Extreme degeneracies characterize the module identication problem for complex networks

techniques, such as the one based on optimizing the quantity called modularity, remains poorly characterized. Here, we present a systemic and critical analysis of this method. We show analytically that in addition to the previously observed resolution limit, the modularity function Q exhibits extreme degeneracies, admitting an exponential number of high-modularity but structurally distinct solutions. The pres- ence of hierarchical structure, in which modules organize themselves into modules-of-modules and which is believed to characterize many real-world networks, further exacerbates this problem. We conrm our analytic results using numerical experiments on synthetic networks with either modular or hierarchical structure and on several real-world examples of metabolic networks. These results contradict the widely held assumption that the modularity function typically exhibits a clear global optimum, and imply that modules identied via modularity maximization are unlikely to be unique and should be interpreted with extreme caution. We conclude with a brief discussion of alternative avenues for accurately and objectively identifying modular structure.