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We consider the problems of computing the average degree and the size of a given network in a distributed fashion under quantized communication. We present two distributed algorithms which rely on quantized operation (i.e., nodes process and transmit quantized messages), and are able to calculate the exact solutions in a finite number of steps. Furthermore, during the operation of our algorithms, each node is able to determine in a distributed manner whether convergence has been achieved and thus terminate its operation. To the best of the authors' knowledge these algorithms are the first to find the exact solutions under quantized communication (i.e., there is no error in the final calculation). Additionally, note that our network size calculation algorithm is the first in the literature which calculates the exact size of a network in a finite number of steps without introducing a final error. This error in other algorithms can be either due to quantization or asymptotic convergence. In our case, no error is introduced since the desired result is calculated in the form of a fraction involving a quantized numerator and a quantized denominator. Finally, we demonstrate the operation of our algorithms and their potential advantages.