Time-Dependent Random Walks and the Theory of Complex Adaptive Systems

TL;DR

This paper studies how time-dependent random walks behave, especially their survival probabilities near boundaries, revealing insights into complex adaptive systems and phenomena like self-segregation.

Contribution

It introduces analysis of random walks with time-varying probabilities and connects these dynamics to behaviors in complex adaptive systems such as the evolutionary minority game.

Findings

Unbiased walks have maximum survival with large oscillations in probabilities.

Drifted walks perform best with constant probabilities.

Results explain self-segregation and clustering in adaptive systems.

Abstract

Motivated by novel results in the theory of complex adaptive systems, we analyze the dynamics of random walks in which the jumping probabilities are {\it time-dependent}. We determine the survival probability in the presence of an absorbing boundary. For an unbiased walk the survival probability is maximized in the case of large temporal oscillations in the jumping probabilities. On the other hand, a random walker who is drifted towards the absorbing boundary performs best with a constant jumping probability. We use the results to reveal the underlying dynamics responsible for the phenomenon of self-segregation and clustering observed in the evolutionary minority game.