Cyclic structures are fundamental topological features in graphs, playing critical roles in network robustness, information flow, community structure, and various dynamic processes. Algorithmic tools that can efficiently probe and analyze these cyclic topologies are increasingly vital for tasks in graph mining, network optimization, bioinformatics, and social network analysis. A core primitive for quantitative analysis of cycles is finding the Minimum Weight Cycle (MWC), representing the shortest cyclic path in a weighted graph. However, computing the MWC efficiently remains a challenge, particularly compared to shortest path computations. This paper introduces a novel deterministic algorithm for finding the MWC in general weighted graphs. Our approach adapts the structure of Dijkstra's algorithm by introducing and minimizing a \textit{composite distance} metric, effectively translating the global cycle search into an iterative node-centric optimization. We provide a rigorous proof of correctness based on loop invariants. We detail two mechanisms for accelerating the search: a provable node discarding technique based on intermediate results, and a highly effective graph pruning heuristic. This heuristic dynamically restricts the search to relevant subgraphs, leveraging the principle of locality often present in complex networks to achieve significant empirical speedups, while periodic resets ensure global optimality is maintained. The efficiency of the proposed MWC algorithm enables its use as a core component in more complex analyses focused on cyclic properties. We illustrate this through a detailed application case study: accelerating the computation of the Loop Modulus, a measure of cycle richness used in advanced network characterization. Our algorithm dramatically reduces the runtime of the iterative constraint-finding bottleneck in this computation.