Abstract
The Lavalette function is generalized to a 2-exponent function in order to represent data looking like a sigmoid on semi-log plots. A Mandelbrot trick is suggested for further investigations, if more fit parameters are needed. The analyzed data is that of the number of adherents to the main religions in the XXth century.
I. INTRODUCTION
Recently, there have several studies of activities related to society dynamics based on physics ideas [1] [2] [3] . This is part of an aim of physicists toward applying their concepts and methods in sociological fields [4] [5] [6] Among these are religion/religious movement aspects.
Let a few recent papers be mentioned [7] [8] [9] [10] [11] [12] [13] in order to set a frame and pretend that one aspect among others has not been examined. True the evolution of adherents to religious movements has been studied through the (statistical) distribution of the number of adherents. However, one cannot (apparently) distinguish between various analytical forms, neither to say whether the distribution is stable. There is a report on some "heavy tail" in [10] , but the discussion and analysis are quite reduced to the shortest arguments for some data snapshot.
One aim of this paper is to report another (simple) approach, the "least effort method" [14] about the rank-size law of religious movements, through the number of adherents for a set of religions, selected through published data. However, a second aim pertains to the findings suggesting that the discovered law might be more general than exemplified here, -in fact, as being valid for many cases of sets measured through the number of members/agents. It is found that a Lavalette law [15] , generalized in quite another field [16] , is rather robust, and represents a fine data envelope, equivalent to a mean field approximation of the rank-size relationship asymptotic behavior. Thereby, hinting that such a law, even possibly further generalized, as suggested below, can serve to a huge array of data interpretation, models, etc., in many possible research fields, in physics and outside, -and could be usefully examined on basic principles, -outside the present aims however.
In Sect. II, the (religious movement adherent) data is presented and analyzed along the conventional ranksize method, classically bearing upon Zipf-MandelbrotPareto (ZMP) ideas. Moreover, introducing a Mandelbrot trick onto the generalized Lavalette function, is possible and likely of interest. Such a generalization and subsequent successful applications seem indicating that possible "super-generalizations", see Sect. III, have a wide, ∼ universal, content. In fact, it is readily related to the Feller-Pareto distribution function (for a chosen set of exponents, see below). Such a finding suggests some conclusion found in Sect. IV.
II. DATA AND ANALYSIS
Due to the unusual type of data hereby considered for a "scaling theory" like analysis, some introduction to the analyzed WCE data [17] is here given. Indeed, a warning is necessary: it is unclear how much distinction was made in the WCE surveys concerning denominations and sects so called adstrated to the main religions. Nevertheless, admitting some flexibility in the definition of religion and ambiguity in that of adhesion, the data are as equally valid as many of those reported about agent behavior in the recent sociophysics literature and examined along statistical mechanics approaches.
The data set is taken from the World Christian Encyclopedia (WCE) [17] : it gives information on the number of adherents of the world's main religions and their main denominations (56 religions overall). I have desaggregated the data for the less general denominations, removing also data on so called "doubly-affiliated", "disaffiliated", and "doubly-counted religionists", as listed in [17] . Moreover some data refers to Atheists and N onreligious persons. Thereafter for conciseness, those three sets are also called "religions".
All religions do not exist in every country. Countries can be ranked according to the number of religions which have local adherents. This leads to a so called Pareto plot for the cumulative distribution of the (56) religious movements in the (258) countries: Fig.1 .
On this log-log plot, giving the relationship between the number of nations ranked according to the number of religions which can be found locally, it is obvious that the size-frequency distribution is surely not a power law, -not even shown; the data has visually a convex shape, and also hardly looks like a Poisson (exponential) form (green dash lines) the (2 parameter) exponential case (Exp2)
However, a fit by a generalized Lavalette function (Lav3) [16] (red dot line), i.e. a decaying power law with a power law cut-off,
gives a quite good regression coefficient R 2 0.981; the exponents have values −φ = 0.001 and ψ = 2.315 For completeness, let it be recalled that Eq. (2) is a 2-exponent (thus 3-parameter) generalization of the basic 2-parameter (1-exponent), (Lav2), introduced by Lavalette [15] , when discussing impact factor distributions:
In the WCE data set [17] , further information is given about changes in the number of adherents of each (main) religion from 1900 till 2000, measured over a 5 year span. In Fig.2 , the 1900 data distribution is emphasized: there were 55 "major denominations", available to a few billions people. There are nowadays 56 available "major denominations" for about twice as many possible adherents. The 56-th, the "black muslims", came later in the century.
Observe that a fit of the 1900 (green half open/filled squares) data with the 3-parameter free Lavalette function evolves toward the recent (after 1970) data for r > 24. The fit with a Lavalette 3-parameter free function is very stable as a function of time, not only matches very well the data for the denominations with r > 24, but also extends toward and covers very finely the "high rank" regime.
