Recently, an approach to graph signal processing based on graphons was proposed. Here we show how such a graphon-driven approach to the Fourier transform can be used on graphs sampled from a stochastic block model (SBM). In particular, we show how a Fourier basis can be easily calculated from the block sizes and the block probability matrix. Using perturbation theory, we derive bounds on the sensitivity of the basis with respect to variations in the block sizes. We then consider SBMs constructed from weighted Cayley graphs. When block sizes are equal, a nice Fourier basis can be derived from the representation theory of the underlying group. When block sizes are nearly uniform, we demonstrate that this Fourier basis closely approximates the SBM Fourier basis. For highly non-uniform block sizes, the group-based Fourier basis is no longer applicable, though, as we show, the underlying group still provides partial information about the SBM Fourier basis.

翻译：近年来，有研究提出了基于图论（graphon）的图信号处理方法。本文展示了如何将这种图论驱动的傅里叶变换方法应用于从随机块模型（SBM）采样的图中。具体而言，我们证明了如何从块大小和块概率矩阵轻松计算出傅里叶基。利用摄动理论，我们推导了该基对块大小变化的敏感度界限。随后，我们考虑由加权凯莱图构造的SBM。当块大小相等时，可以从底层群的表示论中推导出良好的傅里叶基。当块大小接近均匀时，我们证明该傅里叶基能紧密逼近SBM傅里叶基。对于高度非均匀的块大小，基于群的傅里叶基不再适用，但如我们所示，底层群仍能为SBM傅里叶基提供部分信息。