Shiping Liu

Ollivier’s Ricci Curvature, Local Clustering and Curvature-Dimension Inequalities on Graphs

In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally related to the presence of triangles containing those vertices, or more precisely, the local clustering coefficient, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This suggests to derive lower Ricci curvature bounds on graphs in terms of such local clustering coefficients. We also study curvature-dimension inequalities on graphs, building upon previous work of several authors.

Multi-way dual Cheeger constants and spectral bounds of graphs

We introduce a set of multi-way dual Cheeger constants and prove universal higher-order dual Cheeger inequalities for eigenvalues of normalized Laplace operators on weighted finite graphs. Our proof proposes a new spectral clustering phenomenon deduced from metrics on real projective spaces. We further extend those results to a general reversible Markov operator and find applications in characterizing its essential spectrum.

Spectral classes of regular, random, and empirical graphs

Abstract We define a (pseudo-)distance between graphs based on the spectrum of the normalized Laplacian. Since this quantity can be computed easily, or at numerically estimated, it is suitable for comparing in particular large graphs. Numerical experiments demonstrate that the spectral distance provides a practically useful measure of graph dissimilarity. The asymptotic behavior of the Laplacian spectrum furthermore yields a tool for classifying families of graphs in such a way that the distance of two graphs from the same family is bounded by O ( 1 / n ) in terms of size n of their vertex sets.

Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians

We discuss a Cheeger constant as a mixture of the frustration index and the expansion rate, and prove the related Cheeger inequalities and higher order Cheeger inequalities for graph Laplacians with cyclic signatures, discrete magnetic Laplacians on finite graphs and magnetic Laplacians on closed Riemannian manifolds. In this process, we develop spectral clustering algorithms for partially oriented graphs and multi-way spectral clustering algorithms via metrics in lens spaces and complex projective spaces. As a byproduct, we give a unified viewpoint of Harary’s structural balance theory of signed graphs and the gauge invariance of magnetic potentials.

Spectral distances on graphs

By assigning a probability measure via the spectrum of the normalized Laplacian to each graph and using L p Wasserstein distances between probability measures, we define the corresponding spectral distances d p on the set of all graphs. This approach can even be extended to measuring the distances between infinite graphs. We prove that the diameter of the set of graphs, as a pseudo-metric space equipped with d 1 , is one. We further study the behavior of d 1 when the size of graphs tends to infinity by interlacing inequalities aiming at exploring large real networks. A monotonic relation between d 1 and the evolutionary distance of biological networks is observed in simulations.

Liouville theorems for f-harmonic maps into Hadamard spaces

In this paper, we study harmonic functions on weighted manifolds and harmonic maps from weighted manifolds into Hadamard spaces introduced by Korevaar and Schoen. We prove various Liouville theorems for these har- monic maps.

The Geometric Meaning of Curvature: Local and Nonlocal Aspects of Ricci Curvature

Curvature is a concept originally developed in differential and Riemannian geometry. There are various established notions of curvature, in particular sectional and Ricci curvature. An important theme in Riemannian geometry has been to explore the geometric and topological consequences of bounds on those curvatures, like divergence or convergence of geodesics, convexity properties of distance functions, growth of the volume of distance balls, transportation distance between such balls, vanishing theorems for Betti numbers, bounds for the eigenvalues of the Laplace operator or control of harmonic functions. Several of these geometric properties turn out to be equivalent to the corresponding curvature bounds in the context of Riemannian geometry. Since those properties often are also meaningful in the more general framework of metric geometry, in recent years, there have been several research projects that turned those properties into axiomatic definitions of curvature bounds in metric geometry. In this contribution, after developing the Riemannian geometric background, we explore some of these axiomatic approaches. In particular, we shall describe the insights in graph theory and network analysis following from the corresponding axiomatic curvature definitions.

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.