A Mathematical Structure for Examining Graph Algorithms' Computational Complexity in Big-Scale Networks

Abstract

The demand for scalable graph algorithms has increased due to the quick expansion of graph-structured data in fields like social networks, biological systems, and communication infrastructures. A unified mathematical framework for evaluating the computational complexity of graph algorithms in massive networks is presented in this paper. With a focus on Dijkstra's algorithm, PageRank, Kruskal's Minimum Spanning Tree, and Breadth-First Search (BFS), the framework derives theoretical time and space complexity bounds and empirically evaluates them on synthetic and real-world graphs with 1 million to 100 million edges. According to our analysis, algorithm performance scales nonlinearly with network density and edge count, with PageRank experiencing the highest memory and computational demands and BFS demonstrating the most consistent efficiency. Memory profiling demonstrates that for dense or iterative algorithms, space complexity is a crucial limitation. Sensitivity testing also demonstrates that performance is considerably worsened by increasing the average degree, particularly for Dijkstra and PageRank. Significant performance improvements—up to 81% runtime reduction—were achieved through the use of parallelization and algorithmic optimizations, especially for compute-intensive tasks. All things considered, the suggested framework provides precise, topology-aware performance forecasts as well as useful advice for choosing algorithms and designing systems. It establishes the foundation for upcoming extensions to dynamic, streaming, and distributed graph processing models and is a useful tool for researchers and engineers working with large-scale graph analytics.