  We review the problem of how to compute the spectral density of sparse
symmetric random matrices, i.e. weighted adjacency matrices of undirected
graphs. Starting from the Edwards-Jones formula, we illustrate the milestones
of this line of research, including the pioneering work of Bray and Rodgers
using replicas. We focus first on the cavity method, showing that it quickly
provides the correct recursion equations both for single instances and at the
ensemble level. We also describe an alternative replica solution that proves to
be equivalent to the cavity method. Both the cavity and the replica derivations
allow us to obtain the spectral density via the solution of an integral
equation for an auxiliary probability density function. We show that this
equation can be solved using a stochastic population dynamics algorithm, and we
provide its implementation. In this formalism, the spectral density is
naturally written in terms of a superposition of local contributions from nodes
of given degree, whose role is thoroughly elucidated. This paper does not
contain original material, but rather gives a pedagogical overview of the
topic. It is indeed addressed to students and researchers who consider entering
the field. Both the theoretical tools and the numerical algorithms are
discussed in detail, highlighting conceptual subtleties and practical aspects.
