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 sub graph 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 a cyclic 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 a cyclic nature of directed a cyclic networks. Karrer and Newman introduced an algorithm that does preserve the directed a cyclic property but which may create multiple edges. This paper introduces alternative null-models that maintain the degree sequences, directed a cyclic 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 a cyclic 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.