Introduction to the Modeling and Analysis of Complex Systems

12.4. RENORMALIZATION GROUP ANALYSIS TO PREDICT PERCOLATION... 219Mean-field approximation is a technique that ignores spatial relationships amongcomponents. It works quite well for systems whose parts are fully connected orrandomly interacting with each other. It doesn’t work if the interactions are local ornon-homogeneous, and/or if the system has a non-uniform pattern of states. In suchcases, you could still use mean-field approximation as a preliminary, “zeroth-order”approximation, but you should not derive a final conclusion from it.In some sense, mean-field approximation can serve as a reference point to understandthe dynamics of your model system. If your model actually behaves the same way aspredicted by the mean-field approximation, that probably means that the system is notquite complex and that its behavior can be understood using a simpler model.Exercise 12.7 Apply mean-field approximation to the Game of Life 2-D CAmodel. Derive a difference equation for the average state density p t , and predictits asymptotic behavior. Then compare the result with the actual density obtainedfrom a simulation result. Do they match or not? Why?12.4 Renormalization Group Analysis to Predict PercolationThresholdsThe next analytical method is for studying critical thresholds for percolation to occur inspatial contact processes, like those in the epidemic/forest fire CA model discussed inSection 11.5. The percolation threshold may be estimated analytically by a method calledrenormalization group analysis. This is a serious mathematical technique developed andused in quantum and statistical physics, and covering it in depth is far beyond the scopeof this textbook (and beyond my ability anyway). Here, we specifically focus on the basicidea of the analysis and how it can be applied to specific CA models.In the previous section, we discussed mean-field approximation, which defines theaverage property of a whole system and then describes how it changes over time. Renormalizationgroup analysis can be understood in a similar lens—it defines a certain propertyof a “portion” of the system and then describes how it changes over “scale,” not time.We still end up creating an iterative map and then studying its asymptotic state to understandmacroscopic properties of the whole system, but the iterative map is iterated overspatial scales.