Zipfian and Lotkaian continuous concentration theory

Abstract

In this article concentration (i.e., inequality) aspects of the functions of Zipf and of Lotka are studied. Since both functions are power laws (i.e., they are mathematically the same) it suffices to develop one concentration theory for power laws and apply it twice for the different interpretations of the laws of Zipf and Lotka. After a brief repetition of the functional relationships between Zipf's law and Lotka's law, we prove that Price's law of concentration is equivalent with Zipf's law. A major part of this article is devoted to the development of continuous concentration theory, based on Lorenz curves. The Lorenz curve for power functions is calculated and, based on this, some important concentration measures such as the ones of Gini, Theil, and the variation coefficient. Using Lorenz curves, it is shown that the concentration of a power law increases with its exponent and this result is interpreted in terms of the functions of Zipf and Lotka. IntroductionThe historical law of Lotka (1926) is the basis of modern informetrics and is expressed as follows: the number of authors with n (n ϭ 1,2,3, . . .) publications is proportional to where a Ͼ 0. In other words, there is a constant C Ͼ 0 such that (1) where f(n) denotes the number of authors with n publications. More generally, f(n) can denote the number of sources (authors, journals, word types...) with n items (publications, articles, word occurrences, respectively); henceforth, this dual source/item terminology will be used (see also Egghe, 1989Egghe, , 1990Egghe and Rousseau, 1990a).A totally different informetric formulation, originating from linguistics (in terms of word types and word occurrences) is given by the Zipf law: If we rank the sourcesaccording to their number of items (starting with the source with the highest number of items, hence giving to this source the rank r ϭ 1) then the number of items in the source on rank r (r ϭ 1, 2, 3, . . .) is proportional to where b Ͼ 0. In other words, denoting by g(r) this number of items in the source on rank r, there exists a constant D Ͼ 0 such thatAlthough their informetric definitions are different (they are dual in the sense that, in f and g, the roles of sources and items are interchanged), the functions 1 and 2 are mathematically the same, namely decreasing power laws. A power law is the most occurring regularity in informetrics and far beyond (e.g., also found in economics and sociology, including the description of social networks, see e.g., Egghe and Rousseau, 2003 and references therein) and has the characterizing property (see Roberts, 1979) that, if the argument (x) is multiplied by a constant, k, we obtain the same power law with the same exponent: for we have (ϳ denotes is proportional to). This self-similar property explains its widespread occurrence in real-life examples and also its use in the description of power type informetrics in terms of self-similar fractals (see Feder, 1988). Intuitively, the self-similarity expresses that, independent of the scale at which we examine an "object...

Egghe's construction of Lorenz curves resolved

“…(Note that, following Egghe (2005aEgghe ( , 2005b, attention is restricted to the continuous formulation in what follows. )…”

Doze anos de estudo da Revista de Administração Pública à luz da bibliometria e da rede social

“…A bibliometria desenvolveu-se mediante a elaboração de leis empíricas sobre o comportamento da literatura (EGGHE, 2005). Nesse cenário, é importante conhecer as três leis básicas da bibliometria para o melhor entendimento dos dados.…”

“…Nesse cenário, é importante conhecer as três leis básicas da bibliometria para o melhor entendimento dos dados. Elas são oriundas respectivamente de três pesquisadores que se destacam por suas importantes descobertas -Bradford, Zipf e Lotka -, sendo as mais comumente utilizadas e relacionadas à produtividade científica (EGGHE, 2005).…”

“…Já a Lei de Lotka descreve a produtividade e as citações de autores por meio de um modelo de distribuição de tamanho-frequência em um conjunto de pesquisas, evidenciando aspectos de coautoria (AMBONI; CAMINHA; ANDRADE, 2012). E a Lei de Zipf calcula a quantidade de ocorrências das palavras em vários textos, gerando uma lista de terminações de um determinado assunto ou palavra, sendo utilizada para observar qual tema científico é mais evidenciado nos trabalhos (EGGHE, 2005 Com isso, a análise bibliométrica e de rede social do referido artigo foi feita mediante os seguintes indicadores: (I) características de autoria; (II) autores com maior produção; (III) referências mais citadas; (IV) IES com maior produção; (V) rede social dos autores; (VI) rede social das IES; (VII) abordagens metodológicas; e (VIII) frequência das palavras -no título e nas palavras-chave. …”

“…Notase certa similaridade entre as duas figuras: o destaque das palavras "gestão" e "pública". Remete-se à Lei de Zipf, que calcula a frequência de ocorrências das palavras, com o intuito de criar e evidenciar o tema científico mais contemplado nas pesquisas (EGGHE, 2005).…”