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MODELING LAW REVIEW IMPACT FACTORS AS AN EXPONENTIAL DISTRIBUTION

Jim Chen*

Of all the miracles available for inspection, none is more striking than the fact that the real world may be understood in terms of the real numbers, time and space and flesh and blood and dense primitive throbbings sustained somehow and brought to life by a network of secret mathematical nerves, the juxtaposition of the two, throbbings on the one hand, those numbers on the other, unsuspected and utterly surprising, almost as if some somber mechanical puppet proved capable of articulated animation by means of a distant sneeze or sigh.

DAVID BERLINSKI, A TOUR OF THE CALCULUS (1996)**

Any sufficiently advanced technology is indistinguishable from magic.

ARTHUR C. CLARKE, PROFILES OF THE FUTURE (1962)***

* *Associate Dean and James L. Krusemark Professor of Law, University of Minnesota Law School <[email protected]>. Carl Bergstrom, Dan L. Burk, Paul H. Edelman, Gil Grantmore, and Brett H. McDonnell provided helpful comments. Jody Ward supplied very capable research assistance. Suzanne Thorpe’s help and the University of Minnesota Law School Library provided invaluable support throughout the composition of this article. Special thanks to Kathleen Chen.* **DAVID BERLINSKI, A TOUR OF THE CALCULUS, at xiii (1996).* ***ARTHUR C. CLARKE, PROFILES OF THE FUTURE: AN INQUIRY INTO THE LIMITS OF THE POSSIBLE 36 (Holt, Rinehart and Winston 1984) (1st ed. 1962); accord, e.g., ERIC S. RAYMOND, THE CATHEDRAL & THE BAZAAR: MUSINGS ON LINUX AND OPEN SOURCE BY AN ACCIDENTAL REVOLUTIONARY 139 (1999).

2 LAW REVIEW IMPACT FACTORS [Vol. ___

I. A BIBLIOMETRIC MANIFESTO

Truth, though elusive, sometimes allows itself to be manifested by mathematical means. Some things, in other words, can be measured and articulated through numbers. Peer-based evaluations of educational quality are among those things. This article proposes some steps toward quantifying the otherwise intangible sense of academic quality. Specifically, this article argues that reputational differences among law schools can and should be rigorously analyzed according to quantitative tools. Law review impact factors and citation rates are among the most salient and least manipulable sources of evidence bearing on the impact of law schools as research institutions. This bibliometric data should therefore figure prominently in any effort to gauge differences in prestige and influence within legal education.

This article proposes the unapologetic embrace of bibliometrics as a fundamental tool of academic assessment. No less than postmodern criticism,1 bibliometric thinking should come naturally to legal academics. As social scientists who have nurtured “something like a third culture” between science and literature in order to improve the circumstances under which real “human beings are living,”2 legal academics enjoy a special opportunity to unite the literary culture’s “canon of works and expressive techniques” with the scientific culture’s “guiding principles of quantitative thought and strict logic.”3

More than one century after Oliver Wendell Holmes declared that “the man of the future is the man of statistics and the master of economics,”4 and nearly two decades after Richard Posner celebrated the decline of law as an autonomous discipline,5

1 ?Cf. STANLEY EUGENE FISH, DOING WHAT COMES NATURALLY: CHANGE, RHETORIC, AND THE PRACTICE OF THEORY IN LITERARY AND LEGAL STUDIES (1990).2 ?C.P. SNOW, THE TWO CULTURES: AND A SECOND LOOK 70 (2d ed. 1965).3 ?Frank Wilczek, The Third Culture, 424 NATURE 997, 997 (2003).4 ?Oliver Wendell Holmes, The Path of the Law, 10 HARV. L. REV. 457, 470 (1897), reprinted in 110 HARV. L. REV. 990, 1001 (1997).5 ?Richard A. Posner, The Decline of Law as an Autonomous Discipline: 1962-1987, 100 HARV. L. REV. 761 (1987).

2006] THE POWER AND THE GLORY 3

the legitimacy ─ perhaps even the primacy ─ of empiricism and quantitative analysis in law lies beyond dispute. This project demands (and secures) a considerable commitment to improving the data and tools available to quantitatively inclined scholars. The “absence of reliable data” lamentably circumscribes the ability of researchers to explain the workings of institutions such as the Supreme Court.6 Having applied mathematical tools to their study of other institutions, legal scholars should now proceed without hesitation to train these instruments on themselves.

The mathematical instinct comes to us at play as well as at work. Within an country whose “legal culture” and perhaps even its very commitment to the “rule of law” arises from its “national past time” [sic],7 baseball combines mathematical rigor with a respect for tradition.8 Just as sabermetrics represents “the search for objective knowledge about baseball,”9 bibliometrics represents the quest to quantify texts, information, and the academic pursuit of truth.10 As with baseball and sabermetrics, the preeminence of mathematics transforms bibliometrics into a hopeful, uplifting enterprise. Baseball as “a ritual . . . of hope” traces its inspirational power to “the clarity of the sport ─ a kind of mathematical absoluteness that spills over into moral

