Analysis of Relaxation Processes: The Two-Node Two-Label Case

Several relaxation processes are analyzed in the simple case of two nodes, each having two possible labels. It is shown that the choice of coefficients is very important. For certain values of the coefficients, some processes will have a single nontrivial convergence point regardless of the initial labeling. For other choices of the coefficients, there can be more than one possible convergence point, and different solutions can be obtained for different initial labelings. In the probabilistic approach where the coefficients are predefined in terms of joint probabilities, there are always two nontrivial convergence points for all possible coefficients. The results are also compared to the Bayesian analysis that can be obtained in this simple case of two nodes. Since certain selections of coefficients can give unacceptable results even in this simple case, it can be expected that the proper selection of coefficients will be much more important in the general case involving larger numbers of nodes and labels.