In this talk I will present a solution of the so-called Gaussian level set percolation problem on random graphs. It addresses the question whether a multivariate Gaussian defined on the vertices of a random graph (with inverse covariance matrix defined, e.g., in terms of a weighted graph Laplacian) exhibits a macroscopic contiguous cluster of sites on which the Gaussian exceeds a given level h, or whether on the contrary all such clusters are finite. Because of the correlations encoded in the multivariate Gaussian, the problem is considerably more complicated than the case of independent Bernoulli percolation, and a full solution has, to the best of my knowledge, not been available in the literature. It turns out that the solution of the problem requires control over the microscopic heterogeneity of the percolation problem, i.e. , over distributions of node dependent percolation probabilities on random graphs.