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Unformatted text preview: 1 On Modularity Clustering Ulrik Brandes 1 , Daniel Delling 2 , Marco Gaertler 2 , Robert G¨orke 2 , Martin Hoefer 1 , Zoran Nikoloski 3 , Dorothea Wagner 2 Abstract — Modularity is a recently introduced quality measure for graph clusterings. It has immediately received considerable attention in several disciplines, and in particular in the complex systems literature, although its properties are not well under- stood. We study the problem of finding clusterings with maximum modularity, thus providing theoretical foundations for past and present work based on this measure. More precisely, we prove the conjectured hardness of maximizing modularity both in the general case and with the restriction to cuts, and give an Integer Linear Programming formulation. This is complemented by first insights into the behavior and performance of the commonly applied greedy agglomerative approach. Index Terms — Graph Clustering, Graph Partitioning, Modu- larity, Community Structure, Greedy Algorithm I. INTRODUCTION Graph clustering is a fundamental problem in the analysis of relational data. Studied for decades and applied to many settings, it is now popularly referred to as the problem of partitioning networks into communities. In this line of research, a novel graph clustering index called modularity has been proposed recently . The rapidly growing interest in this measure prompted a series of follow-up studies on various applications and possible adjust- ments (see, e.g., , , , , ). Moreover, an array of heuristic algorithms has been proposed to optimize modularity. These are based on a greedy agglomeration , , on spectral division , , simulated annealing , , or extremal optimization  to name but a few prominent examples. While these studies often provide plausibility arguments in favor of the resulting partitions, we know of only one attempt to characterize properties of clusterings with maximum modularity . In partic- ular, none of the proposed algorithms has been shown to produce optimal partitions with respect to modularity. In this paper we study the problem of finding clusterings with maximum modularity, thus providing theoretical foundations for past and present work based on this measure. More precisely, we proof the conjectured hardness of maximizing modularity both in the general case and the restriction to cuts, and give an integer linear programming formulation to facilitate optimization without enumeration of all clusterings. Since the most commonly employed heuristic to optimize modularity is based on greedy agglomeration, we investigate its worst-case behavior. In fact, we give a graph family for which the greedy approach yields an This work was partially supported by the DFG under grants BR 2158/2- 3, WA 654/14-3, Research Training Group 1042 ”Explorative Analysis and Visualization of Large Information Spaces” and by EU under grant DELIS (contract no. 001907)....
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- Spring '08
- Graph Theory, complete graph, Regular graph, maximum modularity