Curvature Of Graph Research Articles

Long-Scale Ollivier Ricci Curvature of Graphs

Abstract We study the long-scale Ollivier Ricci curvature of graphs as a function of the chosen idleness. Similarly to the previous work on the short-scale case, we show that this idleness function is concave and piecewise linear with at most 3 linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long-scale curvature for the Cartesian product of two regular graphs.

Uniform Approximation of the Curvature of Smooth Plane Curves with the Use of Partial Fourier Sums

An error bound for the approximation of the curvature of graphs of periodic functions from the class Wr for r ≥ 3 in the uniform metric is obtained with the use of the simplest approximation technique for smooth periodic functions, which is approximation by partial sums of their trigonometric Fourier series. From the mathematical point of view, the interest in this problem is connected with the specific nonlinearity of the graph curvature operator on the class of smooth functions Wr on a period or a closed interval for r ≥ 2. There are several papers on curvature approximation for plane curves in the mean-square and Chebyshev norms. In previous works, the approximation was performed by partial sums of trigonometric series (in the L2 norm), interpolation splines with uniform knots, Fejer means of partial sums of trigonometric series, and orthogonal interpolating wavelets based on Meyer wavelets (in the C∞ norm). The technique of this paper, based on the lemma, can possibly be generalized to the Lp metric and other approximation methods.

Comparative analysis of two discretizations of Ricci curvature for complex networks

We have performed an empirical comparison of two distinct notions of discrete Ricci curvature for graphs or networks, namely, the Forman-Ricci curvature and Ollivier-Ricci curvature. Importantly, these two discretizations of the Ricci curvature were developed based on different properties of the classical smooth notion, and thus, the two notions shed light on different aspects of network structure and behavior. Nevertheless, our extensive computational analysis in a wide range of both model and real-world networks shows that the two discretizations of Ricci curvature are highly correlated in many networks. Moreover, we show that if one considers the augmented Forman-Ricci curvature which also accounts for the two-dimensional simplicial complexes arising in graphs, the observed correlation between the two discretizations is even higher, especially, in real networks. Besides the potential theoretical implications of these observations, the close relationship between the two discretizations has practical implications whereby Forman-Ricci curvature can be employed in place of Ollivier-Ricci curvature for faster computation in larger real-world networks whenever coarse analysis suffices.

Ollivier--Ricci Idleness Functions of Graphs

We study the Ollivier--Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most three linear parts, and at most two linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartesian product of two regular graphs is completely determined by the idleness functions of the factors.

Volume doubling, Poincaré inequality and Gaussian heat kernel estimate for non-negatively curved graphs

Abstract Studying the heat semigroup, we prove Li–Yau-type estimates for bounded and positive solutions of the heat equation on graphs. These are proved under the assumption of the curvature-dimension inequality CDE ′ ⁢ ( n , 0 ) {\mathrm{CDE}^{\prime}(n,0)} , which can be considered as a notion of curvature for graphs. We further show that non-negatively curved graphs (that is, graphs satisfying CDE ′ ⁢ ( n , 0 ) {\mathrm{CDE}^{\prime}(n,0)} ) also satisfy the volume doubling property. From this we prove a Gaussian estimate for the heat kernel, along with Poincaré and Harnack inequalities. As a consequence, we obtain that the dimension of the space of harmonic functions on graphs with polynomial growth is finite. In the Riemannian setting, this was originally a conjecture of Yau, which was proved in that context by Colding and Minicozzi. Under the assumption that a graph has positive curvature, we derive a Bonnet–Myers-type theorem. That is, we show the diameter of positively curved graphs is finite and bounded above in terms of the positive curvature. This is accomplished by proving some logarithmic Sobolev inequalities.

Curvature-dependent Energies

We report our recent results from [1, 2] on the total curvature of graphs of curves in high codimension Euclidean space. We introduce the corresponding relaxed energy functional and provide an explicit representation formula. In the case of continuous Cartesian curves, i.e., of graphs \({c_{u}}\) of continuous functions u on an interval, the relaxed energy is finite if and only if the curve \({c_{u}}\) has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We also deal with the "elastic" case, corresponding to a superlinear dependence on the pointwise curvature. Different phenomena w.r.t. the "plastic" case are observed. A p-curvature functional is well-defined on continuous curves with finite relaxed energy, and the relaxed energy is given by the length plus the p-curvature. We treat the wider class of graphs of one-dimensional BV-functions.