An interesting point, confirming the warning about data precision, has to be made: it should be emphasized that the jump in N , the number of adherents, between r=24 and 25 is reminiscent of the observation in [10] that it is "somewhat difficult" (author's emphasis) for major denominations to measure their true number of adherents, -in fact (daring author's interpretation ?) likely preferring to overestimate such an adhesion in order to appear "big". A contrario, it seems common sense to consider that "less popular" denominations have a better measurement of their number of adherents. The biggersmaller border occurs at ∼ 10 7 adherents, corresponding to r ∼ 24.
The more recent data is analyzed in two ways in Figs.3-4 , either within the Lavalette "model" (Lav3) or within a mere exponential behavior (Exp2). It is more simple to display the parameters in a table, i.e. R 2 values are better for the Lav3 than for the exponential law. Let it be observed that the β decay rate is pretty stable (∼ 0.173) in modern times. The amplitude b follows the evolution of the world population, but Λ decreases, -indicating that people likely become less adherent to (the main 56) religions, -which can be an "artificial result" due to the fact that the number of small religious movements itself increases, but these are not considered in the WCE surveys. The power law exponent, φ, measuring the decay, steadily decreases. However the exponent φ of the cut-off at high rank present a minimum (at some time in the middle of the century) and increases nowadays. Note that -(ψ + φ) measures the slope at half range on a semi-log plot, and approximately follows the (increasing nowadays) behavior of ψ, i.e. the power law cut-off. This indicates that proportionally the number of adherents to the main religions in fact increases. Another Matthew effect [18] in some sense: the "winning" religions are always the same ones, and stay more at the top than others, even though the number of adherents in the population falls down.
III. HYPER-GENERALIZED LAVALETTE FUNCTION
The pure power-law distribution, known as the zeta distribution, or discrete Pareto distribution is expressed as
where: k is a positive integer usually measuring some variable of interest, p(k) is the probability of observing the value k; γ is the power-law exponent; ζ(γ) ≡ ∞ k=1 k −γ is the Riemann zeta function. It is important to note, from this definition, that γ > 1 for the Riemann zeta function to be finite. Such a mere power law is of no use here. However, an adaptation of Eq. (4) was suggested for text appraisal by Mandelbrot,
in order to take into account a flattening of the data at low rank, on a log-log plot; it is known as the Zipf- Mandelbrot-Pareto 3-parameter free function (ZMP3) for describing the "queen effect" [19] .
Note that the role of r as independent variable, in Eq. (5), is taken by the ratio r/(N − r + 1) between the descending and the ascending ranking numbers in Eqs.
(2)-(3). Moreover, observe that, on a semi-logarithmic graph, Lav2 and Lav3 follow a characteristic (flipped) sigmoidal S-shape which by no means can be provided by a power law or by ZMP3 [20] .
Moreover, note that N (as a factor of r, in Eq. (3)) is not really needed; in fact it can be usefully replaced by some simple factor having the order of magnitude of y(r M /2). This can be seen through other ways of writing of the 2-parameter Lavalette form, introducing the scaling factorsκ or Λ.
Finally, if the data fits show such a need, -this is not presently needed, an introduction of Mandelbrot trick into the Lavalette function could be made in order to obtain a (6-parameter free) hyper-generalized Lavalette function, i.e.
• in its most generalized form, with a power law cutoff
• ... or alternatively with an exponential cut-off Table I In the latter formula, m 5 is not officially needed: it can be incorporated into A. But, beyond such possibilities, it remains to have some interpretation of the parameters, thus much theoretical work ahead! Nevertheless, for concluding this section, observe that the above generalized in (2-exponent) Lavalette function can find a root in the Feller-Pareto function:
when φ > 0 and ψ < 0; with u ≡ r/(N + 1). To represent a true density function, Λ should then be the inverse of a B (Beta) function, for normalization purpose. The Feller-Pareto function is known to describe random walk distributions, 1-D electronic density of states, moving averages intersections, etc. However, the Feller-Pareto function has neither the flexibility of the generalized nor the hyper-generalized Lavalette functions [20] .
IV. CONCLUSION
When employing combined functions, an excellent fit of rank-frequency distributions is usually ensured. However, improving the quality of the approximation at the expense of an increase in the number of fitting parameters does not provide the best solution toward inventing an appropriate model. Having a large number of fitting parameters causes difficulties in the extraction of their implication (meaning) and increases their dependency on each other, in worst cases. The present work, devoted to developing a fitting function that provides (i) an optimal approximation of rank-probability distributions and (ii) a correlation of its fitting parameters with features of an interesting modern data base, has emphasized that the Lavalette function is of interest in order to represent a sigmoid function with few parameters on semi-log plots. Yet, the basic Lavalette function does not provide enough flexibility for fitting curves in the range of small rank values. Therefore a 2-exponent function has been used, and a Mandelbrot trick suggested for further investigations.

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