Spectral methods for testing cluster structure of graphs
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In the framework of graph property testing, we study the problem of determining if a graph admits a cluster structure. We say that a graph is (k, ϕ)-clusterable if it can be partitioned into at most k parts such that each part has conductance at least ϕ. We present an algorithm that accepts all graphs that are (2, ϕ)-clusterable with probability at least 2/3 and rejects all graphs that are ϵ-far from (2, ϕ^*)-clusterable for ϕ^* <μϕ^2 ϵ^2 with probability at least 2/3 where μ > 0 is a parameter that affects the query complexity. This improves upon the work of Czumaj, Peng, and Sohler by removing a n factor from the denominator of the bound on ϕ^* for the case of k=2. Our work was concurrent with the work of Chiplunkar et al. who achieved the same improvement for all values of k. Our approach for the case k=2 relies on the geometric structure of the eigenvectors of the graph Laplacian and results in an algorithm with query complexity O(n^1/2+O(1)μ·poly(1/ϵ, 1/ϕ, n)).
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