# Levy Flight

Levy flights are a modification of the standard random walk which uses a random step length as well as a random direction. The characteristic of Levy flights is that the step lengths are chosen from a "heavy tailed" probability distribution, which means the decreasing probabilities of longer lengths are not small enough to overpower the increasing lengths; technically, a heavy tailed distribution has infinite variance (possible length).

The probability distribution I've used here is called the Pareto distribution, and it's roughly 1/(x^A). This gives us a more complex search algorithm than our standard random walks, but the advantage is that various length scales are used. The result of having small distances with a high probability and long distances with a low probability is that each search will generally check a few solutions in an area before moving on to a far-off area to repeat the process. This is much more efficient than our constant-length random walks, which have to make a tradeoff between checking nearby and checking far away. Levy flights do both! We do still end up with a parameter to tweak (the constant "A" in the 1/(x^A) behaviour, or alpha in the Wikipedia article), but this can be chosen quite roughly based on the problem, rather than fiddled-with based on how badly the simulation's doing (which is usually the case for random walks' step sizes).

In this example, like with random walks, the fitness of a particular location is shown by how light the grey colour is. This is randomly generated when the page loads.

Each (x, y) point in the square is a solution, and our goal is to find the best (lightest). We start in the centre and move a random distance at a random angle at each step, with distances following a Pareto distribution and angles being uniform.

Like the other algorithms, the edges will wrap around. For simplicity, when this happens we don't draw a line. Sometimes it may look like the search has teleported from one place to another, but it's actually gone off the edge and wrapped around!