Formal convergence analysis for bio-inspired topology control in MANETs

We present a convergence analysis of a genetic algorithm based topological control mechanism for the decision making process of evolutionary and autonomous systems that adaptively reconfigures spatial configuration in mobile ad hoc networks (MANETs). Mobile nodes adjust their speed and direction using information collected from the local neighborhood environment in unknown geography. We extend the stochastic model of the genetic operators (i.e., selection, crossover and mutation) called the dynamical system model that represents the behavior of a single node's decision mechanism in the network viewed as a stochastic variable. We introduce an ergodic homogeneous Markov chain to analyze the convergent nature of multiple mobile nodes running our algorithm, called the Force-based Genetic Algorithm (FGA). Here, a state represents an instantaneous spatial configuration of nodes in a MANET. It is shown that the Markov chain model of our FGA is ergodic and its convergence is shown using Dobrushin's contraction coefficients. It is observed that scenarios where nodes have small communication ranges compared to their movement range converge quicker than larger ones due the limited information they have of their neighborhood, making movement decisions simpler, thus conserving energy.