Demonstration of the fundamental relationship between the progress rate g(x) and the career longevity pdf P(x). The progress rate g(x) represents the probability of moving forward in the career to position x + 1 from position x. The small value of g(x) for small x captures the difficulty in making progress at the beginning of a career. The progress rate increases with career position x, capturing the role of the Matthew effect. We plot five g(x) curves with fixed x_c = 10³ and different values of the parameter α. The parameter α emerges from the small-x behavior in g(x) as the power-law exponent characterizing P(x). (Inset) Probability density functions P(x) resulting from inserting g(x) with varying α into Eq. 5. The value α_c ≡ 1 separates two distinct types of longevity distributions. The distributions resulting from concave career development α < 1 exhibit monotonic statistical regularity over the entire range, with an analytic form approximated by the Gamma distribution Gamma(x; α,x_c). The distributions resulting from convex career development α > 1 exhibit bimodal behavior. In the bimodal case, one class of careers is stunted by the difficulty in making progress at the beginning of the career, analogous to a “potential” barrier. The second class of careers forges beyond the barrier and is approximately centered around the crossover x_c on a log–log scale.

Extremely right-skewed pdfs P(x) of career longevity in several high-impact scientific journals and several major sports leagues. We analyze data from American baseball (Major League Baseball) over the 84-year period 1920–2004, Korean Baseball (Korean Professional Baseball League) over the 25-year period 1982–2007, American basketball (National Basketball Association and American Basketball Association) over the 56-year period 1946–2004, and English soccer (Premier League) over the 15-year period 1992–2007, and several scientific journals over the 42-year period 1958–2000. Solid curves represent least-squares best-fit functions corresponding to the functional form in Eq. 5. (A) Baseball fielder longevity measured in at-bats (pitchers excluded): we find α ≈ 0.77, x_c ≈ 2,500 (Korea) and x_c ≈ 5,000 (United States). (B) Basketball longevity measured in minutes played: we find α ≈ 0.63, x_c ≈ 21,000 minutes. (C) Baseball pitcher longevity measured in IPO: we find α ≈ 0.71, x_c ≈ 2,800 (Korea), and x_c ≈ 3,400 (United States). (D) Soccer longevity measured in games played: we find α ≈ 0.55, x_c ≈ 140 games. (E and F) High-impact journals exhibit similar longevity distributions for the “journal career length,” which we define as the duration between an author’s first and last paper in a particular journal. Deviations occur for long careers due to dataset limitations (for comparison, least-square fits are plotted in E with parameters α ≈ 0.40, x_c = 9 years and in F with parameters α ≈ 0.10, x_c = 11 years). These statistics are summarized in SI Appendix (Table S2).