Rugged Landscapes and Timescale Distributions in Complex Systems

We review a simple model of random walks on a rugged landscape, representing energy or (negative) fitness, and with many peaks and valleys. This paradigm for trapping on long timescales by metastable states in complex systems may be visualized as a terrain with lakes in the valleys whose water level depends on the observational timescale. The “broken ergodicity” structure in space and time of trapping in the valleys can be analyzed using invasion percolation, picturesquely in terms of “ponds and outlets” as water flows to a distant sea. Two main conclusions concern qualitative dependence on the spatial dimension d of the system landscape. The first is that the much-used example of a one-dimensional rugged landscape is entirely misleading for any larger d. The second is that once d is realistically large (above 6 seems to suffice), there are many nonmerging paths to the sea; this may be relevant for the issue of contingency vs. convergence in macro-evolution.