Extended Data Figure 1. Displacement of normalized N-dimensional random walks.

a, Trajectory of a 2-dimensional random-walk represents the absolute abundance of two species x_1, x₂. The initial state is marked by a red circle and the first 100 steps are shown. The solid black line is the 1-dimensional simplex upon which the locations are projected to obtain the relative abundances x̃₁, x̃₂. The dotted lines starting at the origin represent the projection process: all the points in a dotted line have the same relative abundances and they are all projected to the intersection of the dotted line and the simplex (e.g. the solid red and green circles are projected to the red and green open circles, respectively). We define a new coordinate z̃(t) ≡ x̃₂(t)− x̃₁(t) for the location of normalized relative abundance on the simplex. The displacement of the normalized random walk after t steps is then z̃(t) − z̃(0), where z̃(0) is the projected location of the initial state (see, as an example, the distance between the green and the red open circles in a). b, Distributions of displacement of an ensemble of 1,000 random walks after t steps (t=1, 5, 10, 100, 1000). For small t, the displacement distributions depend on t, while for large t (t=100, 1000) the distributions are the same. c, Symbols represent the average displacement of 1,000 N-dimensional normalized random walks (here we set N=50), measured as D_rJSD, and the error-bars represent the standard deviation. Each random walk is forced to stay on the positive orthant, i.e. if $x_i^{(t)}<0$ we set $x_i^{(t)}=0$ . The D_rJSD was calculated using all N coordinates, setting $x_i^{(t)}={10}^{-4}$ as a pseudo count for $x_i^{(t)}=0$. Where t is small, the distance grows with increasing t, however, the distance saturates for large t. The dashed red and green lines represent the average distance between two random locations (green) and between the final locations (x^(t=1,000)) of the random walks (red).