

![0_image_0.png](0_image_0.png)

![0_image_1.png](0_image_1.png)

Table 1: Takeover time for different rectangular grid

![0_image_2.png](0_image_2.png) shapes.

Takeover Time

Grid shape **Avg Min Max**

64 × 64 83.41.9 **79 87**

32 × 128 117.82.4 **114 123** 16 × 256 225.03.8 **219 232**

8 × 512 449.76.3 **437 463**

4 × 1024 937.19.9 **921 960**

2 × 2048 2101.229.9 **2045 2155**

## 2. Anisotropic Selection

This section introduces our contribution: the **Anisotropic**
Selection **method where the neighbors of a cell may be selected with different probabilities.**

## 2.1 Von Neumann Fuzzy Neighborhood

The Von Neumann neighborhood of a cell C **is defined as**
the ball of radius 1 in Manhattan distance centered at C.

Using the Von Neumann Fuzzy Neighborhood**, we assign different probabilities to choose one cell in the neighborhood**
according to the directions (see Figure 4). The probability pc to choose the center cell C **is set at** 15 as for Von Neumann neighborhood. Let pns **denote the probability to choose the**
cell N or S and pew **denote the probability to choose the**
cell E or W. Let α ∈ [−**1; 1] be the control parameter, the**
anisotropy degree. When α = −1, we have pew = 1−pc and pns = 0, when α = 0 we have pns = pew and when α **= 1 we**
have pns = 1 − pc and pew **= 0. Thus, the probabilities** pns and pew **can be described as:**

$$p_{n s}={\frac{(1-p_{e})}{2}}(1+\alpha)$$
$$p_{e w}={\frac{(1-p_{e})}{2}}(1-\alpha)$$

The case α **= 0 correspond to the standard Von Neumann**
neighborhood (pns = pew =
2 5
) and α **= 1 is the limiting**
case for fuzzy neighborhood where pns =
4 5 and pew **= 0. In**
the latter case, there is a vertical neighborhood with three neighbors only1.

2.2 Definition The AS operator exploits the Von Neumann Fuzzy Neighborhood. It works as follows: for each cell k **individuals are**
selected accordingly to the probabilities pns, pew and pc (k stands in the range [1, **5]) within the cell neighborhood. The**
k **individuals participate to a tournament and the winner replaces the old individual if it is fitter or with probability 0.5**
if fitnesses are equal. The control parameter α **is a measure**
of anisotropy: α **= 0 corresponds to standard selection, and**
α **= 1 is the limiting case with the utmost anisotropy. We**
conjecture that selective pressure decreases when anisotropy increases.

## 3. Selective Pressure And Anisotropic Selection

In this section we study the relationship between selective pressure and AS. First, we measure the takeover time for different anisotropy degrees, then we take the study further by considering the growth curve of the best individual.

We measure the effect of different anisotropy degrees on the takeover time. In our experiments, the anisotropic selection is based on tournament selection of size k **= 2 on**
the square grid of side 64. All the 4096 cells are updated synchronously. For each cell, we substitute the selected individual to the one already present in that cell, either systematically if the selected individual is fitter, or with probability 0.5 if fitnesses are equal. For each value of α**, we**
perform 103independent runs. When α **= 0, no direction**
is privileged and AS is equivalent to the standard selection method. When α **= 1, only one direction is exploited, the**
grid can not be filled and the takeover time is not defined.

Figure 5 shows the influence of AS on the takeover time : it increases with the parameter α**. These results are fairly**
consistent with our expectation that selection intensity decreases when the anisotropic degree increases. However, the correlation between takeover and anisotropy is not linear; it rapidly increases after the value α = 0.9.

1using the grid symmetry we will consider α ∈ [0; 1] only