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Counting independent motifs in probabilistic networks

Biological networks provide great potential to understand how cells function. Network motifs, frequent topological patterns, are key structures through which biological networks operate. Counting independent (i.e. non-overlapping) copies of a given motif however remains to be a computationally hard problem. Motif counting problem becomes computationally even harder for biological networks as biological interactions are uncertain events. The central challenge behind this problem is that different embeddings of a given motif in a network can share edges. Such edges can create complex computational dependencies between different instances of the given motif. In this paper, we develop a novel algorithm for counting independent copies of a given motif topology in probabilistic biological networks. We propose a novel mathematical model to capture the dependency between each embedding and all the other embeddings, which it overlaps with. We prove the correctness of this model. Our experiments on real and synthetic networks demonstrate that our method counts non-overlapping embeddings in practical time for a broad range of networks with different probability, and topology models as well as reasonable range of network sizes.