6 ?LEE EPSTEIN ET AL., THE SUPREME COURT COMPENDIUM: DATA, DECISIONS AND DEVELOPMENTS 1 (2d ed. 1996).7 ?Paul Finkelman, Baseball and the Rule of Law, 46 CLEV. ST. L. REV. 239, 239 (1998). See generally STEPHEN JAY GOULD, TRIUMPH AND TRAGEDY IN MUDVILLE: A LIFELONG PASSION FOR BASEBALL (2003).8 ?See Note, Losing Sight of Hindsight: The Unrealized Tradition of Law and Sabermetrics, 117 HARV. L. REV. 1703 (2004); cf. Neil B. Cohen & Spencer Weber Waller, Taking Pop-Ups Seriously: The Jurisprudence of the Infield Fly Rule, 82 WASH. U.L.Q. 453, 458 n.35 (2004) (tracing the origins of sabermetrics to Branch Rickey, Goodbye to Some Old Baseball Ideas, LIFE, Aug. 2, 1954, at 78).9 ?BILL JAMES, THE BILL JAMES BASEBALL ABSTRACT 1987, at 287 (1987); accord, e.g., David Grabiner, The Sabermetric Manifesto, http://www.baseball1.com/bb-data/grabiner/manifesto.html. The word sabermetrics is derived from the acronym of the Society for American Baseball Research (SABR). See http://www.sabr.org.10 ?See generally Farideh Osareh, Bibliometrics, Citation Analysis and Co-Citation Analysis: A Review of Literature I, 46 LIBRI 149 (1996).


absoluteness.”11

Bibliometrics does differ from its sabermetric counterpart in one crucial respect. Whereas law aspires to define itself as the grand “enterprise of subjecting human conduct to the governance of rules,”12 baseball affects nothing besides the happiness of devoted individuals who play or follow “a game with increasingly heightened anticipation of increasingly limited action.”13 “Sports,” after all, “are entertainment.” They “do not often change our world; rather they serve as a distraction from our world.”14

Statistical evaluation of baseball is fun precisely because it is frivolous. As Bill James, the founder of modern sabermetrics, has presciently observed, sabermetric “[a]nalysis is fascinating exactly because nothing is at stake ─ which allows a clearer view of certain issues being played out within a game.”15 Though law takes itself seriously and legal education even more so, we can dedicate ourselves to quantitative rigor without forgetting to have fun. Failing to find the pleasure in serious academic work, even when it is focused on the academy itself, represents one of the most common ways in which a generally humorless legal professoriate misses the “play of intelligence.”16

The playfulness rightly associated with baseball and sabermetrics should infuse legal academics’ efforts to engage in the quantitative evaluation of their own profession. As much as academics rightly despise rankings, higher education demands comparative evaluations of institutions and of individual researchers. “[T]he only coin worth having” in academia remains the “applause” of our peers.17 The dominant source of law school

11 ?John Lahr, Play at the Plate, NEW YORKER, July 22, 2002, at 80, 80.12 ?LON L. FULLER, THE MORALITY OF LAW 122 (rev. ed. 1969).13 ?JOHN IRVING, A PRAYER FOR OWEN MEANY 31 (1989).14 ?DAVID J. BERRI, MARTIN B. SCHMIDT & STACEY L. BROOK, THE WAGES OF WINS: TAKING MEASURE OF THE MANY MYTHS OF MODERN SPORTS 1 (2006).15 ?Bill James, Hits and Errors in Everyday Life, FORBES ASAP, Dec. 1, 1997, http://www.forbes.com/asap/97/1201/096.htm.16 ?See generally Daniel A. Farber, Missing the “Play of Intelligence,” 36 WM. & MARY L. REV. 147 (1994).17 ?Paul A. Samuelson, Economists and the History of Ideas, 52 AM. ECON. REV. 1, 18 (1962).


rankings is the U.S. News and World Report’s annual guide to graduate schools. Within a discipline marked by the absence of objective criteria and, in some quarters, by an ideological commitment to disagreement as a measure of personal virtue, dissatisfaction with the U.S. News rankings may be the lone point on which all members of the legal academy agree.

As though united by U.S. News as a common enemy,18 law professors have risen to the challenge posed by the dismal project of evaluating themselves and their employers. For the better part of the last decade, leading scholars have devoted considerable effort to the project of quantifying the reputation and impact of American law schools.19 In 2006 the Indiana Law Journal devoted an entire issue to dissecting the practice of law school rankings.20

In a celebrated review essay on Michael Lewis’s best-selling report on the business of baseball,21 Paul Caron and Rafael Gely suggested that law schools might do well to apply sabermetric lessons to their own managerial problems.22 This article now proposes its own set of contributions to the burgeoning game of “Moneylaw.”

This paper endeavors to add one more tool to the burgeoning toolkit for improving the quantitative assessment of legal education in the United States. Since humans measure their happiness in relative rather than absolute terms, comparisons between individuals and institutions will necessarily emerge as part of the hierarchical structuring of all human relations. As Karl Marx observed in The German Ideology, human society begins with the production of means to satisfy the need for physical

18 ?Cf. FRANK HERBERT, DUNE 230 (Ace Books, 1990) (1st ed. 1965) (“What do you despise? By this are you truly known.”).19 ?See, e.g., Theodore Eisenberg & Martin T. Wells, Ranking and Explaining the Scholarly Impact of Law Schools, 27 J. LEG. STUD. 373 (1998); Brian Leiter, Measuring the Academic Distinction of Law Faculties, 29 J. LEG. STUD. 451 (2000).20 ?See generally Symposium, The Next Generation of Law School Rankings, 81 IND. L.J. 1 (2006).21 ?MICHAEL LEWIS, MONEYBALL: THE ART OF WINNING AN UNFAIR GAME (2003).22 ?See Paul L. Caron & Rafael Gely, What Law Schools Can Learn from Billy Beane and the Oakland Athletics, 82 TEX. L. REV. 1483 (2004).


sustenance.23 If cooperation to secure the production of means represents the first step toward civilization, however, conflict to control those means surely represents the second.24 Insofar as the competitive treadmill on which all humanity walks condemns us to choose some measure of differences in educational quality and reputation, we should discharge this duty with some degree of competence.

