  In this paper we study the concentration properties for the eigenvalues of
kernel matrices, which are central objects in a wide range of kernel methods
and, more recently, in network analysis. We present a set of concentration
inequalities tailored for each individual eigenvalue of the kernel matrix with
respect to its known asymptotic limit. The inequalities presented here are of
relative type, meaning that they scale with the eigenvalue in consideration,
which results in convergence rates that vary across the spectrum. The rates we
obtain here are faster than the typical $\O(\frac{1}{\sqrt n})$ and are often
exponential, depending on regularity assumptions of Sobolev type. One key
feature of our results is that they apply to non positive kernels, which is
fundamental in the context of network analysis. We show how our results are
well suited for the study of dot product kernels, which are related to random
geometric graphs on the sphere, via the graphon formalism. We illustrate our
results by applying them to a variety of dot product kernels on the sphere and
to the one dimensional Gaussian kernel.
