This master's thesis embarks on an extensive exploration of Anderson localization, a profound phenomenon in condensed matter physics. Our investigation encompasses a balanced trifecta, including the calculation of the number of fermions, the analysis of the participation ratio, and an examination of fractal properties. All three facets dynamically evolve as we systematically vary the number of neighbors (\(k\)), offering fresh perspectives on the complex interplay between disorder and quantum mechanics in the context of Anderson localization. The introductory chapter lays the theoretical foundation for our study, delving into the core principles of Anderson localization and its broad significance in the field of physics. We elucidate the mathematical models underpinning our research, with a specific focus on the Anderson model and its manifestation in one dimension. Furthermore, we provide a survey of the wide-ranging applications of the Anderson model in diverse areas of physics. The heart of our thesis lies in the multifaceted analysis of Anderson localization. We employ a comprehensive approach, considering the behavior of the number of fermions, the participation ratio, and fractal properties as we systematically vary \(k\). This holistic perspective allows us to uncover the multifaceted characteristics of our data set, revealing intricate patterns and self-similarities. The results chapter offers a detailed account of our numerical findings. We present a comprehensive overview of how the number of fermions, the participation ratio, and fractal properties respond to variations in \(k\). Our precise calculations provide profound insights into the multifaceted nature of the data set, offering a deeper understanding of its underlying complexity. The dynamic interplay between these three components and \(k\) underscores the system's sensitivity to disorder and its nuanced scaling properties. This thesis not only delves into the captivating realm of Anderson localization but also showcases the power of multifaceted analysis in unraveling the mysteries of complex systems. By examining the behavior of the number of fermions, the participation ratio, and fractal properties as we manipulate \(k\), we contribute to the broader discourse surrounding disorder, quantum mechanics, and multifractality. Our findings have far-reaching implications, not only within condensed matter physics but also in advancing our comprehension of complex systems across various scientific disciplines. In conclusion, this thesis represents an illuminating exploration into Anderson localization and multifaceted analysis. Our holistic approach, considering the number of fermions, the participation ratio, and fractal properties, provides a comprehensive understanding of Anderson localization as we manipulate the number of neighbors (\(k\)). It serves as a testament to the richness of complex systems in physics and the multifaceted perspectives they offer.
