Measuring the hierarchy of feedforward networks

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• dc.contributor.author Corominas Murtra, Bernatca
• dc.contributor.author Rodríguez Caso, Carlosca
• dc.contributor.author Goñi, Joaquínca
• dc.contributor.author Solé Vicente, Ricard, 1962-ca
• dc.date.accessioned 2015-12-16T14:23:36Z
• dc.date.available 2015-12-16T14:23:36Z
• dc.date.issued 2011
• dc.description.abstract In this paper we explore the concept of hierarchy as a quantifiable descriptor of ordered structures, departing from the definition of three conditions to be satisfied for a hierarchical structure: order, predictability, and pyramidal structure. According to these principles, we define a hierarchical index taking concepts from graph and information theory. This estimator allows to quantify the hierarchical character of any system susceptible to be abstracted in a feedforward causal graph, i.e., a directed acyclic graph defined in a single connected structure. Our hierarchical index is a balance between this predictability and pyramidal condition by the definition of two entropies: one attending the onward flow and the other for the backward reversion. We show how this index allows to identify hierarchical, antihierarchical, and nonhierarchical structures. Our formalism reveals that departing from the defined conditions for a hierarchical structure, feedforward trees and the inverted tree graphs emerge as the only causal structures of maximal hierarchical and antihierarchical systems respectively. Conversely, null values of the hierarchical index are attributed to a number of different configuration networks; from linear chains, due to their lack of pyramid structure, to full-connected feedforward graphs where the diversity of onward pathways is canceled by the uncertainty (lack of predictability) when going backward. Some illustrative examples are provided for the distinction among these three types of hierarchical causal graphs.ca
• dc.description.sponsorship This work was supported by the EU 6th framework project ComplexDis (NEST-043241, C.R.-C. and J.G.), the UTE project CIMA (J.G.), Fundaci ́ón Marcelino Botín (C.R.-C.), the James McDonnell Foundation (B.C.-M. and R.V.S.), and the Santa Fe Institute (R.V.S.).
• dc.format.mimetype application/pdfca
• dc.identifier.citation Corominas-Murtra B, Rodríguez-Caso C, Goñi J, Solé R. Measuring the hierarchy of feedforward networks. Chaos. 2011. 21(1): 016108. DOI 10.1063/1.3562548ca
• dc.identifier.doi http://dx.doi.org/10.1063/1.3562548
• dc.identifier.issn 1054-1500
• dc.identifier.uri http://hdl.handle.net/10230/25447
• dc.language.iso engca
• dc.publisher American Institute of Physics (AIP)ca
• dc.relation.ispartof Chaos. 2011. 21(1): 016108
• dc.relation.projectID info:eu-repo/grantAgreement/EC/FP6/43241
• dc.rights © American Institute of Physics. This article appeared in Corominas-Murtra et al., Chaos, Vol. 21(1), 2011 and may be found at http://dx.doi.org/10.1063/1.3562548ca
• dc.rights.accessRights info:eu-repo/semantics/openAccessca
• dc.subject.other Infografiaca
• dc.title Measuring the hierarchy of feedforward networksca
• dc.type info:eu-repo/semantics/articleca
• dc.type.version info:eu-repo/semantics/publishedVersionca