“Law schools don’t have football teams,” proclaims a sign in the offices of the Harvard Law Review. “They have law reviews.” The exceptional institution of the student-edited law review ─ an anomaly, even an embarrassment, within an academic universe in which the professors of every other discipline get to edit the journals and the students are made to teach class ─ offers one way of gauging academic reputation without relying on surveys that rely almost entirely on popularity contests, readily (and routinely) falsified employment data for recent graduates, and affirmative inefficiency in law school spending. To the extent that hiring and promotion in legal academia still depends on article placement and citations within law reviews, measuring relative influence among law reviews offers one way of assessing the prestige of the law schools that publish these journals.

As the total information load in the world explodes,25

“documentary chaos” threatens to overwhelm scholars, librarians, and others who attempt to swallow this seemingly unstoppable stream of information.26 What information science now understands as “Bradford’s Law” predicts that “journal scatter,” or the concentration of influence within a nucleus of core journals, will emerge as the only way of bringing discipline to documentary chaos.27 As a result, a “surprisingly small number of journals 23 ?See Karl Marx, The German Ideology, in THE MARX-ENGELS READER 110, 114 (Robert C. Tucker ed. 1972).24 ?See Jim Chen, The American Ideology, 48 VAND. L. REV. 809, 817 (1995).25 ?See generally, e.g., Peter Lyman & Hal R. Varian, How Much Information 2003?, at <http://www.sims.berkeley.edu/research/projects/how-much-info-2003/execsum.htm#summary> (Oct. 27, 2003).26 ?See SAMUEL C. BRADFORD, DOCUMENTATION 144-59 (1948) (describing “the documentary chaos” created by the explosion of information).27 ?See id.; Samuel C. Bradford, Sources of Information on Specific Subjects, 137 ENGINEERING 85 (Jan. 26, 1934). For an evaluation of the effects


generate the majority of both what is cited and what is published.”28 To the extent “that publication and citation patterns in the scientific literature are highly skewed,” a small core of journals will dominate each scientific discipline.29

To measure the influence of journals within their disciplines, most scientists rely on an admittedly flawed metric, the impact factor.30 Eugene Garfield invented the impact factor in 1955 as part of an exercise to determine which journals should be documented for purposes of tracking scientific citations.31 Despite frequent (and justifiable) criticism,32 the impact factor remains the dominant statistic of bibliometrics. As computed by Garfield, the impact factor for a journal in year y represents all citations to that journal in years y ─ 1 and y ─ 2, divided by the total number of articles published in that journal in years y ─ 1 and y ─ 2.33 Using impact factor rather than total citations equalizes the playing field between large and small journals by according more weight to a journal with fewer but more influential articles than a competitor that has built its reputation through the sheer number of articles it publishes.34

Legal scholars have already begun examining impact factors among law reviews as a measure of influence among these journals and the schools that publish them. John Doyle, of Bradford’s Law in the broader context of information policy and free speech jurisprudence, see Jim Chen, Mastering Eliot’s Paradox: Fostering Cultural Memory in an Age of Illusion and Allusion, 89 MINN. L. REV. 1361 (2005).28 ?Eugene Garfield, The Significant Scientific Literature Appears in a Small Core of Journals, 10:17 THE SCIENTIST 13, 13 (Sept. 2, 1996).29 ?Philip M. Davis, Patterns in Electronic Journal Usage: Challenging the Composition of Geographic Consortia, 63 COLL. & RESEARCH LIBS. 484, 486 (2002).30 ?See generally Eugene Garfield, The History and Meaning of the Journal Impact Factor, 295 J.A.M.A. 90 (2006).31 ?See Eugene Garfield, Citation Indexes to Science: A New Dimension in Documentation Through Association of Ideas, 122 SCIENCE 108 (1955).32 ?See, e.g., Philip G. Altbach, The Tyranny of Citations, INSIDE HIGHER EDUC., May 8, 2006; Delayed Impact, 7 NATURE CELL BIOL. 925 (2005); Not-So-Deep Impact, 435 NATURE 1003 (2005).33 ?See Garfield, supra note 29, at 90.34 ?See id.


director of the law library at Washington and Lee University, has compiled citation statistics for a wide range of English-language law journals, including total citations in law journals, total citations in judicial opinions, and journal impact factors.35 Doyle’s impact factor calculation differs from Garfield’s in that the Doyle measure encompasses seven rather than two years of publication data. In every other respect it represents a straightforward application of the traditional impact factor to law journals. Ronen Perry of the University of Haifa has developed a measure of law school prestige based on the law review citation statistics reported in the Doyle database.36

