Voting rules

or M if M is the von Neumann, von Neumann with center-site excluded, Moore or Moore with center-site excluded neighborhoods, respectively. 

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The many-to-one mapping that may arise in voting rules. Several different combinations of states may give rise to the same state in the next cycle. 

The coagulation of cells in a five-state cellular automata model as a result of the use of voting rules. 

Coagulation in a two-state system with random voting. 

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Of particular interest is the long-term behavior of voting-rule systems, which turns out to very strongly depend on the initial density of sites with value cr = 1 (= p). While all such systems eventually become either stable or oscillate with period-two, they approach this final state via one of two different mechanisms either through a percolation or nucleation process. Figure 3.60 shows a few snapshots of a Moore-neighborhood voting rule > 4 for p = 0.1, 0.15, 0.25 and 0.3. 

Fig. 3.61 Some snapshots of the evolution of a 5-neighbor von Neumann neighborhood percolating voting rule V3 the initial densities are (a) p = 0.35 < pc and (b) p = 0.50 = pc-...

Voting rule systems approaching their final state through percolation display much of this same behavior. There is a critical initial density, pc, such that for p > Pc, a connected network of cr = 1 valued sites percolates through the lattice. If p < pc, on the other hand, a similar a 0 valued lattice-spanning structure percolates through the lattice. In either case, the set of sites with the minority value consists of a disconnected sea of isolated islands, and a finite number of islands persist to the system s final state as long as the initial density p > 0. 

Figure 3.61 shows a few steps in the contrasting evolutions of the percolating voting-rule system V3 when p < pc and p Pc-... 

That these are the only two possible fates of voting-rule systems is due to a theorem by Goles [goles85] it is discussed in chapter 5. 

Fig. 3.62 An example of overlapping convex-hulls in the 4-site (center-site excluded) von Neumann neighborhood voting rule with threshold = 2 see equation 3.77.

Table 3.7 Critical densities pc and types of processes for a ft w selectefi voting rules defined on von Neumann and Moore neighboriioods.

Fig. 3.64 Some snapshots of the evolution of a twisted form of the 9-neighbor voting rule M5 (see equation 3.79) the initial density p = 0.50.

Voting rules are irreversible as several different configurations of cells in the neighborhood can lead to the same set of states in the following cycle. In other words, there is a many-to-one mapping of states (Figure 6.12). 

The application of voting rules leads unavoidably to pattern formation, since its effect is to delete cells in a state that is poorly represented in the local environment and replace them by cells in a state that is common therefore, cells of the same state collect in neighboring regions of the simulation space. Figure 6.13 shows an example. 

The Circuit Court panel, by a 2-1 vote, ruled that the Controlled Substances Act was intended to apply to interstate commercial trafficking in illicit drugs. It did not apply to drugs produced entirely within a state for personal medical purposes. The panel ordered the case returned to the district court for final determination, with a preliminary injunction issued barring enforcement of the CSA. Meanwhile, federal prosecutors appealed to a full panel of the Ninth Circuit. 

Coomans, D. and Massart, D.L. (1982) Alternative K-nearest neighbour rules in supervised pattern recognition. Part 1 K-nearest neighbour classification by using alternative voting rules Anal. Chim. Acta 136, 15-27. 

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