Invasion percolation and global optimization.

Flow in a porous medium, a problem with important practical applications, has motivated a large number of theoretical and experimental studies [1]. Aiming to understand the complex interplay between the dynamics of flow processes and randomness characterizing the porous medium, a number of models have been introduced that capture different aspects of various experimental situations. One of the most investigated models in this respect is invasion percolation [2], which describes low flow rate drainage experiments or secondary migration of oil during the formation of underground oil reservoirs [1,3]. When a wetting fluid (e.g., water) is injected slowly into a porous medium saturated with a nonwetting fluid (e.g., oil), capillary forces, inversely proportional to the local pore diameter, are the major driving forces determining the motion of the fluid. The invasion bond percolation (IBP) model captures the basic features of this invasion process. Consider a two dimensional square lattice and assign random numbers pij [ f0, 1g to bonds connecting the nearest neighbor vertices xi and xj . Here pij mimic the randomness of the porous medium, corresponding to the random diameter of the pores, and vertices correspond to throats. Invasion bond percolation without trapping is defined by the following steps: (i) Choose a vertex on the lattice. (ii) Find the bond with the smallest pij connected to the occupied vertex and occupy it. At this point the IBP cluster has two vertices and one bond. (iii) In any subsequent step find the empty bond with the smallest pij connected to the occupied vertices, and occupy the bond and the vertex connected to it. The various versions of the model are useful in matching the simulated dynamics to the microscopic effects acting as fluids with different wetting properties and compressibility are considered. Originally introduced to model fluid flow, lately invasion percolation is viewed as a key model in statistical mechanics, investigated for advancing our understanding of irreversible and nonequilibrium growth processes with generic scaling properties [3]. Finding the shortest spanning tree of a weighted random graph is a well known problem in graph theory [4]. Consider a connected nondirected graph G of n vertices and m bonds (links connecting vertices), with costs pij associated with every bond sxi , xjd. A spanning tree on