In this paper, the concepts and techniques for global graph clustering are examined, or the process of locating related clusters of vertices within a graph. We introduce the construction of a graph clustering technique based on an eigenvector embedding and a local graph clustering method based on stochastic exploration of the graph. Then, the developed implementations of both methods are presented and assessed in terms of performance. In addition, the difficulties associated with assessing clusterings and benchmarking cluster algorithms are explored where PageRank and EigEmbed algorithms are utilized. The experiments show that the EigEmbed outperformed PageRank across all experiments as it detected more communities with the same number of clusters. Ultimately, we apply both algorithms to a real-world graph representing Twitter network and the followers and tweets therein.