Sparse eigenvectors of graphs

In order to analyze signals defined over graphs, many concepts from the classical signal processing theory have been extended to the graph case. One of these concepts is the uncertainty principle, which studies the concentration of a signal on a graph and its graph Fourier basis (GFB). An eigenvector of a graph is the most localized signal in the GFB by definition, whereas it may not be localized in the vertex domain. However, if the eigenvector itself is sparse, then it is concentrated in both domains simultaneously. In this regard, this paper studies the necessary and sufficient conditions for the existence of 1, 2, and 3-sparse eigenvectors of the graph Laplacian. The provided conditions are purely algebraic and only use the adjacency information of the graph. Examples of both classical and real-world graphs with sparse eigenvectors are also presented

A Spectral Graph Uncertainty Principle

The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain. Using the eigenvectors of the graph Laplacian as a surrogate Fourier basis, quantitative definitions of graph and spectral "spreads" are given, and a complete characterization of the feasibility region of these two quantities is developed. In particular, the lower boundary of the region, referred to as the uncertainty curve, is shown to be achieved by eigenvectors associated with the smallest eigenvalues of an affine family of matrices. The convexity of the uncertainty curve allows it to be found to within $\varepsilon$ by a fast approximation algorithm requiring $O(\varepsilon^{-1/2})$ typically sparse eigenvalue evaluations. Closed-form expressions for the uncertainty curves for some special classes of graphs are derived, and an accurate analytical approximation for the expected uncertainty curve of Erd\H{o}s-R\'enyi random graphs is developed. These theoretical results are validated by numerical experiments, which also reveal an intriguing connection between diffusion processes on graphs and the uncertainty bounds.Comment: 40 pages, 8 figure

Localization and universality of eigenvectors in directed random graphs

Although the spectral properties of random graphs have been a long-standing focus of network theory, the properties of right eigenvectors of directed graphs have so far eluded an exact analytic treatment. We present a general theory for the statistics of the right eigenvector components in directed random graphs with a prescribed degree distribution and with randomly weighted links. We obtain exact analytic expressions for the inverse participation ratio and show that right eigenvectors of directed random graphs with a small average degree are localized. Remarkably, the critical mean degree for the localization transition is independent of the degree fluctuations. We also show that the dense limit of the distribution of the right eigenvectors is solely determined by the degree fluctuations, which generalizes standard results from random matrix theory. We put forward a classification scheme for the universality of the eigenvector statistics in the dense limit, which is supported by an exact calculation of the full eigenvector distributions. More generally, this paper provides a theoretical formalism to study the eigenvector statistics of sparse non-Hermitian random matrices.Comment: 7 pages and 4 figure

Community detection thresholds and the weak Ramanujan property

Decelle et al.\cite{Decelle11} conjectured the existence of a sharp threshold for community detection in sparse random graphs drawn from the stochastic block model. Mossel et al.\cite{Mossel12} established the negative part of the conjecture, proving impossibility of meaningful detection below the threshold. However the positive part of the conjecture remained elusive so far. Here we solve the positive part of the conjecture. We introduce a modified adjacency matrix $B$ that counts self-avoiding paths of a given length $\ell$ between pairs of nodes and prove that for logarithmic $\ell$, the leading eigenvectors of this modified matrix provide non-trivial detection, thereby settling the conjecture. A key step in the proof consists in establishing a {\em weak Ramanujan property} of matrix $B$. Namely, the spectrum of $B$ consists in two leading eigenvalues $\rho(B)$, $\lambda_2$ and $n-2$ eigenvalues of a lower order $O(n^{\epsilon}\sqrt{\rho(B)})$ for all $\epsilon>0$, $\rho(B)$ denoting $B$'s spectral radius. $d$-regular graphs are Ramanujan when their second eigenvalue verifies $|\lambda|\le 2 \sqrt{d-1}$. Random $d$-regular graphs have a second largest eigenvalue $\lambda$ of $2\sqrt{d-1}+o(1)$ (see Friedman\cite{friedman08}), thus being {\em almost} Ramanujan. Erd\H{o}s-R\'enyi graphs with average degree $d$ at least logarithmic ($d=\Omega(\log n)$) have a second eigenvalue of $O(\sqrt{d})$ (see Feige and Ofek\cite{Feige05}), a slightly weaker version of the Ramanujan property. However this spectrum separation property fails for sparse ($d=O(1)$) Erd\H{o}s-R\'enyi graphs. Our result thus shows that by constructing matrix $B$ through neighborhood expansion, we regularize the original adjacency matrix to eventually recover a weak form of the Ramanujan property