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.

翻译：源定位是图信息传播的逆问题，具有广泛的实际应用。然而，信息传播固有的复杂性和不确定性带来了巨大挑战，而源定位问题的不适定性质进一步加剧了这些挑战。近年来，深度生成模型，特别是受经典非平衡热力学启发的扩散模型，取得了显著进展。尽管扩散模型在解决逆问题和生成高质量重建结果方面已被证明具有强大能力，但直接将其应用于源定位存在两个障碍。首先，在大规模网络上计算后验传播结果以进行迭代去噪采样是不可行的，这将导致巨大的计算成本。其次，在该领域的现有方法中，训练数据本身具有不适定性（多对一），因此简单迁移扩散模型只会导致局部最优。为解决这些挑战，我们提出了一种两阶段优化框架——源定位去噪扩散模型（SL-Diff）。在粗粒度阶段，我们设计源接近度作为监督信号来生成粗粒度源预测，旨在高效初始化下一阶段，显著减少其收敛时间并校准收敛过程。此外，该训练方法中引入级联时序信息将多对一映射关系转化为一对一关系，完美解决了不适定问题。在细粒度阶段，我们设计了一种针对图逆问题的扩散模型，能够量化传播中的不确定性。大量实验表明，所提出的SL-Diff在合理采样时间内取得了出色的预测结果。