A recipe for an optimal power law tailed walk

Abstract

Lévy-like movements, which are an asymptotic power law tailed distribution with an upper cutoff, are known to represent an optimal search strategy in an unknown environment. Organisms seem to show a Lévy walk when μ ≈ 2.0. In the present study, I investigate how such a walk can emerge as a result of the decision making process of a single walker. In my proposed algorithm, a walker avoids a certain direction; this may be related to the emergence of a Lévy walk. Instead of remembering all visited positions, the walker in my algorithm uses and remembers only the direction from which it has come. Moreover, the walker sometimes reconsiders and alters the directions it avoids if it experiences some directional inconsistencies in a series of recent directional moves, i.e., the walker moves in a different direction from the previous one. My results show that a walker can demonstrate power law tailed movements over a long period with an optimal μ.

Functional duality in group criticality via ambiguous interactions

“…First, we briefly discuss Le ´vy walks observed in animal behavior. A Le ´vy walk is an optimal search strategy that balances exploration and exploitation [21][22][23]. It is a special type of a random walk and is in contrast to a Brownian walk, which exploits the surroundings of a specific location, as well as to a ballistic movement, which involves no exploitation and only exploration.…”

“…Overall, macro-scale criticality maximizes the benefits of group membership in various ways [6][7][8]. In contrast, micro-scale criticality is the one that occurs for an element or individual as represented by a Le ´vy walk [16][17][18][19][20][21][22][23], which is an optimal search strategy (i.e., the optimal balance between exploration and exploitation) for a given space. Some researchers have also suggested that Le ´vy walks contribute to smooth communication between individuals in flocks [24,25].…”

Generalized distribution interpolated between the exponential and power-law distributions and applied to pill bug (Armadillidium vulgare) walking data

“…Lévy walks are found in the migratory behavior of organisms at various levels, from bacteria and T cells to humans [1][2][3][4][5][6]. Lévy walks are a type of random walk in which the frequency of occurrence of a linear step length l follows a power law distribution ( ) Generally, Lévy walks with exponents close to two have been observed in the migratory behavior of organisms, and attention has been paid to why such patterns occur [7][8][9][10][11]. The Lévy flight foraging hypothesis (LFFH) [12] [13] states that if food is sparse and randomly scattered and predators have no information (memory) about the food, Lévy walks, as a random search, will be the optimal foraging behavior and will be evolutionarily advantageous [14].…”