Fluctuation of the Largest Eigenvalue of a Kernel Matrix with application in Graphon-based Random Graphs

In this article, we explore the spectral properties of general random kernel matrices [K(U_i,U_j)]_1≤ i≠ j≤ n from a Lipschitz kernel K with n independent random variables U_1,U_2,…, U_n distributed uniformly over [0,1]. In particular we identify a dichotomy in the extreme eigenvalue of the kernel matrix, where, if the kernel K is degenerate, the largest eigenvalue of the kernel matrix (after proper normalization) converges weakly to a weighted sum of independent chi-squared random variables. In contrast, for non-degenerate kernels, it converges to a normal distribution extending and reinforcing earlier results from Koltchinskii and Giné (2000). Further, we apply this result to show a dichotomy in the asymptotic behavior of extreme eigenvalues of W-random graphs, which are pivotal in modeling complex networks and analyzing large-scale graph behavior. These graphs are generated using a kernel W, termed as graphon, by connecting vertices i and j with probability W(U_i, U_j). Our results show that for a Lipschitz graphon W, if the degree function is constant, the fluctuation of the largest eigenvalue (after proper normalization) converges to the weighted sum of independent chi-squared random variables and an independent normal distribution. Otherwise, it converges to a normal distribution.