  Source localization is the inverse problem of graph information dissemination
and has broad practical applications.
  However, the inherent intricacy and uncertainty in information dissemination
pose significant challenges, and the ill-posed nature of the source
localization problem further exacerbates these challenges. Recently, deep
generative models, particularly diffusion models inspired by classical
non-equilibrium thermodynamics, have made significant progress. While diffusion
models have proven to be powerful in solving inverse problems and producing
high-quality reconstructions, applying them directly to the source localization
is infeasible for two reasons. Firstly, it is impossible to calculate the
posterior disseminated results on a large-scale network for iterative denoising
sampling, which would incur enormous computational costs. Secondly, in the
existing methods for this field, the training data itself are ill-posed
(many-to-one); thus simply transferring the diffusion model would only lead to
local optima.
  To address these challenges, we propose a two-stage optimization framework,
the source localization denoising diffusion model (SL-Diff). In the coarse
stage, we devise the source proximity degrees as the supervised signals to
generate coarse-grained source predictions. This aims to efficiently initialize
the next stage, significantly reducing its convergence time and calibrating the
convergence process. Furthermore, the introduction of cascade temporal
information in this training method transforms the many-to-one mapping
relationship into a one-to-one relationship, perfectly addressing the ill-posed
problem. In the fine stage, we design a diffusion model for the graph inverse
problem that can quantify the uncertainty in the dissemination. The proposed
SL-Diff yields excellent prediction results within a reasonable sampling time
at extensive experiments.
