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**Unformatted text preview: **Finding local community structure in networks Aaron Clauset Department of Computer Science, University of New Mexico, Albuquerque NM 87131 aaron@cs.unm.edu (Dated: February 21, 2005) 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 ( k 2 d ) 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 ( k ). We show that on computer-generated graphs this technique compares favorably to algorithms that require global knowledge. We also use this algorithm to extract meaningful local clustering information in the large recommender network of an online retailer and show the existence of mesoscopic structure. I. INTRODUCTION Recently, physicists have become increasingly inter- ested in representing the patterns of interactions in com- plex systems as networks [14]. Canonical examples in- clude the Internet [5], the World Wide Web [6], social networks [7], citation networks [8, 9] and biological net- works [10]. In each case, the system is modeled as a graph with n vertices and m edges, e.g., physical con- nections between computers, friendships between people and citations among academic papers. Within these networks, the global organization of ver- tices into communities has garnered broad interest both inside and beyond the physics community. Convention- ally, a community is taken to be a group of vertices in which there are more edges between vertices within the group than to vertices outside of it. Although the par- titioning of a network into such groups is a well-studied problem, older algorithms tend to only work well in spe- cial cases [1115]. Several algorithms have recently been proposed within the physics community, and have been shown to reliably extract known community structure in real world networks [1621]. Similarly, the computer sci- ence community has proposed algorithms based on the concept of flow [22]. However, each of these algorithms require knowledge of the entire structure of the graph. This constraint is prob- lematic for networks like the World Wide Web, which for all practical purposes is too large and too dynamic to ever be known fully, or networks which are larger than can be accommodated by the fastest algorithms [21]. In spite of these limitations, we would still like to make quantita- tive statements about community structure, albeit con- fined to some accessible and known region of the graph in question. For instance, we might like to quantify the local communities of either a person given their social network, or a particular website given its local topology...

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