Graph signal processing is an emerging field which aims to model processes that exist on the nodes of a network and are explained through diffusion over this structure. Graph signal processing works have heretofore assumed knowledge of the graph shift operator. Our approach is to investigate the question of graph filtering on a graph about which we only know a model. To do this we leverage the theory of graphons proposed by L. Lovasz and B. Szegedy. We make three key contributions to the emerging field of graph signal processing. We show first that filters defined over the scaled adjacency matrix of a random graph drawn from a graphon converge to filters defined over the Fredholm integral operator with the graphon as its kernel. Second, leveraging classical findings from the theory of the numerical solution of Fredholm integral equations, we define the Fourier-Galerkin shift operator. Lastly, using the Fourier-Galerkin shift operator, we derive a graph filter design algorithm which only depends on the graphon, and thus depends only on the probabilistic structure of the graph instead of the particular graph itself. The derived graphon filtering algorithm is verified through simulations on a variety of random graph models.

中文翻译：

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图形滤波器：极限图形信号处理


图信号处理是一个新兴领域，旨在对网络节点上存在的过程进行建模，并通过该结构上的扩散来解释。迄今为止，图信号处理工作都假设了解图移位算子。我们的方法是研究我们只知道模型的图上的图过滤问题。为此，我们利用 L. Lovasz 和 B. Szegedy 提出的图基理论。我们对新兴的图信号处理领域做出了三项关键贡献。我们首先表明，在从图元绘制的随机图的缩放邻接矩阵上定义的过滤器收敛到在以图元为核的 Fredholm 积分算子上定义的过滤器。其次，利用 Fredholm 积分方程数值解理论的​​经典发现，我们定义了傅里叶-伽辽金移位算子。最后，使用傅里叶-伽辽金移位算子，我们推导了一种仅依赖于图子的图滤波器设计算法，因此仅依赖于图的概率结构而不是特定图本身。通过对多种随机图模型的仿真验证了推导的图子过滤算法。