This article evaluates the underlying mathematics of impact factors among law journals as reported by the Doyle database. It presumes that the reader has access to the Doyle database and can generate a ranked list of 901 journals that are (1) not exclusively online and (2) not “unranked.” I have discovered that law journal impact factors follow the sort of stretched exponential distribution that characterizes many “right-skewed” distributions found in the social and natural sciences. Indeed, a simple exponential distribution ─ that is, a stretched exponential distribution with an exponent of 1 ─ suffices to describe the probability density function of impact factors among law reviews. Mindful of physicist Hermann Weyl’s admonition that any necessary choice between truth and beauty should favor beauty,37

I freely admit to sacrificing some marginal improvement in the descriptive accuracy of my model in order to develop the elegant mathematics of the simple exponential distribution. Further elaboration of this model of law review impact factors as an exponential distribution allows us to calculate the Gini coefficient

35 ?http://lawlib.wlu.edu/LJ/index.aspx36 ?See Ronen Perry, The Relative Value of American Law Reviews: A Critical Appraisal of Ranking Methods, 11 VA. J.L. & TECH. (forthcoming 2006) (available at http://ssrn.com/abstract=806144); Ronen Perry, The Relative Value of American Law Reviews: Refinement and Implementation, 39 CONN. L. REV. (forthcoming 2006) (available at http://ssrn.com/abstract=897063).37 ?Obituaries, 177 NATURE 457, 458 (1956) (quoting Weyl: “My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful.”), quoted in EDWARD O. WILSON, BIOPHILIA 61 (1984).


of a stylized legal literature in which each journal’s influence is expressed by its impact factor. The striking result of this “inequality” computation is that the Gini coefficient of the legal literature modeled according to a simple exponential distribution is exactly 1/2, an outcome that is determined analytically rather than empirically. I conclude that modeling law review impact factors according to an exponential distribution gives rise to a powerful mathematical tool for assessing influence among law journals and law schools.

II. MODELING RIGHT-SKEWED DISTRIBUTIONS: POWER LAWS, STRETCHED EXPONENTIALS, AND SIMPLE EXPONENTIAL DISTRIBUTIONS

Many statistical distributions revolve around a typical size or “scale,” evocative of the so-called “bell curve” that characterizes the standard Gaussian distribution. The histogram of speeds on a highway at any one time, for instance, would cluster near the average speed of cars on that highway. Many other histograms, however, would be significantly right-skewed in the sense that the distribution consists largely of fairly small items but also contains a small number of much larger items. One of the oldest of these right-skewed distributions is Zipf’s law, which describes the rank-frequency of words in natural languages.38 Earthquakes,39 cities,40

38 ?See GEORGE KINSLEY ZIPF, HUMAN BEHAVIOR AND THE PRINCIPLE OF LEAST EFFORT: AN INTRODUCTION TO HUMAN ECOLOGY (1949); GEORGE KINSLEY ZIPF, SELECTIVE STUDIES AND THE PRINCIPLE OF RELATIVE FREQUENCY IN LANGUAGE (1932).39 ?See DONALD L. TURCOTTE, FRACTALS AND CHAOS IN GEOLOGY AND GEOPHYSICS (1997); Didier Sornette et al., Rank-Ordering Statistics of Extreme Events: Application to the Distribution of Large Earthquakes, 101 J. GEOPHYS. RESEARCH 13,883 (1996).40 ?See PAUL KRUGMAN, DEVELOPMENT, GEOGRAPHY, AND ECONOMIC THEORY 42-46 (1997); Felix Auerbach, Das Gesetz der Bevölkerungskonzentration, 59 PETERMANNS GEOGRAPHISCHE MITTEILUNGEN 74 (1913); Xavier Gabaix, Zipf’s Law for Cities: An Explanation, 114 Q.J. ECON. 739 (1999).


meteorites,41 personal incomes,42 and for-profit businesses,43

pages on the World Wide Web,44 and legal precedents45 have also been demonstrated to follow right-skewed distributions.46

Much of the current literature on right-skewed distributions treats these phenomena as expressions of power laws, relationships in which one quantity can be expressed as some power of another.47 In practical terms, replotting a histogram by taking the natural logarithm of the horizontal and vertical axes can reveal a power law distribution at work. If the histogram, thus replotted, “follows quite closely a straight line,” this “remarkable pattern” provides the most vivid visual evidence of a power law.48

