Graph filters play an important role in graph signal processing, in which the data is analyzed with respect to the underlying network (graph) structure. As an extension to classical signal processing, graph filters are generally constructed as a polynomial (FIR), or a rational (IIR) function of the underlying graph operator, which can be implemented via successive shifts on the graph. Although the graph shift is a localized operation, it requires all nodes to communicate synchronously, which can be a limitation for large scale networks. To overcome this limitation, this study proposes a node-asynchronous implementation of rational filters on arbitrary graphs. In the proposed algorithm nodes follow a randomized collect-compute-broadcast scheme: if a node is in the passive stage it collects the data sent by its incoming neighbors and stores only the most recent data. When a node gets into the active stage at a random time instance, it does the necessary filtering computations locally, and broadcasts a state vector to its outgoing neighbors. For the analysis of the algorithm, this study first considers a general case of randomized asynchronous state recursions and presents a sufficiency condition for its convergence. Based on this result, the proposed algorithm is proven to converge to the filter output in the mean-squared sense when the filter, the graph operator and the update rate of the nodes satisfy a certain condition. The proposed algorithm is simulated using rational and polynomial filters, and its convergence is demonstrated for various different cases, which also shows the robustness of the algorithm to random communication failures.

中文翻译：

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具有随机节点异步更新的图上的 IIR 过滤

图过滤器在图信号处理中发挥着重要作用，其中数据是根据底层网络（图）结构进行分析的。作为经典信号处理的扩展，图滤波器通常构造为多项式 (FIR) 或底层图算子的有理 (IIR) 函数，可以通过图上的连续移位来实现。虽然图移位是一个局部操作，但它需要所有节点同步通信，这对于大规模网络来说可能是一个限制。为了克服这个限制，本研究提出了在任意图上节点异步实现有理滤波器。在所提出的算法中，节点遵循随机收集-计算-广播方案：如果一个节点处于被动阶段，它会收集其传入邻居发送的数据并仅存储最近的数据。当节点在随机时间实例进入活动阶段时，它会在本地进行必要的过滤计算，并向其传出邻居广播状态向量。对于算法的分析，本研究首先考虑随机异步状态递归的一般情况，并给出其收敛的充分条件。基于此结果，证明了当滤波器、图算子和节点的更新率满足一定条件时，所提出的算法收敛到均方意义上的滤波器输出。所提出的算法使用有理和多项式滤波器进行模拟，并在各种不同情况下证明了其收敛性，