Fuzzy Generalized Fractal Dimensions on Sierpiński and Social Network Graphs

Fractal theory is the propelled technique to analyze non-linear systems and complex graphs. The quantification of complexity in Sierpiński and social network graphs requires the estimation of Generalized Fractal Dimensions (GFD), where complexity refers to the greater inconsistency and uncertain nature of the systems. This study introduces the fuzzy version of GFD and compares the Fuzzy GFD (FGFD) with the usual GFD for extended Sierpiński and social network graphs. The computational results indicate that the complexity of the graphical structure increases with the number of iterations due to self-similarity, as fractal-based measure values increase with iterations for generalized Sierpiński graphs. The FGFD values are consistently higher than the usual GFD, demonstrating its ability to capture more structural information. Thus, FGFD provides a more effective method for estimating non-linearity and analyzing Sierpiński and real-time graphical networks. The proposed fuzzy-based multifractal measures better quantify complexity levels compared to traditional multifractal measures.