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Dynamic allocation of resources to the best link in large multiuser networks offers considerable improvement in spectral efficiency. This gain, often referred to as multiuser diversity gain, can be cast as double-logarithmic growth of the network throughput with the number of users. In this paper, we consider large cognitive networks granted concurrent spectrum access with license-holding users. The primary network affords to share its underutilized spectrum bands with the secondary users. We assess the optimal multiuser diversity gain in the cognitive networks by quantifying how the sum-rate throughput of the network scales with the number of secondary users. For this purpose, we look at the optimal pairing of spectrum bands and secondary users, which is supervised by a central entity fully aware of the instantaneous channel conditions, and show that the throughput of the cognitive network scales double-logarithmically with the number of secondary users and linearly with the number of available spectrum bands , i.e., . We then propose a distributed spectrum allocation scheme, which does not necessitate a central controller or any information exchange among different secondary users and still obeys the optimal throughput scaling law. This scheme requires that some secondary transmitter-receiver pairs exchange information bits among themselves. We also show that the aggregate amount of information exchange between secondary transmitter-receiver pairs is asymptotically equal to . Finally, we show that our distributed scheme guarantees fairness among the secondary users, meaning that they are equally likely to get access to an available spectrum band.