Hierarchical benchmark graphs for testing community detection algorithms

Hierarchical organization is an important, prevalent characteristic of complex systems; in order to understand their organization, the study of the underlying (generally complex) networks that describe the interactions between their constituents plays a central role. Numerous previous works have shown that many real-world networks in social, biologic and technical systems present hierarchical organization, often in the form of a hierarchy of community structures. Many artificial benchmark graphs have been proposed in order to test different community detection methods, but no benchmark has been developed to throughly test the detection of hierarchical community structures. In this study, we fill this vacancy by extending the Lancichinetti-Fortunato-Radicchi (LFR) ensemble of benchmark graphs, adopting the rule of constructing hierarchical networks proposed by Ravasz and Barab\'asi. We employ this benchmark to test three of the most popular community detection algorithms, and quantify their accuracy using the traditional Mutual Information and the recently introduced Hierarchical Mutual Information. The results indicate that the Ravasz-Barab\'asi-Lancichinetti-Fortunato-Radicchi (RB-LFR) benchmark generates a complex hierarchical structure constituting a challenging benchmark for the considered community detection methods.