To understand the structure of a network, it can be useful to break it down into its constituent pieces. This is the approach taken in a multitude of successful network analysis methods, such as motif analysis. These methods require one to enumerate or sample small connected subgraphs of a network, which can be computationally intractable if naive methods are used. Efficient algorithms exists for both enumeration and uniform sampling of subgraphs, and here we generalize the ESU algorithm for a very general notion of multilayer networks. We show that multilayer network subnetwork enumeration introduces nontrivial complications to the existing algorithm, and present two different generalized algorithms that preserve the desired features of unbiased sampling and trivial parallelization. We evaluate these algorithms in synthetic networks and with real-world data, and show that neither of the algorithms is strictly more efficient but rather the choice depends on the features of the data. Having a general algorithm for finding subnetworks makes advanced multilayer network analysis possible, and enables researchers to apply a variety of methods to previously difficult-to-handle multilayer networks in a variety of domains and across many different types of multilayer networks.

翻译：为了理解网络的结构，将其分解为组成单元是一种有效方法，这也是许多成功网络分析方法（如模体分析）所采用的策略。这些方法要求对网络中的小型连通子图进行枚举或采样，若使用朴素方法可能导致计算困难。目前已有针对子图枚举和均匀采样的高效算法，本文针对多层网络这一非常广义的概念，对ESU算法进行了泛化推广。研究表明，多层网络子网络枚举为现有算法引入了非平凡的复杂性，我们提出两种不同的泛化算法，它们保留了无偏采样和简单并行化的理想特性。我们在合成网络和真实数据上对这两种算法进行了评估，结果表明两者并非严格意义上的效率优劣之分，而是取决于数据特征。拥有通用的子网络查找算法使高级多层网络分析成为可能，能使研究者将多种方法应用于先前难以处理的多层网络，这些网络涵盖不同领域和多种类型。