Spectral graph multisection through orthogonality

Authors:
Huanyang Zheng;Jie Wu
Affiliations:
Temple University, Philadelphia, PA;Temple University, Philadelphia, PA
Venue:
Proceedings of the 4th MultiClust Workshop on Multiple Clusterings, Multi-view Data, and Multi-source Knowledge-driven Clustering
Year:
2013

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Abstract

Although the spectral modularity optimization algorithm works well in most cases, it is not perfect, due to the characteristic of its recursive bisection, which loses "global" view. In this paper, we propose a spectral multisection algorithm, which cuts the graph into multisections directly, with acceptable time complexity. Instead of using −1 and +1 in the modularity bisection algorithm, we propose using orthogonal vectors of the Hadamard matrix, as to denote the group assignments in the graph division. Then the modularity matrix is "inflated" to higher order through the Kronecker product, which is able to coordinate with the vectors that represent the group assignments of the nodes. The relaxation method is also employed in our algorithm. The eigenvector, which corresponds to the largest eigenvalue of the inflated modularity matrix, reflects the final group assignment of the nodes. The proposed algorithm can be viewed as a natural extension of the original bisection algorithm, which also succeeds its properties. In sparse graphs, the time complexity of the proposed algorithm is O(K4n2), where K is a carefully designed input parameter that reveals the estimated number of communities. Finally, the simulations show that the proposed algorithm achieves outstanding performances in the LFR benchmarks of different settings.