Cheeger Constant Research Articles

On a Cheeger type inequality in Cayley graphs of finite groups

Let G be a finite group. It was remarked in Breuillard et al. (2015) that if the Cayley graph C(G,S) is an expander graph and is non-bipartite then the spectrum of the adjacency operator T is bounded away from −1. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval −1+h(G)4γ,1−h(G)22d2, where h(G) denotes the (vertex) Cheeger constant of the d regular graph C(G,S) with respect to a symmetric set S of generators and γ=29d6(d+1)2.

Orbigraphs: a graph-theoretic analog to Riemannian orbifolds

A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold that is locally modeled on $R^n$ modulo the action of a finite group. Orbifolds have proven interesting in a variety of settings. Spectral geometers have examined the link between the Laplace spectrum of an orbifold and the singularities of the orbifold. One open question in this field is whether or not a singular orbifold and a manifold can be Laplace isospectral. Motivated by the connection between spectral geometry and spectral graph theory, we define a graph theoretic analogue of an orbifold called an orbigraph. We obtain results about the relationship between an orbigraph and the spectrum of its adjacency matrix. We prove that the number of singular vertices present in an orbigraph is bounded above and below by spectrally determined quantities, and show that an orbigraph with a singular point and a regular graph cannot be cospectral. We also provide a lower bound on the Cheeger constant of an orbigraph.

Proof of the honeycomb asymptotics for optimal Cheeger clusters

We prove that, in the limit as k→+∞, the hexagonal honeycomb solves the optimal partition problem in which the criterion is minimizing the largest among the Cheeger constants of k mutually disjoint cells in a planar domain. As a by-product, the same result holds true when the Cheeger constant is replaced by the first Robin eigenvalue of the Laplacian.

On random walk on growing graphs

Nous considérons un modèle de marche aléatoire sur un graphe dynamique. Pour une suite de graphes finis croissant vers un graphe limite infini, nous montrons une borne supérieure pour la probabilité de transition. Cela donne un critère de transience pour la marche simple, pour des graphes à croissante lente, à partir du volume et de la constante de Cheeger de chaque graphe. Pour des cas plus particuliers, nous montrons une borne inférieure du même ordre et déduisons un critère de récurrence (dans un sens faible). Nous répondons aussi à la question de la récurrence directement, en lien avec une conjecture d’universalité de (Electron. J. Probab. 19 (2014) Article ID 106). Nous répondons aussi négativement à une question reliée de (Ann. Probab. 39 (2011) 1161–1203, Problem 1.8), à propos du « plongement inhomogène ».

Buser's inequality on infinite graphs

In this paper, we establish Buser type inequalities, i.e., upper bounds for eigenvalues in terms of Cheeger constants. We prove the Buser's inequality for an infinite but locally finite connected graph with Ricci curvature lower bounds. Furthermore, we derive that the graph with positive curvature is finite, especially for unbounded Laplacians. By proving Poincaré inequality, we obtain a lower bound for Cheeger constant in terms of positive curvature.

Tighter spectral bounds for the cut size, based on Laplacian eigenvectors

The cut-set ∂V in a graph is defined as the set of all links between a set of nodes V and all other nodes in that graph. Finding bounds for the size of a cut-set |∂V| is an important problem, and is related to mixing times, connectedness and spreading processes on networks. A standard way to bound the number of links in a cut-set |∂V| relies on Laplacian eigenvalues, which approximate the largest and smallest possible cut-sets for a given size of the set V. In this article, we extend the standard spectral approximations by including information about the Laplacian eigenvectors. This additional information leads to provably tighter bounds compared to the standard spectral bounds. We apply our new method to find improved spectral bounds for the well-known Cheeger constant, the Max Cut problem and the expander mixing lemma. We also apply our bounds to study cut sizes in the hypercube graph, and describe an application related to the spreading of epidemics on networks. We further illustrate the performance of our new bounds using simulations, revealing that a significant improvement over the standard bounds is possible.

