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The largest eigenvalue and associated eigenvector of a random matrix play a major role in many diverse applications: from multivariate data analysis and graph theory to chemistry and quantum mechanics. However, statistical properties of eigenvectors constitute a less explored area of Random Matrix Theory, and in particular results for sparse matrices are rather scarce.
Building on the pioneering work by Kabashima and collaborators, we compute the statistics of the largest eigenvalue and of the corresponding top eigenvector for some ensembles of sparse symmetric matrices, i.e. (weighted) adjacency matrices of Erdös-Rényi graphs with bounded degree distribution, weighted random regular graphs, and Markov transition matrices. The top eigenpair problem can be recast in terms of the optimisation of a quadratic Hamiltonian on the sphere: introducing the associated Gibbs-Boltzmann distribution and a fictitious inverse temperature, the top eigenvector represents the ground state of the system, which is attained in the zero-temperature limit.
By employing both the cavity and replica methods - borrowed from the statistical mechanics of disordered systems -  we compute the typical largest eigenvalue and the density of the top eigenvector’s components: the two procedures yield consistent results. They rely on self-consistent equations for auxiliary probability density functions, which can be efficiently solved via population dynamics. Their physical meaning, and connection to the underlying graph structure, is elucidated thanks to mutual insights gained from the two complementary methods. Furthermore, this derivation allows us to identify and unfold the individual contributions to the top eigenvector's components coming from nodes of degree ‘k’. The analytical results are corroborated via numerical simulations showing excellent agreement.