Limitations in the spectral method for graph partitioning: Detectability threshold and localization of eigenvectors

Abstract

Investigating the performance of different methods is a fundamental problem in graph partitioning. In this paper, we estimate the so-called detectability threshold for the spectral method with both un-normalized and normalized Laplacians in sparse graphs. The detectability threshold is the critical point at which the result of the spectral method is completely uncorrelated to the planted partition. We also analyze whether the localization of eigenvectors affects the partitioning performance in the detectable region. We use the replica method, which is often used in the field of spin-glass theory, and focus on the case of bisection. We show that the gap between the estimated threshold for the spectral method and the threshold obtained from Bayesian inference is considerable in sparse graphs, even without eigenvector localization. This gap closes in a dense limit.

Localized eigenvectors of the non-backtracking matrix

“…The value of IPR is close to 1 if v is localized and is of O(N −1 ) for a graph with size N if v is an extended vector. Although localized eigenvectors have been studied for decades, there is still much scope for a more thorough theoretical understanding [6,7,8,9,5,10,11,12]. The localized eigenvectors of adjacency matrix A in complex networks that exist owing to the hub structure have been frequently discussed in the literature [13,11,14]; in addition, it is known that the localized eigenvectors of the modularity matrix M also exists because of the hub structure [10].…”

Spectral partitioning in equitable graphs

“…A different ensemble of random graphs with a block structure is the regular stochastic block models, studied in [3,35], where the probability measure is the same as in stochastic block models but a regularity constraint is imposed to all nodes. Moreover, the form of the constraints in (2) allow edges to be drawn independently for each pair of blocks, and, for the case of blocks of the same size, it is possible to sample equitable graphs simply by assembling regular graphs: between each pair of blocks the edges are drawn according to a k-regular graph, where the value of k equals the corresponding element of the connectivity matrix, then the total set of edges is given by the union of the sets of edges for each of the m regular graphs and m * (m − 1) bi-regular graphs.…”

Optimal Segmentation of Directed Graph and the Minimum Number of Feedback Arcs

“…The digraph D-segmentation problem is closely related to the partitioning problem of an undirected graph [9], whose objective is to split a graph into two disconnected parts of comparable sizes by cutting the minimum number of edges. This later problem has been extensively investigated by the replica and the cavity method of statistical physics (see, e.g., [10,11,12,13,14]). A major new feature of the digraph case is that not all the arcs between two layers need to be deleted but only those upward arcs from the lower layer to the higher layer.…”

“…This is the physical reason why the free energy contributions from all the arcs should be subtracted in Eq. (14). The free energy density f is simply f ≡ F N .…”