The cycles approach uses graph theory and linear algebra to study models of knowledge, characterized by a state space, a set of players and their partitions. In finite state spaces, there is a simple formula for the cyclomatic number; i.e., the dimension of cycle spaces of a model. We prove that the cyclomatic number is the minimum number of cycle equations that must be checked to guarantee the existence of a common prior, and explain why some cycle equations are automatically satisfied. If the cyclomatic number is zero, a common prior always exists, regardless of the probabilistic information given by players.posteriors. There is an isomorphism taking cycles into cycle equations; adding cycles is the counterpart of multiplying the corresponding cycle equations. With these tools, we study the processes of learning and forgetting, as well as properties of sub models (i.e., restricting attention to a proper subset of players), and decompositions of the set of players in subsets. We analyze how individual learning translates into more common knowledge or cycle destruction.