The structure of many complex networks includes edge directionality and weights on top of their topology. Network analysis that can seamlessly consider combination of these properties are desirable. In this paper, we study two important such network analysis techniques, namely, centrality and clustering. An information-flow based model is adopted for clustering, which itself builds upon an information theoretic measure for computing centrality. Our principal contributions include a generalized model of Markov entropic centrality with the flexibility to tune the importance of node degrees, edge weights and directions, with a closed-form asymptotic analysis. It leads to a novel two-stage graph clustering algorithm. The centrality analysis helps reason about the suitability of our approach to cluster a given graph, and determine `query' nodes, around which to explore local community structures, leading to an agglomerative clustering mechanism. The entropic centrality computations are amortized by our clustering algorithm, making it computationally efficient: compared to prior approaches using Markov entropic centrality for clustering, our experiments demonstrate multiple orders of magnitude of speed-up. Our clustering algorithm naturally inherits the flexibility to accommodate edge directionality, as well as different interpretations and interplay between edge weights and node degrees. Overall, this paper thus not only makes significant theoretical and conceptual contributions, but also translates the findings into artifacts of practical relevance, yielding new, effective and scalable centrality computations and graph clustering algorithms, whose efficacy has been validated through extensive benchmarking experiments.