Curvature-dependent energies: a geometric and analytical approach

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

On nonlocal quasilinear equations and their local limits

We introduce a new class of quasilinear nonlocal operators and study equations involving these operators. The operators are degenerate elliptic and may have arbitrary growth in the gradient. Included are new nonlocal versions of p-Laplace, ∞-Laplace, mean curvature of graph, and even strongly degenerate operators, in addition to some nonlocal quasilinear operators appearing in the existing literature. Our main results are comparison, uniqueness, and existence results for viscosity solutions of linear and fully nonlinear equations involving these operators. Because of the structure of our operators, especially the existence proof is highly non-trivial and non-standard. We also identify the conditions under which the nonlocal operators converge to local quasilinear operators, and show that the solutions of the corresponding nonlocal equations converge to the solutions of the local limit equations. Finally, we give a (formal) stochastic representation formula for the solutions and provide many examples.

Transcriptional Responses to Ultraviolet and Ionizing Radiation: An Approach Based on Graph Curvature.

More than half of all cancer patients receive radiotherapy in their treatment process. However, our understanding of abnormal transcriptional responses to radiation remains poor. In this study, we employ an extended definition of Ollivier-Ricci curvature based on LI-Wasserstein distance to investigate genes and biological processes associated with ionizing radiation (IR) and ultraviolet radiation (UV) exposure using a microarray dataset. Gene expression levels were modeled on a gene interaction topology downloaded from the Human Protein Reference Database (HPRD). This was performed for IR, UV, and mock datasets, separately. The difference curvature value between IR and mock graphs (also between UV and mock) for each gene was used as a metric to estimate the extent to which the gene responds to radiation. We found that in comparison of the top 200 genes identified from IR and UV graphs, about 20~30% genes were overlapping. Through gene ontology enrichment analysis, we found that the metabolic-related biological process was highly associated with both IR and UV radiation exposure.

Curvature-dependent energies: The elastic case

We continue our analysis on functionals depending on the curvature of graphs of curves in high codimension Euclidean space. We deal with the “elastic” case, corresponding to a superlinear dependence on the pointwise curvature. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. Different phenomena w.r.t. the “plastic” case, i.e. to the relaxation of the total curvature functional, are observed. A p-curvature functional is well-defined on continuous curves with finite relaxed energy, and the relaxed energy is given by the length plus the p-curvature. The wider class of graphs of one-dimensional BV-functions is treated.

Geometric elliptic functionals and mean curvature

We introduce an extended notion of mean curvature for graphs via the Euler-Lagrange equation of a geometric elliptic functional. We then draw some geometric conclusions for Killing graphs with prescribed weighted and anisotropic mean curvatures with the aid of a general form of the weak maximum principle and a sufficient condition for an appropriate notion of parabolicity.

Graph Curvature for Differentiating Cancer Networks

Cellular interactions can be modeled as complex dynamical systems represented by weighted graphs. The functionality of such networks, including measures of robustness, reliability, performance, and efficiency, are intrinsically tied to the topology and geometry of the underlying graph. Utilizing recently proposed geometric notions of curvature on weighted graphs, we investigate the features of gene co-expression networks derived from large-scale genomic studies of cancer. We find that the curvature of these networks reliably distinguishes between cancer and normal samples, with cancer networks exhibiting higher curvature than their normal counterparts. We establish a quantitative relationship between our findings and prior investigations of network entropy. Furthermore, we demonstrate how our approach yields additional, non-trivial pair-wise (i.e. gene-gene) interactions which may be disrupted in cancer samples. The mathematical formulation of our approach yields an exact solution to calculating pair-wise changes in curvature which was computationally infeasible using prior methods. As such, our findings lay the foundation for an analytical approach to studying complex biological networks.

Li-Yau inequality on graphs

AbstractWe prove the Li-Yau gradient estimate for the heat kernel on graphs. The onlyassumption is a variant of the curvature-dimension inequality, which is purely local,and can be considered as a new notion of curvature for graphs. We compute thiscurvature for lattices and trees and conclude that it behaves more naturally than thealready existing notions of curvature. Moreover, we show that if a graph has non-negative curvature then it has polynomial volume growth.We also derive Harnack inequalities and heat kernel bounds from the gradient esti-mate, and show how it can be used to strengthen the classical Buser inequality relatingthe spectral gap and the Cheeger constant of a graph. 1 Introduction and main ideas In their celebrated work [15] Li and Yau proved an upper bound on the gradient ofpositive solutions of the heat equation. In its simplest form, for an n-dimensional com-pact manifold with non-negative Ricci curvature the Li-Yau gradient estimate statesthat a positive solution uof the heat equation (∆−∂

