Fig. 3.

Power-law behaviors of “exact” multiplication factors from case (iii) local PWAs. a Log-log plots of the “exact” multiplication factors (${\mu }_P^{〈N_{\mathit{ID}}=200〉}\left[\Delta L\right]$, ordinate) against the local PWA size (∆L, abscissa), showing nearly perfect power-law behaviors. Although this panel shows the results under λ _I : λ _D = 1 : 1 only, the power-law approximation is actually very good also under λ _I : λ _D = 1 : 3 and λ _I : λ _D = 3 : 1 (Additional file 1: Table S2). Panels b and c show the power-law exponent (γ) and the coefficient (A), respectively, as functions of the distance ((λ _I + λ _D)(t − t _I) indels/site, abscissa) and the rate ratio (λ _I : λ _D, different curves). Here, we assumed the approximate power-law relation, ${\mu }_P^{〈N_{\mathit{ID}}=200〉}\left[\Delta L\right]\approx A{\left(\Delta L\right)}^{-\gamma }$. (See Additional file 1: Table S2 also for the results of correlation and regression analyses.) Note that the results apply also to case (ii) local PWAs with due modifications