Random walk model that universally generates inverse square Lévy walk by eliminating search cost minimization constraint

The L\'evy walk, a type of random walk characterized by linear step lengths that follow a power-law distribution, is observed in the migratory behaviors of various organisms, ranging from bacteria to humans. Notably, L\'evy walks with power exponents close to two are frequently observed, though their underlying causes remain elusive. This study introduces a simplified, abstract random walk model designed to produce inverse square L\'evy walks, also known as Cauchy walks and explores the conditions that facilitate these phenomena. In our model, agents move toward a randomly selected destination in multi-dimensional space, and their movement strategy is parameterized by the extent to which they pursue the shortest path. When the search cost is proportional to the distance traveled, this parameter effectively reflects the emphasis on minimizing search costs. Our findings reveal that strict adherence to this cost minimization constraint results in a Brownian walk pattern. However, removing this constraint transitions the movement to an inverse square L\'evy walk. Therefore, by modulating the prioritization of search costs, our model can seamlessly alternate between Brownian and Cauchy walk dynamics. This model has the potential to be utilized for exploring the parameter space of an optimization problem.

Inverse square Levy walk emerging universally in goal-oriented tasks

The Levy walk in which the frequency of occurrence of step lengths follows a power-law distribution, can be observed in the migratory behavior of organisms at various levels. Levy walks with power exponents close to 2 are observed, and the reasons are unclear. This study aims to propose a model that universally generates inverse square Levy walks (called Cauchy walks) and to identify the conditions under which Cauchy walks appear. We demonstrate that Cauchy walks emerge universally in goal-oriented tasks. We use the term "goal-oriented" when the goal is clear, but this can be achieved in different ways, which cannot be uniquely determined. We performed a simulation in which an agent observed the data generated from a probability distribution in a two-dimensional space and successively estimated the central coordinates of that probability distribution. The agent has a model of probability distribution as a hypothesis for data-generating distribution and can modify the model such that each time a data point is observed, thereby increasing the estimated probability of occurrence of the observed data. To achieve this, the center coordinates of the model must be close to those of the observed data. However, in the case of a two-dimensional space, arbitrariness arises in the direction of correction of the center; this task is goal oriented. We analyze two cases: a strategy that allocates the amount of modification randomly in the x- and y-directions, and a strategy that determines allocation such that movement is minimized. The results reveal that when a random strategy is used, the frequency of occurrence of the movement lengths shows a power-law distribution with exponent 2. When the minimum strategy is used, the Brownian walk appears. The presence or absence of the constraint of minimizing the amount of movement may be a factor that causes the difference between Brownian and Levy walks.

Lévy walks derived from a Bayesian decision-making model in non-stationary environments

L\'evy walks are found in the migratory behaviour patterns of various organisms, and the reason for this phenomenon has been much discussed. We use simulations to demonstrate that learning causes the changes in confidence level during decision-making in non-stationary environments, and results in L\'evy-walk-like patterns. One inference algorithm involving confidence is Bayesian inference. We propose an algorithm that introduces the effects of learning and forgetting into Bayesian inference, and simulate an imitation game in which two decision-making agents incorporating the algorithm estimate each other's internal models from their opponent's observational data. For forgetting without learning, agent confidence levels remained low due to a lack of information on the counterpart and Brownian walks occurred for a wide range of forgetting rates. Conversely, when learning was introduced, high confidence levels occasionally occurred even at high forgetting rates, and Brownian walks universally became L\'evy walks through a mixture of high- and low-confidence states.