Eigenvalues of Random Power Law Graphs (DRAFT)

Many graphs arising in various information networks exhibit the “power law” behavior – the number of vertices of degree k is proportional to k−β for some positive β. We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1+o(1)) √ m where m is the maximum degree. When 2 < β < 2.5, the largest eigenvalue is heavily concentrated at cm3−β for some constant c depending on β 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 d̃, the weighted average of the squares of the expected degrees. We show that λ is almost surely (1 + o(1))max{d̃,√m} provided some minor condition is satisfied. Our results have implications on the usage of spectral techniques in many areas related to pattern detection and information retrieval.