The Cheeger constant of curved tubes

We compute the Cheeger constant of spherical shells and tubular neighbourhoods of complete curves in an arbitrary dimensional Euclidean space.

Explicit Computation of Cheeger Constants for some Classes of Graphs

This article deals with the explicit computation of the Cheeger constant and its connection to understanding the properties of some classes of graphs. Cheeger constant is a measure of the connectivity or disconnectivity of graphs and gives the best possible way to cut a graph. More precisely, we deal with the problem of splitting the graph into two large components of approximately equal volumes by making a small cut, which is the idea of Cheeger constant of a graph. We computed the Cheeger constants for simple classes of graphs such as 2-comb graphs, cycle graphs, complete graphs, and cube graphs. We further analyzed the dynamics of Cheeger constants of the graphs under vertex-edge transformation.

Growing in time IDLA cluster is recurrent

We show that Internal Diffusion Limited Aggregation (IDLA )on $\mathbb{Z} ^{d}$ has near optimal Cheeger constant when the growing cluster is large enough. This implies, through a heat kernel lower bound derived previously in [11], that simple random walk evolving independently on growing in time IDLA cluster is recurrent when $d\ge 3$.

Stable Gabor Phase Retrieval and Spectral Clustering

AbstractWe consider the problem of reconstructing a signalffrom its spectrogram, i.e., the magnitudes |Vφf| of its Gabor transformurn:x-wiley:00103640:media:cpa21799:cpa21799-math-0001Such problems occur in a wide range of applications, from optical imaging of nanoscale structures to audio processing and classification.While it is well‐known that the solution of the above Gabor phase retrieval problem is unique up to natural identifications, the stability of the reconstruction has remained wide open. The present paper discovers a deep and surprising connection between phase retrieval, spectral clustering, and spectral geometry. We show that the stability of the Gabor phase reconstruction is bounded by the reciprocal of theCheeger constantof the flat metric on ℝ2, conformally multiplied with |Vφf|. The Cheeger constant, in turn, plays a prominent role in the field of spectral clustering, and it precisely quantifies the “disconnectedness” of the measurementsVφf.It has long been known that a disconnected support of the measurements results in an instability—our result for the first time provides a converse in the sense that there are no other sources of instabilities.Due to the fundamental importance of Gabor phase retrieval in coherent diffraction imaging, we also provide a new understanding of the stability properties of these imaging techniques: Contrary to most classical problems in imaging science whose regularization requires the promotion of smoothness or sparsity, the correct regularization of the phase retrieval problem promotes the “connectedness” of the measurements in terms of bounding the Cheeger constant from below. Our work thus, for the first time, opens the door to the development of efficient regularization strategies. © 2018 Wiley Periodicals, Inc.

Reverse Faber–Krahn and Mahler inequalities for the Cheeger constant

We prove a reverse Faber–Krahn inequality for the Cheeger constant, stating that every convex body in ℝ2 has an affine image such that the product between its Cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle. An analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square. We also prove a Mahler-type inequality for the Cheeger constant, stating that every axisymmetric convex body in ℝ2 has a linear image such that the product between its Cheeger constant and the Cheeger constant of its polar body is not larger than the same quantity for the square.

The Allen–Cahn equation on closed manifolds

We study global variational properties of the space of solutions to $$-\varepsilon ^2\Delta u + W'(u)=0$$ on any closed Riemannian manifold M. Our techniques are inspired by recent advances in the variational theory of minimal hypersurfaces and extend a well-known analogy with the theory of phase transitions. First, we show that solutions at the lowest positive energy level are either stable or obtained by min–max and have index 1. We show that if $$\varepsilon $$ is not small enough, in terms of the Cheeger constant of M, then there are no interesting solutions. However, we show that the number of min–max solutions to the equation above goes to infinity as $$\varepsilon \rightarrow 0$$ and their energies have sublinear growth. This result is sharp in the sense that for generic metrics the number of solutions is finite, for fixed $$\varepsilon $$ , as shown recently by G. Smith. We also show that the energy of the min–max solutions accumulate, as $$\varepsilon \rightarrow 0$$ , around limit-interfaces which are smooth embedded minimal hypersurfaces whose area with multiplicity grows sublinearly. For generic metrics with $$\mathrm{Ric}_M>0$$ , the limit-interface of the solutions at the lowest positive energy level is an embedded minimal hypersurface of least area in the sense of Mazet–Rosenberg. Finally, we prove that the min–max energy values are bounded from below by the widths of the area functional as defined by Marques–Neves.

