Detecting local network motifs

Studying the topology of so-called real networks, that is networks obtained from sociological or biological data for instance, has become a major field of interest in the last decade. One way to deal with it is to consider that networks are built from small functional units called motifs, which can be found by looking for small subgraphs whose numbers of occurrences in the whole network are surprisingly high. In this article, we propose to define motifs through a local overrepresentation in the network and develop a statistic to detect them without relying on simulations. We then illustrate the performance of our procedure on simulated and real data, recovering already known biologically relevant motifs. Moreover, we explain how our method gives some information about the respective roles of the vertices in a motif.

💡 Research Summary

The paper introduces a novel statistical framework for detecting network motifs based on local over‑representation rather than global frequency excess. Traditional motif detection methods rely on counting how often a small subgraph appears in the whole network and then comparing this count to a distribution obtained from thousands of simulated random graphs that preserve certain global properties (e.g., degree sequence). While statistically sound, this approach is computationally expensive and only captures motifs that are globally abundant.

In contrast, the authors define a motif as a subgraph that is significantly enriched in the neighbourhood of a particular vertex. For each vertex (v) and each candidate (k)-node subgraph (H), they consider the event that a copy of (H) lies entirely within the (r)-hop neighbourhood of (v). Under a null model given by the configuration model (a random graph that preserves the degree sequence of the observed network), they derive an analytical expression for the expected number (\lambda_v) of such local occurrences. By assuming that the actual count (X_v) follows a Poisson distribution with mean (\lambda_v), they construct a standardized statistic

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