MOTIVATION Laplacian matrices capture the global structure of networks and are widely used to study biological networks. However, the local structure of the network around a node can also capture biological information. Local wiring patterns are typically quantified by counting how often a node touches different graphlets (small, connected, induced sub-graphs). Currently available graphlet-based methods do not consider whether nodes are in the same network neighbourhood. To combine graphlet-based topological information and membership of nodes to the same network neighbourhood, we generalize the Laplacian to the Graphlet Laplacian, by considering a pair of nodes to be 'adjacent' if they simultaneously touch a given graphlet. RESULTS We utilize Graphlet Laplacians to generalize spectral embedding, spectral clustering and network diffusion. Applying Graphlet Laplacian-based spectral embedding, we visually demonstrate that Graphlet Laplacians capture biological functions. This result is quantified by applying Graphlet Laplacian-based spectral clustering, which uncovers clusters enriched in biological functions dependent on the underlying graphlet. We explain the complementarity of biological functions captured by different Graphlet Laplacians by showing that they capture different local topologies. Finally, diffusing pan-cancer gene mutation scores based on different Graphlet Laplacians, we find complementary sets of cancer-related genes. Hence, we demonstrate that Graphlet Laplacians capture topology-function and topology-disease relationships in biological networks. AVAILABILITY AND IMPLEMENTATION http://www0.cs.ucl.ac.uk/staff/natasa/graphlet-laplacian/index.html. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.

中文翻译：

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Graphlet Laplacians用于拓扑功能和拓扑疾病关系。

动机拉普拉斯矩阵捕获网络的整体结构，并被广泛用于研究生物网络。但是，节点周围网络的本地结构也可以捕获生物信息。通常通过计算节点多长时间接触一次不同的小图（小的，连接的，诱导的子图）来量化局部布线方式。当前可用的基于图集的方法不考虑节点是否在同一网络邻居中。为了将基于图的拓扑信息和节点成员资格组合到同一个网络邻居，我们将拉普拉斯算子概括为图小子拉普拉斯算子，方法是将一对节点同时触摸给定的图子，将它们视为“相邻”节点。结果我们利用Graphlet Laplacians来概括频谱嵌入，频谱聚类和网络扩散。应用基于Graphlet Laplacian的光谱嵌入，我们直观地证明了Graphlet Laplacians捕获生物学功能。通过应用基于Graphlet Laplacian的光谱聚类可以量化此结果，该聚类发现了丰富的生物学功能相关的簇，这些簇依赖于底层的Graphlet。我们通过说明不同的Graphlet Laplacian捕获的生物功能的互补性来说明它们捕获了不同的局部拓扑。最后，根据不同的Graphlet Laplacians扩散泛癌基因突变评分，我们发现了与癌症相关的基因的互补集合。因此，我们证明了Graphlet Laplacians捕获了生物网络中的拓扑功能和拓扑疾病关系。可用性和实现http://www0.cs.ucl.ac.uk/staff/natasa/graphlet-laplacian/index.html。