the number of users n. The first scenario requires the cognitive users to have the transmitter-receiver (Tx-Rx) distance bounded, but it can be arbitrarily large. Then with high probability, any network realization has the throughput scaling linearly with n. The second scenario allows the cognitive Tx-Rx distance to grow with the network at a feasible exponent as a function of the path loss and the power scaling factors. In this case, the average network throughput grows at least linearly with n and at most as n log(n). These results suggest that single-hop transmission may be a suitable choice for cognitive transmission.