In formal terms, the appearance of a straight line on a log-log plot suggests that ln p(x) = ─ α ln x + c, where α and c are constants.49 Taking the exponential of both sides of this equation 41 ?See A.Z. Mekjian, Model of a Fragmentation Process and Its Power-Law Behavior, 64 PHYS. REV. LETTERS 2125 (1990).42 ?See VILFREDO PARETO, COURS D’ECONOMIE POLITIQUE (1896); David G. Champernowne, A Model of Income Distribution, 63 ECON. J. 318 (1953).43 ?See Robert L. Axtell, Zipf Distribution of U.S. Firm Sizes, 293 SCIENCE 1818 (2001); M.H.R. Stanley et al., Zipf’s Plots and the Size Distribution of Firms, 49 ECON. LETTERS 453 (1995).44 ?See BERNARDO A. HUBERMAN, THE LAWS OF THE WEB: PATTERNS IN THE ECOLOGY OF INFORMATION 19-31 (2001); see also Bernardo A. Huberman & Lada A. Adamic, Growth Dynamics of the World-Wide Web, 401 NATURE 131 (1999); Albert Reka et al., Diameter of the World-Wide Web, 401 NATURE 130 (1999).45 ?See David G. Post & Michael B. Eisen, How Long Is the Coastline of the Law? Thoughts on the Fractal Nature of Legal Systems, 29 J. LEG. STUD. 545 (2000).46 ?For legally literate applications of these distributions to law, see Daniel A. Farber, Probabilities Behaving Badly: Complexity Theory and Environmental Uncertainty, 37 U.C. DAVIS L. REV. 145 (2003); Daniel A. Farber, Earthquakes and Tremors in Statutory Interpretation: An Empirical Study of the Dynamics of Interpretation, 89 MINN. L. REV. 848 (2005).47 ?For comprehensive introductions to power laws in a wide variety of natural, physical, and social systems, see PER BAK, HOW NATURE WORKS: THE SCIENCE OF SELF-ORGANIZED CRITICALITY (1996); MANFRED SCHROEDER, FRACTALS, CHAOS, POWER LAWS: MINUTES FROM AN INFINITE PARADISE 103-19 (1991).48 ?M.E.J. Newman, Power Laws, Pareto Distributions and Zipf’s Law, 46 CONTEMP. PHYSICS 323, 323 (2005).49 ?See id.


allows us to re-express the probability density function p(x) in simple terms:

p(x) = Cx─α

with C = ec.50 Of the two constants in a typical power law relationship, the exponent α is more interesting, since the requirement that the cumulative distribution function equal 1 will dictate the value of c once α is computed.51

A brief methodological elaboration is warranted. The power law is so seductively elegant that it dominates the literature on complex adaptive systems. It suffers from the assumption that the fractal, “scale-free” properties of a power law distribution ─ as evidenced by the linear appearance of the log-log plot ─ continue infinitely. Because real-world phenomena occur within finite systems, analysts can scarcely afford to live within an “Asymptopia” where computer simulations, temperatures, gravity, and the like can be extended and observed infinitely.52

The appearance of a curve in a log-log curve suggests that a slightly different model, that of the stretched exponential, is more appropriate.53 The signature characteristic of a stretched exponential is that its histogram is a linear function of the natural logarithm of rank n:54

Ync = ─ a ln n + b

The stretched exponential therefore bears a strong resemblance to the power law. In order to convert a stretched exponential to a linear model, the independent variable (x-axis) alone is plotted on a logarithmic scale. Yn, the histogram, should then be raised to

50 ?See id.51 ?See id. at 323-24.52 ?See Peter Fulde & Richard A. Ferrell, Superconductivity in a Strong-Spin Exchange Field, 135 PHYS. REV. A550 (1964).53 ?See J. Laherrère & D. Sornette, Stretched Exponential Distributions in Nature and Economy: “Fat Tails” with Characteristic Scales, 2 EUR. PHYS. J. B 525, 526 (1998).54 ?See id. at 527.


some power c, with c 1. The exponent c expresses the degree of curvature that can be seen in a log-log plot.55 As c approaches 1, a stretched exponential model approaches an ordinary exponential distribution.56 Many right-skewed distributions that are more appropriately modeled according to a stretched exponential may nevertheless be modeled with a power law for multiple orders of magnitude of Euler’s constant.57

This methodological debate is pertinent to bibliometrics, the science of measuring citations in academic literature. At least one pair of proponents of stretched exponentials favors this finite model over power laws for evaluating citation statistics among physicists.58 Another study of citation patterns within the broader catalog of the Institute for Scientific Information yields a power law relationship.59 My empirical evaluation of law review impact factors reveals that a simple exponential distribution ─ that is, a stretched exponential distribution with an exponent of 1 ─ comes so close to describing the observed data that any marginal improvement in descriptive accuracy attributable to the adoption of a more elaborate stretched exponential model is simply not worth the additional mathematical complexity.

III. LAW REVIEW IMPACT FACTORS FOLLOW AN EXPONENTIAL DISTRIBUTION

A. The Basic Relationship Between Impact Factor and Journal Rank

Law review impact factor among law journals, as measured by John Doyle’s database on the Washington & Lee University School of Law’s website, range from a high of 12.1 for the Yale

55 ?See id. at 528.56 ?See id. at 526. Another way of expressing the relationship is to describe a stretched exponential as a “curve of the form e─ax^b, where a and b are constants. See Newman, supra note 47, at 330.57 ?See Newman, supra note 47, at 330.58 ?See Laherrère & Sornette, supra note 52, at 536-37.59 ?See S. Redner, How Popular Is Your Paper? An Empirical Study of the Citation Distribution, 4 EUR. PHYS. J. B 131 (1998).


