Spectral Properties of Dense Barab\'asi-Albert Graphs

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The study examines the adjacency spectra of dense Barab\'asi-Albert graphs, revealing that as graph size increases, the expected adjacency matrix converges to a rank-one kernel, with fluctuations forming a random matrix having a calculable variance profile. This finding is significant for understanding the spectral properties of real-world networks characterized by preferential attachment and uneven degree distributions. Notably, the research derives the limiting bulk spectral distribution using the quadratic vector equation approach and identifies the asymptotic location of the leading eigenvalue influenced by the rank-one mean component.

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Exploring the spectral properties of a dense Barabasi-Albert graph reveals insights into the adjacency matrix and its limiting kernel, contributing to the field of mathematics.

Excerpt

arXiv:2606.20816v2 Announce Type: replace Abstract: Preferential attachment graphs model networks whose growth produces highly uneven degree distributions, describing many real-world systems. Their adjacency spectra are important because they allow graph-theoretic questions to be studied through the eigenvalues of matrices. We analyze the adjacency matrix of a dense Barab\'asi-Albert (B-A) multigraph, where the number of edges added at each step is proportional to the final number of vertices. First, we compute the large-$n$ limit of the expected adjacency matrix and show that it is described by a rank-one limiting kernel, viewed as a continuous analogue of the adjacency matrix. After centering and scaling, the fluctuations form a random matrix with a computable variance profile. Using the quadratic vector equation approach, we derive the limiting bulk spectral distribution. We also determine the asymptotic location of the leading eigenvalue generated by the rank-one mean component.