Many graphs arising in various information networks exhibit the "power law" behavior -the number of vertices of degree k is proportional to k-# for some positive #. We show that if # > 2.5, the largest eigenvalue of a random power law graph is almost surely$ (1+ o(1))\sqrt{m} $ where m is the maximum degree. Moreover, the klargest eigenvalues of a random power law graph with exponent # have power law distribution with exponent 2# if the maximum degree is sufficiently large, where k is a function depending on #, mand d, the average degree. When 2 < 2.5, the largest eigenvalue is heavily concentrated at cm3-# for some constant c depending on # and the average degree. This result follows from a more general theorem which shows that the largest eigenvalue of a random graph with a given expected degree sequence is determined by m, the maximum degree, and $ \tilde{d} $, the weighted average of the squares of the expected degrees. We show that the k-th largest eigenvalue is almost surely $ (1+ o(1))\sqrt{m_k} $ where mk is the k-th largest expected degree provided mk is large enough. These results have implications on the usage of spectral techniques in many areas related to pattern detection and information retrieval.
CITATION STYLE
Chung, F., Lu, L., & Vu, V. (2003). Eigenvalues of Random Power law Graphs. Annals of Combinatorics, 7(1), 21–33. https://doi.org/10.1007/s000260300002
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