Law Journal to 0.1 for the Temple Journal of Law, Science & Environmental Law and 118 other journals. In addition to 781 journals with measurably positive impact factors, another 120 journals report an impact factor rounded down to 0. Graph (1) depicts the basic histogram of observed impact factors as reported by the Doyle database:

It is possible to project, with very satisfying accuracy, the impact factor of any given law journal as the product of the average impact factor in the survey (empirically determined to be 2.0) and the difference of the natural logarithm of the total number of “rankable” journals (a stylized number determined to be 782, representing the total number of journals with a positive

1 Law journal impact factors as observed within the Doyle database


impact factor, 781, plus 1) and the natural logarithm of the journal’s rank. The following histogram, as plotted in graph (2), depicts this model’s projected impact factor as a function of journal rank:

Combining these two historgrams reveals the closeness of fit between the projected and observed impact factors:

2 Projected law review impact factors


Law review impact factors therefore follow an exponential distribution taking the form:

where

R represents rank

N represents the number of rankable journals in the Doyle database, plus 1. A “rankable” journal is one that has a positive impact factor (IF). Formally, for R = 1 to N ─ 1, IFR

3 The combined histograms of observed and projected impact factors


> 0.

IF represents impact factor

is the average impact factor of all journals in the Doyle database.

Only R and IF are true variables. N and are empirically determined constants. N, the number of rankable journals in the Doyle database plus 1, is 782. Since a total of 781 journals have positive impact factors (IF 0.1), N equals 782. , rather remarkably, is equivalent to 2.

If we (1) understand IF as a dependent variable, as a function of the independent variable R, and (2) treat N and as constants, the graphic signature of an exponential distribution emerges, even in the algebraic expression of the relationship between rank and impact factor. Recall the formal expression of a simple exponential distribution:

Yn = ─ a ln n + b

This is nothing more than the full expression of a stretched exponential whose exponent c is 1. Substituting IF for the dependent variable Yn and R for the independent variable n reveals the exponential relationship.

Readers wishing to evaluate this relationship on an empirical basis may wish to extract a spreadsheet from the Doyle database, sorted in declining order of journal impact factor, excluding all strictly online journals and all “unranked” journals. The resulting spreadsheet should report 901 journals, of which 781 boast positive impact factors. All other journals in 782d place report an impact factor of 0, which itself is an artifact of Doyle’s decision to report impact factors to a single place after the decimal point. Pearson’s correlation between observed and projected impact factors generated an R2 value of .990. Pearson’s correlation between observed and projected ranks generated an R2 value of .985. These correlations suggest a very strong relationship between projected and observed impact factors and journal ranks.


Basic algebra permits the use this relationship to project a journal’s rank as a function of its observed impact factor:

Likewise, one can project a journal’s impact factor through its observed rank:

In short, the basic formula expressing impact factor as a function of rank and as following an simple exponential distribution allows us to project impact factor from a journal’s observed rank and to project a rank from a journal’s observed impact factor.

The expression of impact factor as a function of the natural logarithm of a journal’s rank warrants one final observation. The impact factor of the top ranked journal (which observation identifies as the Yale Law Journal), formally expressed as IF1, should be approximately the product of the average impact factor and the natural logarithm of the total number of journals with positive impact factors. If the journal’s rank is 1, the natural logarithm of the rank is 0. This implies that the highest impact factor in any literature is constrained by the average impact factor and the total number of rankable journals in that literature. This finding is consistent with other stretched exponential models. For instance, fitting the Raup-Sepkoski “kill curve,” a histogram of biological extinctions over the 600 million years of multicellular life on earth,60 to a stretched exponential model “suggest[s] that there might be a maximum lifespan of about 350 million[] years” for any single species.61

60 ?See David M. Raup & J. John Sepkoski, Jr., Periodicity of Extinctions in the Geologic Past, 81 PROC. NAT’L ACAD. SCI. 801 (1984); David M. Raup & J. John Sepkoski, Jr., Periodic Extinction of Families and Genera, 231 SCIENCE 833 (1986).61 ?See Laherrère & Sornette, supra note 52, at 534.


B. Individual and Cumulative Share of Influence Within Legal Publishing

Slightly more complex mathematical manipulation of this basic formula allows us to project a journal’s share of influence within the overall body of secondary literature in law. It also yields a good estimate of the cumulative share of influence attributable to the top n law journals.

Both of these measures equate influence with the impact factor. Let us assume an academic universe in which every journal contains the same number of pages and articles. This is a realistic assumption insofar as actual variations in journal size among law reviews, whether measured by pages or by the number of discrete articles, are actually quite small. In such a universe, the total number of citations may be expressed as a multiple of the sum of the impact factors for all the journals:

Total citations =

where K represents a constant expressing the relationship between impact factor and total citations in this stylized academic universe. We can simplify by adopting 1 as our value for K.

This relationship also allows us to express total citations as a multiple of the cumulative distribution function for impact factors. In terms of the calculus, total citations may be expressed as the area underneath the curve depicting the histogram of the impact factor. That function, to repeat, is as follows:

Total citations may be expressed as the integral of this function:

Total citations =

Focus on the far right side of this extended equation. The first figure being integrated is a constant. Recall that N is an


empirically determined constant, whose value we have already set at 782. The average impact factor in the Doyle database has also been empirically determined to have a value of 2. may therefore be re-expressed as ∙ ln(N) ∙ R. Thus:

The integral for a function in the form c ln(x) is well known:

Substituting yields this result:

For the special case R = N, in which case ∫ IF expresses the total number of citations in this stylized universe of legal literature, ∫ IF assumes a beautiful identity:

In other words, given the exponential distribution described above, a body of legal literature consisting of equal-sized journals whose impact factors fall into this sort of distribution will contain exactly citations. Integral calculus thus confirms what instinct tells us: the total influence in this system is equal to the average impact factor times the number of “rankable” (and therefore meaningful) journals in the literature.

