Abstract: Many real-world graphs are large and have some characteristic subgraph patterns, 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, establishing graph neural networks (GNNs) that can detect such patterns and run fast on large graphs is demanding.In this study, we propose a new GNN layer, named \emph{graph homomorphism layer}.It enumerates local subgraph patterns that match the predefined set of patterns $\mathcal{P}^\bullet$, applies non-linear transformations to node features, and aggregates them along with the patterns. By stacking these layers, we obtain a deep GNN model called \emph{deep homomorphism network (DHN)}.The expressive power of the DHN is completely characterised by the set of patterns generated from $\mathcal{P}^\bullet$ by graph-theoretic operations;hence, it serves as a useful theoretical tool to analyse the expressive power of many GNN models.Furthermore, the model runs in the same time complexity as the graph homomorphisms, which is fast in many real-word graphs.Thus, it serves as a practical and lightweight model that solves difficult problems using domain knowledge.