Lavalette Function: Flexible Rank-Size Modeling

• The Lavalette function is a flexible parametric model for rank–size distributions, capturing smooth S-shaped deviations from pure power-law or exponential behavior.
• It features adjustable parameters that independently control the head decay and tail cutoff, allowing multiparameter generalizations for detailed empirical fitting.
• Widely applied in economics, linguistics, bibliometrics, and social sciences, it provides a robust framework for modeling finite-size and frequency distributions.

The Lavalette function defines a flexible and compact parametric family for rank–size or rank–frequency distributions, especially suited for empirical data exhibiting smooth, S-shaped departures from strict power-law or exponential laws. Originally introduced as a two-parameter law, it has since been generalized to forms with additional exponents and shifts, allowing precise modeling of both head and tail curvature in ranked datasets. The function’s unique combination of terms provides a built-in, symmetric high-rank cutoff and accommodates various empirical regularities—critical in economic, social, linguistic, and bibliometric contexts.

1. Mathematical Form and Generalizations

The standard Lavalette function (often called “Lav2”) for a ranked set of $N$ items ($r = 1$ highest, $r = N$ lowest) is given by: $$y(r) = \kappa \left(\frac{r}{N - r + 1}\right)^{-\chi} = \kappa\, r^{-\chi}\, (N - r + 1)^{\chi}$$ where:

• $y(r)$: size or score of item at rank $r$,
• $\kappa$: normalization/scale parameter,
• $\chi > 0$: shape (decay) exponent.

A modified, rescaled form used in practical applications such as the Italian municipality aggregated tax income (ATI) dataset introduces a scale factor $10^6$ to stabilize parameter estimation: $$y(r) = \hat\kappa \times 10^6\, \left(\frac{r}{N - r + 1}\right)^{-\chi}$$ with $r = 1$0 and $r = 1$1 as fit parameters (Cerqueti et al., 2014).

The two-exponent Lavalette (“Lav3”) generalization separates the low- and high-rank exponents: $r = 1$2 where $r = 1$3 and $r = 1$4 are independent exponents controlling, respectively, the decay in the head and cutoff in the tail (Ausloos, 2014, Ausloos, 2014).

Further extensions include:

• “Super-generalized” (5-parameter) and “hyper-generalized” (7-parameter) forms with explicit shifts and additional powers, enabling fitting of complex curvature and inflection points on log–log plots (Ausloos, 2014).

2. Parameter Interpretation and Curve Shape

Each parameter in the Lavalette family controls a distinct aspect of the curve:

• $r = 1$5 (or $r = 1$6, $r = 1$7): overall vertical scaling.
• $r = 1$8 (or $r = 1$9, $r = N$0): single or multiple exponents, shaping the steepness of decay (head) and sharpness of the cutoff (tail).

In Lav2 ($r = N$1 only), varying $r = N$2 tunes both tail and head in lockstep. For small $r = N$3, the distribution is flat (weak rank effect); for large $r = N$4, the function decays rapidly at low $r = N$5 and falls off sharply at the tail (Cerqueti et al., 2014). Lav3 forms introduce independent control, allowing, for instance, an S-shaped (sigmoid) curve on semi-log axes, with inflection near $r = N$6 (Ausloos, 2014). Larger $r = N$7 steepens the drop-off from the top ranks, while larger $r = N$8 produces a more pronounced tail cutoff.

The Lav3 normalized form,

$r = N$9

makes the symmetry between head ($$y(r) = \kappa \left(\frac{r}{N - r + 1}\right)^{-\chi} = \kappa\, r^{-\chi}\, (N - r + 1)^{\chi}$$0) and tail ($$y(r) = \kappa \left(\frac{r}{N - r + 1}\right)^{-\chi} = \kappa\, r^{-\chi}\, (N - r + 1)^{\chi}$$1) explicit.

The multi-parameter extensions allow modeling plateau regions, king/queen/vice-roy effects (outlier head cities, for example), and even inflection points on log–log axes by combining multiple exponents, shifts, or sum-of-Lavalette components (Ausloos, 2014).

3. Fitting Methodology and Statistical Considerations

Estimation of Lavalette function parameters is typically accomplished via nonlinear least squares, using robust algorithms such as Levenberg–Marquardt. Preprocessing steps may involve remapping item identities for consistency across time or handling missing/merged items (e.g., Italian municipalities in 2007–2011 adjusted to a uniform 2011 list) (Cerqueti et al., 2014).

The regression is usually performed over the full rank range, optionally omitting king/vice-roy outliers—top-ranked items with anomalously large sizes (e.g., Rome, Milan for Italian ATI). Removing such outliers routinely improves fit quality (measured by $$y(r) = \kappa \left(\frac{r}{N - r + 1}\right)^{-\chi} = \kappa\, r^{-\chi}\, (N - r + 1)^{\chi}$$2) from 0.94 to $$y(r) = \kappa \left(\frac{r}{N - r + 1}\right)^{-\chi} = \kappa\, r^{-\chi}\, (N - r + 1)^{\chi}$$30.99 at the national level and $$y(r) = \kappa \left(\frac{r}{N - r + 1}\right)^{-\chi} = \kappa\, r^{-\chi}\, (N - r + 1)^{\chi}$$4 regionally.