Therefore, the cumulative share of citations in secondary legal literature attributable to all journals ranked at R and higher may be expressed thus:


An equivalent version of expression may help make the relationship between R, N, and the cumulative distribution function a bit more intuitive:

As R increases, the total share of impact factors in the secondary legal literature increases not only at the linear rate of , but also

by an additional logarithmic factor equivalent to times the

natural logarithm of .

The share attributable to a single journal ranked R is even simpler:

Or, in terms of the natural logarithm of ratios involving R and N:


To see these relationships, the reader may find it helpful to work through a concrete example. The Michigan Journal of Law Reform and the Michigan Journal of International Law, remarkably enough, come in a tie for 100th place with a common impact factor of 3.8. Each of these journals accounts for .254% of all citations in a stylized legal literature scaled solely according to observed impact factors. In that stylized literature, the top 100 journals account for 39.3 percent of all citations, again measured according to observed impact factors. Using these empirically observed values allows us to predict:

Projected impact factor

Projected rank

Projected share of the 100th ranked journal

Projected cumulative share for journals ranked 1-100

Moreover, where is the “break-even” point in the secondary legal literature? In formal terms: What value of R in the equation

yields a value of .5? Although this

problem may be stated in simple algebraic terms – namely,


– it defies an analytical solution. Since N and ln(N) are empirically determined constants, we would solve for R in an equation that takes the form . Exponentiation of both sides of that equation yields the equivalent expression , where K represents . Because this class of equations has no closed-form solution, any equation taking this form must be solved numerically. 62 Inserting plausible values for R – essentially every value of R between 1 and inclusive – provides a brute-force solution to this numerical problem. 63 The top 146 journals should contain just a shade over half of all citations in my stylized impact factor-scaled legal literature. That is exactly what the empirical numbers report: all journals from the Yale Law Journal to the American Journal of International Law, a total of 146 journals, contain just over half of the value attributable to impact factors among all journals. That field, to repeat, contains 781 journals with nonzero impact factors, plus 120 other journals.

C. Inequality in Legal Scholarship and Legal Academia:Herein of the Gini Coefficient as Applied to Law Review Impact

Factors

62 ?I thank Paul Edelman for informing me of this mathematical limitation and thereby saving me – and my readers – untold hours of frustration.63 ? almost certainly represents the upper limit on the value of R. Another

way of stating the equation at issue is . As long as the conditions N > 0, R > 0, and N ≥ R are all true, then the expression

must be greater than or equal to 0. Therefore, . For purposes of solving this numerical problem by brute force, it makes more intuitive sense to begin with a proposed value of R at and then to move downward, rather than beginning with 1 as the proposed value of R and moving upward toward

.


1. Predicting the Gini coefficient of the secondary legal literature. Integration of the probability density function for law review impact factors enables us to state the formula for the cumulative distribution function for law review influence by rank, on the not altogether outrageous assumption that law reviews are reasonably equal in size. Let F(R) represents the cumulative share for law journals up to rank R, as follows:

F(R) =

where R is the independent variable representing rank and N is the constant representing the number of journals having nonzero impact factors, plus one.

This cumulative distribution function carries strong implications for inequality among law reviews and, derivatively, for the law schools that publish them. The Lorenz area is the famous “inequality zone” formed by an observed income distribution curve combined with a straight line representing hypothetical equality. The Gini coefficient is the ratio between the Lorenz area and the complete area of the triangle formed by the hypothetical equality line, the X-axis, and a vertical line connecting the X-axis to the top of the curve at 100 percent representation of the population.64 That is, rank is set to that of the last member of the relevant population. The area of this triangle, for a population of N members, must be N/2, since the cumulative distribution function at x = N must report a value of 1.

A Gini coefficient of 0 means that there is no departure from perfect equality, since the observed income distribution is coextensive with the equality line, and the resulting Lorenz area is 0. A Gini coefficient of 1 means that since exactly one member of society holds its wealth, the cumulative distribution function for income is a curve hugging the right angle of the equality line’s triangle, which makes the Lorenz area coextensive with the

64 ?See generally CORRADO GINI, VARIABILITÀ E MUTABILITÀ (1912); Philip M. Dixon, Jacob Weiner, Thomas Mitchell-Olds & Robert Woodley, Bootstrapping the Gini Coefficient of Inequality, 68 ECOLOGY 1548 (1987); Gerald J. Glasser, Variance Formulas for the Mean Difference and Coefficient of Concentration. 57 J. AM. STAT. ASS’N 648 (1962).


equality triangle.The hypothetical equality line is easily represented by a

simple ratio:

In other words, for journals ranked through 19 in this hypothetically equal universe, the first 19 journals would hold 19/N of the prestige in the literature.

The first step in computing the Gini coefficient of legal academia requires the calculation of the Lorenz area at issue. The relevant cumulative distribution function is a little inconvenient in that it is expressed “backwards.” The first-ranked item has the highest “income,” whereas the cumulative distribution functions used in most Lorenz and Gini calculations begin with the poorest member of society. This is a quirk of no real consequence. The hypothetical equality line necessarily bisects the quadrangle defined by x = 0 to N and y = 0 to 1. Stating the cumulative distribution function in classic Lorenz/Gini fashion places it under rather than over the line of equality. But the Lorenz area would be the same in either event.

