Eigenvalues of Random Power Law Graphs

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. Moreover, the k largest eigenvalues of a random power law graph with exponent β have power law distribution with exponent 2β − 1 if the maximum degree is sufficiently large, where k is a function depending on β, m and d, the average 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 the k-th largest eigenvalue is almost surely (1 + o(1)) √ mk 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.