Exact and asymptotic results on coarse Ricci curvature of graphs

Ricci curvature was proposed by Ollivier in a general framework of metric measure spaces, and it has been studied extensively in the context of graphs in recent years. In this paper we obtain the exact formulas for Ollivier’s Ricci-curvature for bipartite graphs and for the graphs with girth at least 5. These are the first formulas for Ricci-curvature that hold for a wide class of graphs, and extend earlier results where the Ricci-curvature for graphs with girth 6 was obtained. We also prove a general lower bound on the Ricci-curvature in terms of the size of the maximum matching in an appropriate subgraph. As a consequence, we characterize the Ricci-flat graphs of girth 5. Moreover, using our general lower bound and the Birkhoff–von Neumann theorem, we give the first necessary and sufficient condition for the structure of Ricci-flat regular graphs of girth 4. Finally, we obtain the asymptotic Ricci-curvature of random bipartite graphs G(n,n,p) and random graphs G(n,p), in various regimes of p.

Ricci curvature, circulants, and a matching condition

The Ricci curvature of graphs, as recently introduced by Lin, Lu, and Yau following a general concept due to Ollivier, provides a new and promising isomorphism invariant. This paper presents a simplified exposition of the concept, including the so-called logistic diagram as a computational or visualization aid. Two new infinite classes of graphs with positive Ricci curvature are identified. A local graph-theoretical condition, known as the matching condition, provides a general formula for Ricci curvatures. The paper initiates a longer-term program of classifying the Ricci curvatures of circulant graphs. Aspects of this program may prove useful in tackling the problem of showing when twisted tori are not isomorphic to circulants.

Mean Curvature, Threshold Dynamics, and Phase Field Theory on Finite Graphs

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study. We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow. We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation.

Numerical Simulation of Anisotropic Mean Curvature of Graphs in Relative Geometry

The aim of this paper is the numerical simulation of anisotropic mean curvature of graphs in the context of relative geometry, developed in (1). We extend results in (4) to our problem; we prove an existence theorem and energy equality. The numerical scheme is based on the method of lines where the spatial derivatives are approximated by finite differences (2). We then solve the resulting ODE system by means of the adaptive Runge- Kutta-Merson method. To show the stability of the scheme we prove the discrete version of the energy equality. Finally, we show experimental order of convergence and results of numerical experiments with various anisotropy settings.

Smooth solutions to a class of mixed type Monge–Ampère equations

We prove the existence of C∞ local solutions to a class of mixed type Monge–Ampere equations in the plane. More precisely, the equation changes type to finite order across two smooth curves intersecting transversely at a point. Existence of C∞ global solutions to a corresponding class of linear mixed type equations is also established. These results are motivated by and may be applied to the problem of prescribed Gaussian curvature for graphs, the isometric embedding problem for 2-dimensional Riemannian manifolds into Euclidean 3-space, and also transonic fluid flow.

Total curvature of graphs after Milnor and Euler

We define a new notion of total curvature, called net total curvature, for finite graphs embedded inR n , and investigate its properties. Two guiding principles ar e given by Milnor's way of measuring the local crookedness of a Jorda n curve via a Crofton-type formula, and by considering the double cover of a given graph as an Eulerian circuit. The strength of combining these ideas in defining the curvature f unctional is (1) it allows us to interpret the singular/non-eulidean behavior at the vertices of the graph as a superposition of vertices of a 1-dimensional manifold, and thus (2) one can compute the total curva- ture for a wide range of graphs by contrasting local and global properties of the graph utilizing the integral geometric representation of the cur vature. A collection of results on upper/lower bounds of the total curvature on isotopy/homeomorphism classes of embed- dings is presented, which in turn demonstrates the effectiveness of net total curvature as a new functional measuring complexity of spatial graphs in differential-geometric terms.

Solutions to the Allen Cahn Equation and Minimal Surfaces

We discuss and outline proofs of some recent results on application of singular perturbation techniques for solutions in entire space of the Allen-Cahn equation Δu + u − u 3 = 0. In particular, we consider a minimal surface Γ in \({\mathbb {R}^9}\) which is the graph of a nonlinear entire function x 9 = F(x 1, . . . , x 8), found by Bombieri, De Giorgi and Giusti, the BDG surface. We sketch a construction of a solution to the Allen Cahn equation in \({\mathbb {R}^9}\) which is monotone in the x9 direction whose zero level set lies close to a large dilation of Γ, recently obtained by M. Kowalczyk and the authors. This answers a long standing question by De Giorgi in large dimensions (1978), whether a bounded solution should have planar level sets. We sketch two more applications of the BDG surface to related questions, respectively in overdetermined problems and in eternal solutions to the flow by mean curvature for graphs.