Describe the difference between models of aggregation, mobility, and percolation. Provide an example for each.

O’Sullivan and Perry, in attempting to create a general typology of simulation models, have identified three basic patterns: aggregation/segregation, mobility, and percolation. Aggregation/segregation models are models which aim to produce a spatial pattern with some degree of spatial homogeneity and spatial heterogeneity at different scales. The “null hypothesis” for this family of models reflects Tobler’s ‘first law of geography,’ which states, approximately, that closer things are more similar to each other than to further things. In other words, other variables tend to vary along with distance: as distance increases, so too will gaps between air humidity or rent per square foot or anything else. This assumption provides a backdrop against which deviations can stand out as interesting and worthy of further investigation. Various aggregation models produce spatial patterns incorporating heterogeneity and homogeneity at different scales: a landscape like the “tiger brush” is simultaneously profoundly heterogenous at one scale and quite homogenous at scales above and below.

Tiger Bush in Niger, showing a spatial pattern incorporating homogeneity and heterogeneity at different scales.

This example also makes a point that the authors emphasize repeatedly: aggregation and segregation are two sides of the same process. This can also be seen in the classic Schelling segregation model. This model is capable of inducing high levels of segregation (or aggregation) through no more than giving each agent a preference to be near agents like it, and the option of moving to maximize their preference.

Mobility models incorporate the use of agents who are capable of moving over a coordinate plane according to a set of rules. The set of rules can be very simple–a purely random walk function–or highly complex, relying on memory and environmental perception. Random walks are often employed as a null hypothesis, and some, particularly Levy flights, with non-finite step lengths, have proven to be remarkably powerful mimics of real world patterns (110). More “realistic” mobility models incorporate some level of self-direction to the agent’s movement: the rules direct the agent towards the pursuit of some resource. These “foraging” rule sets can equip agents with sense perception to locate resources in nearby locations, or a memory that directs them towards locations where resources have been previously found, in addition to other search strategies.

Mobility models have been widely used in different disciplines: mobile agents, for many, define agent-based modeling. An example of a mobility model is the monkey fruit foraging model discussed in the book on page 127. This model explores the relative importance of memory and exploration, it incorporates both a memory function in movement decisions as well as providing a certain level of randomness. Results from this model indicate that ideal strategies lie between the two extremes, employing both memory and exploration: without random moments of exploration, agents become stuck in non-optimal search patterns, returning to picked-over locations rather than going to new locations to find for as-yet-undiscovered jackpots.

Percolation models explore processes of spread. Whether fire through a forest, information through a society, or viruses through a population, many entities move through spatial or networked arrays in ways that evidence certain similarities. In percolation models the medium–be it dirt the ground or people in society–as an important, and heterogenous component of the spread of the fluid. Bits of the medium can be either disconnected, or resistant to the spread, channelling or limiting the percolation process. Percolation models tend to demonstrate a critical value at which the system transitions from a low-spread phase to a high-spread phase: for example with a simple percolation model, randomly selected cells tend to form small, isolated clusters when less than fifty percent of the total possible cells are filled. Above 60% however, the size of clusters expands significantly.

An example of a percolation model is a simple model of forest fire spread. Within a landscape of forested cells, a random cell is ignited, and spreads to adjacent cells (more complicated models include the possibility of non-adjacent spread as well). Even very simple models of fire progression and reforestation are capable of generating landscapes of different age forests that are remarkably like those observed in life.