Multimodality is common in complex Bayesian models and poses a major challenge for Markov chain Monte Carlo (MCMC) convergence assessment. Standard convergence diagnostics, such as the widely used \hat{R} statistic, summarise disagreement between chains through a single global value. While this is useful for detecting that chains should not be pooled, it does not explain how the disagreement is structured. In multimodal posterior distributions, chains may remain in different posterior regions during finite sampling runs, and this behaviour can be scientifically meaningful rather than simply indicating sampling failure. This project develops and evaluates a graph-based convergence diagnostic based on pairwise \hat{R} values. Instead of computing one global diagnostic over all chains, the proposed method computes \hat{R} for every pair of chains and uses these values to construct a graph in which nodes represent chains and edges connect chains that appear to have explored the same posterior region. The connected components of this graph are then used to identify internally consistent groups of chains, isolated chains, and possible posterior “islands”. This provides a more informative summary of MCMC behaviour by distinguishing coherent multimodal structure from unstructured non-convergence or poor mixing. The method is implemented in R using Stan for posterior sampling and igraph for graph construction. The project will evaluate the diagnostic on synthetic posterior geometries, including line-shaped posteriors, Neal’s funnel, and Gaussian mixture targets, as well as Bayesian regression models with mixture likelihoods where multimodality arises from the statistical model. The experiments require repeated independent MCMC runs across different numbers of chains in order to assess the stability of the graph-based summaries and to calibrate the minimum number of chains needed to reliably detect posterior structure. Preliminary work shows that the method can recover known multimodal structure, such as three-component Gaussian mixtures and bimodal mixture-regression posteriors, while also identifying cases where isolated chains indicate problematic posterior geometry rather than meaningful multimodality. The expected outcome is a practical diagnostic framework that complements existing Stan diagnostics by providing structural information about chain disagreement. This can help Bayesian practitioners better interpret MCMC output in multimodal settings and avoid treating all between-chain disagreement as the same type of convergence failure. Main supervisor: Sara Hamis, Assistant Professor, Department of Information Technology, Uppsala University.