Egghe L. "Development of hierarchy theory for digraphs using concentration theory based on a new type of Lorenz curve" MATHEMATICAL AND COMPUTER MODELLING 36 (4-5): 587-602 SEP 2002

[Eugene Garfield]

Leo Egghe : leo.egghe at luc.ac.be


Title     Development of hierarchy theory for digraphs using
          concentration theory based on a new type of Lorenz curve
Author    Egghe L
Journal   MATHEMATICAL AND COMPUTER MODELLING 36 (4-5): 587-602 SEP 2002

 Document type: Article    Language: English
 Cited References: 17       Times Cited: 0


Abstract:
In digraphs one has a hierarchy based on the unidirectional order between
the vertices of the graph. We present a method of measuring degrees of
hierarchy as expressed by the inequality that exists between the vertices'
hierarchical numbers. In order to do so, we need to extend the classical
Lorenz theory of concentration
(curves and measures) for a set of numbers x(1),..., x(N) to the case that
Sigma(i=1)(N) x(i)=0. This is then applied to the set of hierarchical
numbers of the vertices of the graph. A graph has a more concentrated
hierarchy than another one if the Lorenz curve of the first one is above the
Lorenz curve of the second one, hereby expressing that the inequality in
domination in the first case is larger than in the second case, and that the
inequality in subordination in the first case is larger than in the second
case. We also determine maximal and minimal Lorenz curves in this setting
and characterize the graphs that yield these curves. Based on this theory,
we also
determine good measures of hierarchical concentration in graphs.
Applications can be given in the study of organigrams in companies and
administrations and in citation analysis. (C) 2002 Elsevier Science Ltd. All
rights reserved.

Author Keywords:
digraph, hierarchy, Lorenz, concentration theory

KeyWords Plus:
INEQUALITY, METRICS

Addresses:
Egghe L, Limburgs Univ Ctr, Univ Campus, B-3590 Diepenbeek, Belgium
Limburgs Univ Ctr, B-3590 Diepenbeek, Belgium
Univ Instelling Antwerp, B-2610 Wilrijk, Belgium

Publisher:
PERGAMON-ELSEVIER SCIENCE LTD, OXFORD

IDS Number:
600YC

ISSN:
0895-7177


 Cited Author            Cited Work                Volume      Page   Year

 ALLISON PD            AM SOCIOL REV                 43       865      1978
 ATKINSON AB           J ECON THEORY                  2       244      1970
 BOTAFOGO RA           ACM T INFORM SYST             10       142      1992
 DALTON H              ECON J                        30       348      1920
 DEBRA P               SCIENTOMETRICS                47       227      2000
 EGGHE L               INTRO INFORMETRICS Q                            1990
 GINI C                GIORNALE DEGLI EC             11        37      1909
 HARDY GH              INEQUALITIES                                    1952
 HARDY GH              MESSENGER MATHEMATIC          58       145      1929
 LORENZ MO             J AM STAT ASSOC                9       209      1905
 MARSHALL AW           MATH SCI ENG                 143                1979
 MUIRHEAD RF           P EDINBURGH MATH SOC          21       144      1903
 REID RC               ENUMERATION GRAPH CO                            1997
 ROUSSEAU R            ACTA BIOTHEOR                 47         1      1999
 SHANNON C             BELL SYSTEM J                 28       379      1948
 THEIL H               EC INFORMATION THEOR                            1967
 WILSON BJ             INTRO GRAPH THEORY                              1972


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_______________________________________________________________________
Eugene Garfield, PhD.  email: garfield at codex.cis.upenn.edu
home page: www.eugenegarfield.org
Tel: 215-243-2205 Fax 215-387-1266
President, The Scientist LLC. www.the-scientist.com
Chairman Emeritus, ISI www.isinet.com
Past President, American Society for Information Science and Technology
(ASIS&T)  www.asis.org
_______________________________________________________________________