How the site degree influences quantum probability on inhomogeneous substrates

We investigate the effect of the node degree and energy $E$ on the electronic wave function for regular and irregular structures, namely, regular lattices, disordered percolation clusters, and complex networks. We evaluate the dependence of the quantum probability for each site on its degree. For bi-regular structures, we prove analytically that the probability $P_k(E)$ of finding the particle on any site with $k$ neighbors is independent of $E$. For more general structures, the dependency of $P_k(E)$ on $E$ is discussed by taking into account exact results on a one-dimensional semi-regular chain: $P_k(E)$ is large for small values of $E$ when $k$ is also small, and its maximum values shift towards large values of $|E|$ with increasing $k$. Numerical evaluations of $P_k(E)$ for two different types of percolation clusters and the Apollonian network suggest that this feature might be generally valid