Our Lorenz area (represented by the variable LA) is the difference of integrals ─ the integral of the cumulative distribution function of law review impact factors less the integral of the line of equality. Thus:

Multiplying out the factors of the first integral lets us simplify considerably:

The last two items add to zero. All that is left is the integration of two expressions:


Structuring the equation in this fashion makes it easier to disaggregate expressions rendered in N, which after all is an empirically determined constant, from expressions rendered in R, which is our independent variable:

The first expression is easy. The integral of R is R2/2. The second expression takes a little more work. Recall the general integration formula:

This is exactly the sort of equation at issue. Mercifully, the exponent carries the easy value of 1. Substitution yields the following:

All of these fractions can be combined once we re-express the denominator as 2N:


We are solving this integral for R = N. Replacing all values of R with N simplifies the equation considerably:

Or even more simply:

Recall that the area of equality was, by definition, N/2. Dividing the Lorenz area by the area of equality yields the Gini coefficient:

In other words, the degree of (in)equality in law review publishing is exactly halfway between a legal literature where influence is evenly distributed and a literature where exactly one journal monopolizes all the citations. A Gini coefficient of .5 in the real world, coincidentally, corresponds roughly to the level of inequality found in Peru or Malawi.65

This is a stunning conclusion. Notice that the Gini coefficient of 1/2 is a constant. Remarkably, it does not appear to depend on the fortuity that the mean impact factor in legal publishing is equal to 2. Nor would the Gini coefficient vary if we changed the number of journals in the literature. Rather, this appears to be a property of the fact that impact factors in law publishing obey a simple exponential distribution.

2. Evaluating the model for predicting the Gini coefficient of the secondary legal literature. There are two ways to testing this Gini coefficient model against observed law review impact factors and against impact factors projected by the model described in

65 ?See UNITED NATIONS, 2005 HUMAN DEVELOPMENT REPORT 270-74 (2006).


this article. One method turns on the formal statistical definition of a Gini coefficient. For any unordered distribution of impact factors, the Gini coefficient can be computed as the “relative mean difference,” or the mean of the difference between impact factors for every possible pair of journals, divided by the mean impact factor for the literature as a whole.66 This formula may be expressed in the notation used throughout this article:

Arranging the data in order of increasing impact factors enables the considerable simplification of this formula. 67 Thus:

Although this method does report the Gini coefficient for observed or projected impact factors in the secondary legal literature, it is an arguably less satisfying method for assessing this article’s model for projecting law review impact factors. That model reports impact factors in terms of rank and the number of journals in the literature. Having to convert those terms into projected impact factors before calculating the Gini coefficient adds an additional and less than fully elegant step.

I therefore turn to a second method of evaluation, one involving a relatively straightforward application of Riemann

66 ?See Christian Damgaard & Jacob Weiner, Describing Inequality in Plant Size or Fecundity, 81 ECOLOGY 1139 (2000); Dixon et al., supra note Error:Reference source not found.67 ?See Damgaard & Weiner, supra note Error: Reference source not found; Philip M. Dixon, Jacob Weiner, Thomas Mitchell-Olds & Robert Woodley, Erratum to “Bootstrapping the Gini Coefficient of Inequality,” 69 ECOLOGY 1307 (1988).


sums. 68 Within a ranked-order listing of impact factors, whether observed or projected, the Gini coefficient equals the Lorenz area divided by the area under the line of equality. Both the Lorenz area and the area defined by the line of equality may be approximated with Riemann sums. Dividing the Lorenz area so computed by the area under the line of equality yields the Gini coefficient.

The area defined by the line of equality is more simply expressed. It can be approximated by the Riemann sum taking the form, , where N (again) represents the stylized number of “rankable” journals in the secondary legal literature. Since the series is equal to , the expression is equal to

, or more simply . As N increases, this value

approaches . A more formal statement of this proposition follows:

Since we have been ordering impact factors from the largest observed or projected value to the smallest, we must compute the Lorenz area by subtracting the area under the line of equality from integral of the cumulative distribution function. For either observed or projected impact factors, the Lorenz area may be approximated as a Riemann sum:

68 ?For technical explanations of Riemann sums, Riemann integrals, and their use in approximating the value of definite integrals, see HOWARD ANTON, CALCULUS: A NEW HORIZON 324-27 (6th ed. 1999); BERLINSKI, supra note **, at 244-65.


In this equation, F(r) represents the cumulative distribution function for law review impact factors. As described above, the cumulative distribution function for this article’s projection of impact factors according to a journal’s observed rank takes the following form:

The corresponding cumulative distribution function for observed impact factors takes the following form:

Dividing the Lorenz area by the area underneath the line of equality yields the Gini coefficient. Assembling the component equations developed above yields the following formula:


For projected impact factors, this formula yields a Gini coefficient of .498, as close as one can expect to the .500 predicted by the integration of the cumulative distribution function for projected law review impact factors. For observed impact factors, the Gini coefficient is approximately .518. Therefore, analytical prediction of a Gini coefficient for the secondary legal literature closely approximates the Gini coefficient generated by evaluation of observed impact factors for law journals.

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