Hybrid Bridge-Based Memetic Algorithms for Finding Bottlenecks in Complex Networks

Abstract We propose a memetic approach to find bottlenecks in complex networks based on searching for a graph partitioning with minimum conductance. Finding the optimum of this problem, also known in statistical mechanics as the Cheeger constant, is one of the most interesting NP-hard network optimisation problems. The existence of low conductance minima indicates bottlenecks in complex networks. However, the problem has not yet been explored in depth in the context of applied discrete optimisation and evolutionary approaches to solve it. In this paper, the use of a memetic framework is explored to solve the minimum conductance problem. The approach combines a hybrid method of initial population generation based on bridge identification and local optima sampling with a steady-state evolutionary process with two local search subroutines. These two local search subroutines have complementary qualities. Efficiency of three crossover operators is explored, namely one-point crossover, uniform crossover, and our own partition crossover. Experimental results are presented for both artificial and real-world complex networks. Results for Barabasi–Albert model of scale-free networks are presented, as well as results for samples of social networks and protein–protein interaction networks. These indicate that both well-informed initial population generation and the use of a crossover seem beneficial in solving the problem in large-scale.

On the higher Cheeger problem

We develop the notion of higher Cheeger constants for a measurable set $\Omega \subset \mathbb{R}^N$. By the $k$-th Cheeger constant we mean the value \[h_k(\Omega) = \inf \max \{h_1(E_1), \dots, h_1(E_k)\},\] where the infimum is taken over all $k$-tuples of mutually disjoint subsets of $\Omega$, and $h_1(E_i)$ is the classical Cheeger constant of $E_i$. We prove the existence of minimizers satisfying additional "adjustment" conditions and study their properties. A relation between $h_k(\Omega)$ and spectral minimal $k$-partitions of $\Omega$ associated with the first eigenvalues of the $p$-Laplacian under homogeneous Dirichlet boundary conditions is stated. The results are applied to determine the second Cheeger constant of some planar domains.

Intrinsic isoperimetry of the giant component of supercritical bond percolation in dimension two

We study the isoperimetric subgraphs of the giant component $\mathbf{{C}} _n$ of supercritical bond percolation on the square lattice. These are subgraphs of $\mathbf{{C}} _n$ with minimal edge boundary to volume ratio. In contrast to the work of [8], the edge boundary is taken only within $\mathbf{{C}} _n$ instead of the full infinite cluster. The isoperimetric subgraphs are shown to converge almost surely, after rescaling, to the collection of optimizers of a continuum isoperimetric problem emerging naturally from the model. We also show that the Cheeger constant of $\mathbf{{C}} _n$ scales to a deterministic constant, which is itself an isoperimetric ratio, settling a conjecture of Benjamini in dimension two.

The Cheeger constant of a Jordan domain without necks

We show that the maximal Cheeger set of a Jordan domain $\Omega$ without necks is the union of all balls of radius $r = h(\Omega)^{-1}$ contained in $\Omega$. Here, $h(\Omega)$ denotes the Cheeger constant of $\Omega$, that is, the infimum of the ratio of perimeter over area among subsets of $\Omega$, and a Cheeger set is a set attaining the infimum. The radius $r$ is shown to be the unique number such that the area of the inner parallel set $\Omega^r$ is equal to $\pi r^2$. The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake.

