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In a cognitive network, autonomous and adaptive radios select their operating parameters to achieve individual and network-wide goals. The effectiveness of these adaptations depends on the amount of knowledge about the state of the network that is available to the radios. We examine the price of ignorance in topology control in a cognitive network with power- and spectral-efficiency objectives. We propose distributed algorithms that, if radios possess global knowledge, minimize both the maximum transmit power and the spectral footprint of the network. We show that while local (as opposed to global) knowledge has little effect on the maximum transmission power used by the network, it has a significant effect on the spectral performance. Furthermore, we show that due to the high cost of maintaining network knowledge for highly dynamic networks, the cost/performance tradeoff makes it advantageous for radios to operate under some degree of local knowledge, rather than global knowledge.We also propose distributed algorithms for power and frequency adaptations as radios join or leave the network, and assess how partial knowledge impacts the performance of these adaptations.