We study the dynamical behaviour of simple graphs under the iterated application of the clique graph operator k, which transforms each finite graph G into the intersection graph kG of its (maximal) cliques. The graph G is said to be clique divergent if the sequence of the orders o(kⁿG) of the iterated clique graphs of G tends to infinity with n, and G is said to have linear growth if this divergent sequence is bounded by a linear function of n. In this work, we introduce an important family of graphs (the clockwork graphs) which is closed under the clique operator and contains clique divergent graphs with strictly linear growth, i.e., o(kⁿG)=o(G)+rn, where r is any fixed positive integer. We apply our results to give examples of clique divergent graphs having non-strict linear growth.