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Unformatted text preview: EPJ manuscript No. (will be inserted by the editor) Modularity-Maximizing Graph Communities via Mathematical Programming Gaurav Agarwal 1 , 2 and David Kempe 1 1 Computer Science Department, University of Southern California, Los Angeles, CA 90089 2 Google Inc., Hyderabad, India the date of receipt and acceptance should be inserted later Abstract. In many networks, it is of great interest to identify communities , unusually densely knit groups of individuals. Such communities often shed light on the function of the networks or underlying properties of the individuals. Recently, Newman suggested modularity as a natural measure of the quality of a network partitioning into communities. Since then, various algorithms have been proposed for (approximately) maximizing the modularity of the partitioning determined. In this paper, we introduce the technique of rounding mathematical programs to the problem of modularity maximization, presenting two novel algorithms. More specifically, the algorithms round solutions to linear and vector programs. Importantly, the linear programing algorithm comes with an a posteriori approximation guarantee: by comparing the solution quality to the fractional solution of the linear program, a bound on the available room for improvement can be obtained. The vector programming algorithm provides a similar bound for the best partition into two communities. We evaluate both algorithms using experiments on several standard test cases for network partitioning algorithms, and find that they perform comparably or better than past algorithms, while being more efficient than exhaustive techniques. 1 INTRODUCTION Many naturally occurring systems of interacting entities can be conveniently described using the notion of net- works. Networks (or graphs ) consist of nodes (or vertices ) and edges between them . For example, social networks [2, 3] describe individuals and their interactions, such as friendships, work relationships, sexual contacts, etc. Hy- perlinked text, such as the World Wide Web, consists of pages and their linking patterns . Metabolic networks model enzymes and metabolites with their reactions . In analyzing and understanding such networks, it is frequently extremely useful to identify communities , which are informally defined as unusually densely connected sets of nodes. Among the benefits of identifying com- munity structure are the following: 1. Frequently, the nodes in a densely knit community share a salient real-world property. For social networks, this could be a common interest or location; for web pages, a common topic or language; and for biologi- cal networks, a common function. Thus, by analyzing structural features of a network, one can infer semantic attributes....
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This note was uploaded on 05/20/2011 for the course CAP 5515 taught by Professor Ungor during the Spring '08 term at University of Florida.
- Spring '08