In this paper we introduce a novel information propagation method in Twitter, while maintaining a low computational complexity. It exploits the power of Compressive Sensing in conjunction with a Kalman filter to update the states of a dynamical system. The proposed method first employs Joint Complexity, which is defined as the cardinality of a set of all distinct factors of a given string represented by suffix trees, to perform topic detection. Then based on the inherent spatial sparsity of the data, we apply the theory of Compressive Sensing to perform sparsity-based topic classification by recovering an indicator vector, while reducing significantly the amount of information from tweets, possessing limited power, storage, and processing capabilities, to a central server. We exploit datasets in various languages collected by using the Twitter streaming API and achieve better classification accuracy when compared with state-of-the-art methods.