Multi-way spectral partitioning and higher-order cheeger inequalities

Abstract

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheeger's inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero.It has been conjectured that an analogous characterization holds for higher multiplicities: There are k eigenvalues close to zero if and only if the vertex set can be partitioned into k subsets, each defining a sparse cut. We resolve this conjecture positively. Our result provides a theoretical justification for clustering algorithms that use the bottom k eigenvectors to embed the vertices into R k , and then apply geometric considerations to the embedding.We also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of size ≈ n/k and λ k , the kth smallest eigenvalue of the normalized Laplacian, where n is the number of vertices. In particular, we show that in every graph there are at least k/2 disjoint sets (one of which will have size at most 2n/k), each having expansion at most O( √ λ k log k). Louis, Raghavendra, Tetali, and Vempala have independently proved a slightly weaker version of this last result. The √ log k bound is tight, up to constant factors, for the "noisy hypercube" graphs.

Constrained spectral clustering‐based methodology for intentional controlled islanding of large‐scale power systems

“…The theoretical lower bound (defined by the eigenvalue ν 2 /2 of L N [22]) establishes that the quality of the optimal solution (defined by the Cheeger inequality, see [22] for more details) is bounded to 99.87%, whereas the quality of our approximation was 99.84%. …”

“…The matrix L N has the advantage of being scale-independent and, in particular, it allows the comparison of graphs with different weights [18][19][20][21][22].…”

“…To improve the quality of the solution, the vectors x i must be normalised to have length one before applying any clustering technique [19,22]. Thus, we define…”

“…Our methodology has the key advantage of solving the associated eigenproblem only once, even when multiple islands are to be created, which considerably accelerates the determination of an islanding solution, particularly when dealing with large-scale power systems, and potentially increases the quality of the islanding solution [18][19][20][21][22]. 2 1.1478 7.3884 Great Britain-815 3 1.1638 9.7664 Great Britain-815 4 1.1653 9.9415 Great Britain-815 5 1.1753 9.9901 * The superscript indicates the number of created islands…”

“…Nevertheless, recursive bisection is a computationally demanding technique that requires the repeated eigendecomposition of a matrix associated with the graph. Recursive bisection can also affect the quality of the islanding solution [18][19][20][21][22].…”