Lecture : Local Spectral Methods ( 4 of 4 )

That is, rather than look for the best conductance cluster in the entire graph (which we considered before), look instead for the best conductance cluster that contains a specified seed node and that is not too large. Before proceeding, let’s state a version of Cheeger’s Inequality that applies not just to the leading nontrivial eigenvector of L but instead to any “test vector.” Theorem 1. Let x ∈ Rn s.t. xTD~1 = 0. Then there exists a t ∈ [n] such that S ≡ SweepCutt(x) ≡ {i : xi ≥ t} satisfies x TLx xTDx ≥ φ(S) 2 8 .