eprintid: 2897 rev_number: 20 eprint_status: archive userid: 61 dir: disk0/00/00/28/97 datestamp: 2015-11-16 08:48:19 lastmod: 2016-02-02 14:40:32 status_changed: 2015-11-16 08:48:19 type: article metadata_visibility: no_search contact_email: juanignacioperotti@imtlucca.it creators_name: Perotti, Juan I. creators_name: Tessone, Claudio Juan creators_name: Caldarelli, Guido creators_id: juanignacio.perotti@imtlucca.it creators_id: claudio.tessone@business.uzh.ch creators_id: guido.caldarelli@imtlucca.it title: Hierarchical mutual information for the comparison of hierarchical community structures in complex networks ispublished: pub subjects: HA subjects: HG subjects: Q1 subjects: QA subjects: QA75 subjects: QC divisions: CSA full_text_status: public monograph_type: working_paper note: The paper was accepted for publication subject minor revisions. Together with the paper, a software python package was released under the GPL licence 2 (1991), useful for the computation of the hierarchical mutual information. abstract: The quest for a quantitative characterization of community and modular structure of complex networks produced a variety of methods and algorithms to classify different networks. However, it is not clear if such methods provide consistent, robust and meaningful results when considering hierarchies as a whole. Part of the problem is the lack of a similarity measure for the comparison of hierarchical community structures. In this work we give a contribution by introducing the {\it hierarchical mutual information}, which is a generalization of the traditional mutual information, and allows to compare hierarchical partitions and hierarchical community structures. The {\it normalized} version of the hierarchical mutual information should behave analogously to the traditional normalized mutual information. Here, the correct behavior of the hierarchical mutual information is corroborated on an extensive battery of numerical experiments. The experiments are performed on artificial hierarchies, and on the hierarchical community structure of artificial and empirical networks. Furthermore, the experiments illustrate some of the practical applications of the hierarchical mutual information. Namely, the comparison of different community detection methods, and the study of the the consistency, robustness and temporal evolution of the hierarchical modular structure of networks. date: 2015-12 date_type: published publication: Physical Review E number: 92 publisher: American Physical Society place_of_pub: Physical Review E pagerange: 062825 id_number: 10.1103/PhysRevE.92.062825 institution: IMT Institute for Advanced Studies Lucca department: NETWORKS refereed: TRUE issn: 1539-3755 official_url: http://journals.aps.org/pre/abstract/10.1103/PhysRevE.92.062825 related_url_url: http://arxiv.org/abs/1508.04388v1 projects: MULTIPLEX projects: SIMPOL projects: DOLFINS citation: Perotti, Juan I. and Tessone, Claudio Juan and Caldarelli, Guido Hierarchical mutual information for the comparison of hierarchical community structures in complex networks. Physical Review E (92). 062825. ISSN 1539-3755 (2015) document_url: http://eprints.imtlucca.it/2897/1/1508.04388v1.pdf