Abstract: Many real-world graphs have characteristic subgraphs, such as triangles in social networks, cliques in web graphs, and cycles in molecular networks. Detecting such subgraph patterns is important in many applications; therefore, it is necessary to establish graph neural networks (GNNs) that can detect such patterns. In this study, we propose a new GNN layer, named \emph{graph homomorphism layer}, which enumerates local subgraph patterns (homomorphisms) that match the predefined set of patterns $\mathcal{P}^\bullet$, and then aggregate node features along with the patterns. By stacking the homomorphism layers, we obtain a deep GNN model called \emph{deep homomorphism network (DHN)}. The expressive power of the proposed DHN is completely characterised by the set of patterns ``generated'' from $\mathcal{P}^\bullet$; thus, it can be used as a theoretical tool to analyse the expressive power of many GNN models that locally aggregate information. Moreover, the model is evaluated in the same time complexity as the graph homomorphisms; hence, it runs fast in many practical situations. Therefore, it can also be used as a practical and lightweight model that solves difficult problems using domain knowledge.