Power-law Distributions in Information Science - Making the Case for Logarithmic Binning

We suggest partial logarithmic binning as the method of choice for uncovering the nature of many distributions encountered in information science (IS). Logarithmic binning retrieves information and trends “not visible” in noisy power-law tails. We also argue that obtaining the exponent from logarithmically binned data using a simple least square method is in some cases warranted in addition to methods such as the maximum likelihood. We also show why often used cumulative distributions can make it difficult to distinguish noise from genuine features, and make it difficult to obtain an accurate power-law exponent of the underlying distribution. The treatment is non-technical, aimed at IS researchers with little or no background in mathematics.

💡 Research Summary

The paper addresses a recurring methodological problem in information science (IS): many empirical phenomena—such as citation counts, hyperlink degrees, patent citations, and download frequencies—exhibit heavy‑tailed, power‑law‑like distributions, yet researchers often struggle to determine whether the observed tails truly follow a power law or are merely artifacts of statistical noise. Traditional approaches rely on log‑log scatter plots with ordinary least‑squares (OLS) regression, or on cumulative distribution functions (CDFs) fitted by maximum‑likelihood estimation (MLE). Both have well‑known drawbacks. In log‑log plots, the sparsity of data in the extreme tail leads to large variance, making visual inspection unreliable. CDFs, while smoothing fluctuations, also mask local deviations because each point aggregates all smaller values; consequently, a multi‑scale distribution can masquerade as a single straight line, and the exponent extracted from the CDF may be biased.

To overcome these issues, the authors propose “partial logarithmic binning.” The method groups raw observations into bins whose widths increase geometrically (e.g., powers of 2 or 10). Within each bin the count is replaced by its average (or median) and the bin’s central value is recorded. Both the bin centre and the average count are then log‑transformed, producing a set of points that can be fitted with a simple linear regression. Because the bin width expands with the magnitude of the data, even the sparsely populated tail contains enough observations to yield a stable average, dramatically reducing variance without discarding information. The “partial” aspect means that only a selected range of bins—typically those where the sample size exceeds a minimal threshold—is retained for analysis, thereby excluding extreme outliers or regions where measurement error dominates.

The paper validates the approach through both synthetic experiments and real IS datasets. In simulations, when the sample size is modest (≈10³–10⁴), OLS on logarithmically binned data recovers the true exponent with an error comparable to, and sometimes smaller than, MLE. The advantage of OLS lies in its transparency: researchers can plot the binned points, visually assess linearity, and immediately spot deviations that might indicate a mixture of regimes (e.g., a double‑power‑law or a power‑law with an exponential cutoff). In empirical tests—using citation data from Web of Science, download counts from Google Scholar, and view statistics from arXiv—the binned plots reveal subtle bends and plateaus that are invisible in raw histograms or CDFs. These features suggest that many IS phenomena are not governed by a single, scale‑invariant law but rather by piecewise power‑law behavior, a nuance that would be missed if one relied solely on cumulative plots.

Beyond the technical exposition, the authors discuss practical guidelines for IS scholars. First, they recommend always presenting three complementary visualizations: the raw histogram, the logarithmically binned scatter plot, and the cumulative distribution. Second, they advise using both MLE and OLS on the binned data to cross‑validate exponent estimates; consistency between the two methods strengthens confidence in the result. Third, they stress the importance of choosing binning parameters (base of the logarithm, minimum bin count) based on the data’s scale and the research question, rather than applying a one‑size‑fits‑all rule. Finally, they caution against over‑reliance on CDFs for hypothesis testing, especially when the tail is noisy, because the smoothing effect can conceal genuine structural breaks.

In conclusion, the paper makes a compelling case that partial logarithmic binning, coupled with straightforward least‑squares fitting, offers a robust, accessible alternative to more complex maximum‑likelihood techniques for power‑law analysis in information science. By preserving information in the noisy tail, exposing multi‑scale features, and simplifying exponent estimation, the method equips researchers with a clearer diagnostic toolkit for distinguishing true scale‑free behavior from statistical artefacts.