Finding local community structure in networks

Although the inference of global community structure in networks has recently become a topic of great interest in the physics community, all such algorithms require that the graph be completely known. Here, we define both a measure of local community structure and an algorithm that infers the hierarchy of communities that enclose a given vertex by exploring the graph one vertex at a time. This algorithm runs in time $O\left(k^{2}d\right)$ for general graphs when $d$ is the mean degree and $k$ is the number of vertices to be explored. For graphs where exploring a new vertex is time consuming, the running time is linear, $O\left(k\right)$. We show that on computer-generated graphs the average behavior of this technique approximates that of algorithms that require global knowledge. As an application, we use this algorithm to extract meaningful local clustering information in the large recommender network of an online retailer.