Graph structures representing relationships between entries have been studied in statistical analysis, and the results of these studies have been applied to biological networks, whose nodes and edges represent proteins and the relationships between them, respectively. Most of the studies have focused on only graph structures such as scale-free properties and cliques, but the relationships between nodes are also important features since most of the proteins perform their functions by connecting to other proteins. In order to determine such relationships, the problem of network motif discovery has been addressed; network motifs are frequently appearing graph structures in a given graph.

However, the methods for network motif discovery are highly restrictive for the application to biological network because they can only be used to find small network motifs or they do not consider noise and uncertainty in observations. In this study, we introduce a new index to measure network motifs called AR index and develop a novel algorithm called ARIANA for finding large motifs even when the network has noise. Experiments using a synthetic network verify that our method can find better network motifs than an existing algorithm. By applying ARIANA to a real complex biological network, we find network motifs associated with regulations of start time of cell functions and generation of cell energies and discover that the cell cycle proteins can be categorized into two different groups.