Networks are versatile representations of the interactions between entities in complex systems. Cycles on such networks represent feedback processes which play a central role in system dynamics. In this work, we introduce a measure of the importance of any individual cycle, as the fraction of the total information flow of the network passing through the cycle. This measure is computationally cheap, numerically well-conditioned, induces a centrality measure on arbitrary subgraphs and reduces to the eigenvector centrality on vertices. We demonstrate that this measure accurately reflects the impact of events on strategic ensembles of economic sectors, notably in the US economy. As a second example, we show that in the protein-interaction network of the plant Arabidopsis thaliana, a model based on cycle-centrality better accounts for pathogen activity than the state-of-art one. This translates into pathogen-targeted-proteins being concentrated in a small number of triads with high cycle-centrality. Algorithms for computing the centrality of cycles and subgraphs are available for download. Recent research on networks has slowly progressed from questions directly concerning individual entities, to questions regarding the dynamics of the system, from the local to the global scale. Already over the course of the development of vertexcentralities, i.e. measures of the importance of individual nodes, it became clear that vertex-neighborhoods, subgraphs and motifs were of paramount importance to understand the evolution of real networks [18,27]. For example, in a recent study of the propagation of economic shocks in input-output networks, Alatriste Contreras and Fagiolo concluded that "the systemic importance of industrial sectors should not be evaluated only by looking at their economic size [i.e. properties of individual