For maximum statistical rigor, alternative techniques include:

• Linear regression on log-rank-transformed data for Lav2/Lav3 models (Fontanelli et al., 2016).
• Maximum likelihood estimation (MLE) for the continuous Lavalette distribution, using the analytic probability density function,

$$y(r) = \kappa \left(\frac{r}{N - r + 1}\right)^{-\chi} = \kappa\, r^{-\chi}\, (N - r + 1)^{\chi}$$5

• Goodness-of-fit via Kolmogorov–Smirnov bootstrapping and comparison with information criteria (AIC) (Fontanelli et al., 2016).

Model comparison frequently demonstrates that Lavalette forms outperform Zipf/Zipf–Mandelbrot–Pareto (ZMP) laws, both globally and in specific subgroups, as measured by higher $$y(r) = \kappa \left(\frac{r}{N - r + 1}\right)^{-\chi} = \kappa\, r^{-\chi}\, (N - r + 1)^{\chi}$$6 and AIC differences favoring Lavalette (Cerqueti et al., 2014, Fontanelli et al., 2016). A notable exception is the Lazio region of Italy, where even after outlier removal, ZMP describes the data better, suggesting sensitivity to administrative/geographic peculiarities (Cerqueti et al., 2014).

4. Functional Properties and Connections to Other Distributions

The Lavalette function articulates a smooth interpolation between a central quasi-exponential regime and power-law–like head/tail behavior, with a built-in cutoff as $$y(r) = \kappa \left(\frac{r}{N - r + 1}\right)^{-\chi} = \kappa\, r^{-\chi}\, (N - r + 1)^{\chi}$$7. This property uniquely contrasts with ZMP-type forms, which lack a self-contained upper cutoff and cannot flexibly fit both head and tail curvature with a single set of exponents.

There is a close analytical relationship between the Lavalette distribution and the lognormal. Over most of the range, Lavalette’s cumulative distribution and the lognormal cdf are nearly indistinguishable—though their tails differ in detail. The analytic moments of the Lavalette distribution exist only up to order $$y(r) = \kappa \left(\frac{r}{N - r + 1}\right)^{-\chi} = \kappa\, r^{-\chi}\, (N - r + 1)^{\chi}$$8; for $$y(r) = \kappa \left(\frac{r}{N - r + 1}\right)^{-\chi} = \kappa\, r^{-\chi}\, (N - r + 1)^{\chi}$$9, the mean is finite, but higher moments diverge rapidly (Fontanelli et al., 2016).

A plausible implication is that for most empirical datasets, distinguishing between Lavalette and lognormal fits may require resolving very extreme (tail) behavior, as the central distributions coincide up to high precision.

5. Empirical Applications and Practical Guidelines

The Lavalette function and its generalizations have been successfully applied across a wide variety of domains:

• Urban economics (city and municipality size distributions; Italian ATI, Bulgarian cities) (Cerqueti et al., 2014, Ausloos, 2014, Ausloos, 2014).
• Linguistics (word frequency ranks), bibliometrics (citation and impact factor distributions), and codon usage (Ausloos, 2014, Fontanelli et al., 2016).
• Social and biological data (adherents to world religions, amino acid mutations, UEFA soccer team coefficients) (Ausloos, 2014, Ausloos, 2014).

A representative summary of the empirical procedure:

1. Fit the data to the standard or generalized Lavalette form (Lav2, Lav3, or higher), selecting parameterization by visual inspection (semi-log or log-log) and hypothesis testing.
2. If systematic outliers are observed at the head or tail, remove king/vice-roy items or employ a super-/hyper-generalized Lavalette with low-/high-rank shifts and independent exponents (Ausloos, 2014).
3. Assess fit quality using $y(r)$0, AIC, or KS–bootstrap. Model selection—when Lav2 and Lav3 are both plausible—can be guided by comparing exponents ($y(r)$1 implies Lav2 suffices; substantial divergence motivates Lav3).
4. Take care with parameter correlations and robustness, especially in heterogeneous datasets or when delineating structural regime shifts (e.g., "top class," "middle class" in football team rankings) (Ausloos, 2014).

6. Strengths, Limitations, and Extensions

Strengths of the Lavalette family include:

• S-shaped (“sigmoid”) form on semi-log plots, adjusting flexibly to head, midrange, and tail curvature with minimal parameters.
• Built-in finite-size cutoff, ensuring the function fades smoothly at high ranks—a limitation of most pure power-law alternatives.
• Modular generalization: higher-parameter forms (up to seven) allow progressively finer adaptation to empirical idiosyncrasies, including inflections on log–log axes.

Limitations:

• Overfitting risks in high-parameterizations; interpretability of exponents and shifts diminishes as parameters multiply (Ausloos, 2014, Ausloos, 2014).
• Inability of the basic two-parameter form to produce inflection points on log–log plots (only achievable via summed or hyper-generalized forms).
• Lack of theoretical grounding for some generalizations (e.g., log-transform tricks, sum-of-Lavalette components).
• Sensitivity to structural outliers; ad hoc division of regime boundaries (“king,” “top class”) remains a subjective, visual process.

In summary, the Lavalette function and its derived forms provide a versatile, empirically validated toolkit for modeling finite rank–size and frequency distributions with curvature. The methodology, analytic properties, and fitting procedures are well established and broadly adopted in modern analysis of scaling phenomena across scientific disciplines (Cerqueti et al., 2014, Fontanelli et al., 2016, Ausloos, 2014, Ausloos, 2014, Ausloos, 2014).