Motifs in directed acyclic networks

Finding motifs is important for understanding the structure of a network in terms of its building blocks. A network motif is a subgraph that appears significantly more often in a real network than expected in randomised networks. This paper looks at motif detection for a special class of directed networks: directed acyclic networks. Normally, randomised networks are obtained using the switching algorithm. This algorithm preserves the in-degree and out-degree of each node. However, it does not preserve the directed acyclic nature of directed acyclic networks. Karrer and Newman introduced an algorithm that does preserve the directed acyclic property but which may create multiple edges. This paper introduces alternative null-models that maintain the degree sequences, directed acyclic property and do not introduce multiple edges. It is shown that there are explicit formulas for the number of occurrences of each possible 3-node pattern in such random networks. Even though the different random network models result in networks with different properties, the patterns that are keyed as network motifs in three real-world directed acyclic networks do not depend on the choice of null-model. However, when using the switching model as a null-model, sometimes anti-motifs are found that contain directed cycles.