The First Cheeger Constant of a Simplex

The coboundary expansion generalizes the classical graph expansion to the case of the general simplicial complexes, and allows the definition of the higher-dimensional Cheeger constants $$h_k(X)$$ for an arbitrary simplicial complex X, and any $$k\ge 0$$ . In this paper we investigate the value of $$h_1(\Delta ^{[n]})$$ —the first Cheeger constant of a simplex with n vertices. It is known, due to the pioneering work of Meshulam and Wallach [12], that $$\begin{aligned} \lceil n/3\rceil \ge h_1(\Delta ^{[n]})\ge n/3, \text { for all } n, \end{aligned}$$ and that the equality $$h_1(\Delta ^{[n]})=n/3$$ is achieved when n is divisible by 3. Here we expand on these results. First, we show that $$\begin{aligned} h_1(\Delta ^{[n]})=n/3, \text { whenever }n\text { is not a power of }2. \end{aligned}$$ So the sharp equality holds on a set whose density goes to 1. Second, we show that $$\begin{aligned} h_1(\Delta ^{[n]})=n/3+O(1/n),\text { when }n\text { is a power of }2. \end{aligned}$$ In other words, as n goes to infinity, the value $$h_1(\Delta ^{[n]})-n/3$$ is either 0 or goes to 0 very rapidly. Our methods include recasting the original question in purely graph-theoretic language, followed by a detailed investigation of a specific graph family, the so-called staircase graphs. These are defined by associating a graph to every partition, and appear to be especially suited to gain information about the first Cheeger constant of a simplex.

A Faber–Krahn Inequality for Solutions of Schrödinger’s Equation on Riemannian Manifolds

We consider a bounded open set with smooth boundary \(\Omega \subset M\) in a Riemannian manifold (M, g), and suppose that there exists a non-trivial function \(u\in C({\overline{\Omega }})\) solving the problem $$\begin{aligned} -\Delta u=V(x)u, \,\, \text{ in }\,\,\Omega , \end{aligned}$$ in the distributional sense, with \(V\in L^\infty (\Omega )\), where \(u\equiv 0\) on \(\partial \Omega .\) We prove a sharp inequality involving \(||V||_{L^{\infty }(\Omega )}\) and the first eigenvalue of the Laplacian on geodesic balls in simply connected spaces with constant curvature, which slightly generalises the well-known Faber–Krahn isoperimetric inequality. Moreover, in a Riemannian manifold which is not necessarily simply connected, we obtain a lower bound for \(||V||_{L^{\infty }(\Omega )}\) in terms of its isoperimetric or Cheeger constant. As an application, we show that if \(\Omega \) is a domain on a m-dimensional minimal submanifold of \({\mathbb {R}}^n\) which lies in a ball of radius R, then $$\begin{aligned} ||V||_{L^{\infty }(\Omega )}\ge \left( \frac{m}{2R}\right) ^{2}. \end{aligned}$$

On the generalized Cheeger problem and an application to 2d strips

In this paper we consider the generalization of the Cheeger problem which comes by considering the ratio between the perimeter and a certain power of the volume. This generalization has been already sometimes treated, but some of the main properties were still not studied, and our main aim is to fill this gap. We will show that most of the first important properties of the classical Cheeger problem are still valid, but others fail; more precisely, long and thin rectangles will give a counterexample to the property of Cheeger sets of being the union of all the balls of a certain radius, as well as to the uniqueness. The shape of Cheeger set for rectangles and strips is then studied as well as their Cheeger constant.

Cheeger N-clusters

In this paper we introduce a Cheeger-type constant defined as a minimization of a suitable functional among all the $N$-clusters contained in an open bounded set $\Omega$. Here with $N$-Cluster we mean a family of $N$ sets of finite perimeter, disjoint up to a set of null Lebesgue measure. We call any $N$-cluster attaining such a minimum a Cheeger $N$-cluster. Our purpose is to provide a non trivial lower bound on the optimal partition problem for the first Dirichlet eigenvalue of the Laplacian. Here we discuss the regularity of Cheeger $N$-clusters in a general ambient space dimension and we give a precise description of their structure in the the planar case. The last part is devoted to the relation between the functional introduced here (namely the $N$-Cheeger constant), the partition problem for the first Dirichlet eigenvalue of the Laplacian and the Caffarelli and Lin's conjecture.