Date of Publication :8th May 2024

Abstract:This study investigates the relationship between network structure and spectral properties, with a particular emphasis on social networks built upon natural numbers and arithmetic. It examines how the Barabási-Albert (BA) model and power-law distributions help explain the formation of hubs in social networks. The research utilizes spectral graph theory – the study of graph properties using matrices, eigenvalues, and eigenvectors – to analyse these networks. The thesis demonstrates how both natural number and arithmetic networks exhibit characteristics of scale-free networks, indicated by power-law degree distributions. Furthermore, it explores how the preferential attachment mechanism in the BA model contributes to the emergence of hubs (highly connected nodes) within social networks. Spectral analysis offers insights into community structures, influential nodes, and the overall resilience of social networks. These findings contribute to a deeper understanding of social dynamics and how factors like information diffusion and community formation operate. Keywords: social networks, spectral graph theory, number theory, Barabási-Albert model, power-law distribution, scale-free networks.

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