Figure 2: Stretched exponential path distributions explained by random walks on networks: sequences of outs in American baseball and symbolic dynamics of the Lorenz attractor.

(a) Outs sequences from the half-innings in the first game of the 2012 season between the Kansas City Athletics and the Anaheim Angels (coordinates perturbed slightly for visibility). (b) The empirical frequencies of outs sequences from all 2012 Major League baseball games (blue) do not conform to a power law, as shown by the poor fit of a least-squares regression line (cyan). (c) A random walk model, with stepping probabilities estimated from the same 2012 data. (d) The empirical path probabilities (blue) scale as the third root of rank, with slope close to that predicted by our theory (red). (e) xy projection of a trajectory of the Lorenz system. Any trajectory can be divided into return paths to the plane x=0 travelling in the direction. Qualitatively, each path comprises one or more loops in the right halfspace (R, or x>0), followed by one or more loops in the left halfspace (L, or x<0). (f) The empirical frequencies of different qualitative paths in a very long simulated trajectory (blue) are not power law. (g) A random walk model of the qualitative dynamics with stepping probabilities estimated from the simulated trajectory. (h) The empirical frequencies scale as the square root of rank, with slope very close to that predicted by our theory (red).