We now focus on quantifying the overall importance of each of the two mechanisms described above, namely, rich-get-richer and external factors. We quantify the effect of each mechanism on a given decision by computing the relative contribution, Grgr and Gext to the expected total number of citations c. That is, of all c citations a decision will have, cGext describes the fraction explained by external factors and cGrgr describes the fraction explained by the rich-get-richer mechanism (according to the model). The fractions Grgr and Gext can be computed analytically for the Bass model from the parameters a, b and c and are given by Ghanbarnejad et al. (2014)
$$ G_{ext}=\frac{a}{bc}\log\left(1+\frac{bc}{a}\right)\, {,} \,G_{rgr}=1-G_{ext}\,\, {.} $$
(3)
The relative contributions quantify whether the use of a decision is driven by the rich-get-richer mechanism, external factors, or a combination of the two. A decision with high Grgr (low Gext) has citations that were obtained gradually throughout time, in a shape that resembles an S curve (see Fig. 4, upper panel for an example). Conversely, a decision with high Gext (low Grgr) characterizes a case where the number of citations were obtained soon after the decision was published, with interest gradually fading (exponential increase after publication, see Fig. 4, lower panel for an example). These contrasting behaviors reflect different ways in which decisions are used by the court.
In Fig. 6 we summarize Gext for all decisions. Analyzing this figure, we note that at around decision rank 50 (approximately 10% of the cases considered) have a value for Gext of 0.5 or larger. The remaining 90% percent of cases are explained primarily by the rich-get-richer mechanism. In fact, for 80% of all decisions, the rich-get-richer mechanism explains more than 70% of all citations. In summary, the majority of citations is well described by a combination of rich-get-richer and external factors, and that the rich-get-richer factors tend to play a larger role in how decisions are cited.
The rich-get-richer mechanism is known to describe well how scientific publications (Wang et al. 2013) and innovations (Bass 1999; 2004) acquire attention. In the context of scientific publications, the reason behind preferential attachment is that the more a scientific publication is known, the more likely it is to be cited, and the more cited it is, the more well known it is.
In legal practice, decisions are decided on the basis of a set of facts specific to the individual case and references should reflect that by specifically referring to earlier cases with a similar fact content. This is the basic content of the classic doctrine Stare DecisisFootnote 4. Therefore, it is not immediately obvious that we should expect to see the rich-get-richer effect in legal practice.
We now make a case to why the rich-get-richer should be observed in this court. Our argument is that with increasing case-load and faster paced societal development, legal practices tend to become increasingly standardized and bureaucratized. This leads to a reliance on general principles of law to decide cases. Herman Oliphant noted this development in the US already in 1928 (Oliphant 1928). Oliphant notes a development from Stare Decisis to what he calls Stare Dictis, on which Judges and Legal Scholars increasingly rely on generalizations from prior cases to form abstract principles of law, which are then more freely applied to new cases.
To see how this practice may have emerged in the European Court of Human Rights, we note that the European Court of Human Rights is overburdened with cases, see Additional file 1. To manage its load, the court relies on various procedures that allows it to decide groups of cases that are perceived by the court to derive from the same underlying (structural problem). By definition, this is Stares Dictis: the underlying argument for deciding for the group is no longer the particularities of the individual cases but rather a common abstract principle of law. Since each of these principles are well represented by a limited number of cases, these cases will be competing for their merit in representing the specific principle. This mechanism is well known to lead to a rich-get-richer effect where the more a case is used, the better it represents the principle.
To illustrate how this dynamics is present in the ECHR, we provide examples of two procedures used by the court to mitigate the overburdeness of the court and that results in cases competing for merit. One procedure is the courts’ use of ‘case law guides’Footnote 5 which is aimed at potential litigants and which sets out to explain the main content of the court’s case law. In these publications, the court refers to its own case law, another practice which may also enforce the rich-get-richer principle by reinforcing the tendency to use case law highlighted in these publications when deciding future cases. Another procedure is the so-called ‘Pilot Judgment Procedure’ Footnote 6 which allows the court to dismiss a whole group of cases at a time. Those cases that are referenced in the pilot judgments will be considered more important and will therefore feature as leading principles (Stare Dicta in Oliphant’s terminology) in the Court’s case law.
In summary, while one should not immediately expect a rich-get-richer effect in legal practice, the specific practical constraints of the European Court of Human Rights can explain its occurrence. We hypothesise that rich-get-richer is observed in other overburden courts that apply Stare Dictis to mitigate it.