A goal in network science is the geometrical characterization of complex networks. In this direction, we have recently introduced Forman's discretization of Ricci curvature to the realm of undirected networks. Investigation of this edge-centric network measure, Forman-Ricci curvature, in diverse model and real-world un…
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48 results for “Forman-Ricci Curvature”
Efficient Decomposition of Forman-Ricci Curvature on Vietoris-Rips Complexes and Data Applicationsmath.GT
Researchers decompose Forman-Ricci curvature for efficient computation in VR complexes.
problem Efficiently computing Forman-Ricci curvature in higher-dimensional data.
method Decomposition and set-theoretical proof for local computation of FRC in VR complexes.
result Reveals critical geometric insights overlooked by conventional techniques.
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…
New method connects curvature and Persistent Homology for networks.
problem Efficient computation of Persistent Homology for complex networks.
method Discrete Morse Theory, Bloch's extension, Forman-Ricci curvature.
result Efficient Persistent Homology scheme using curvature-based approach.
New method uses curvature to improve graph neural networks.
problem Graph Neural Networks struggle with over-smoothing and over-squashing.
method Augmented Forman-Ricci curvature (AFRC) for scalable rewiring.
result AFRC effectively mitigates over-smoothing and over-squashing.
Forman--Ricci Curvature on Contact-Sequence Temporal Networks via Spatiotemporal Prism Complexesmath.DG
Study develops curvature for contact-sequence networks, revealing temporal dynamics.
problem Lack of geometric analysis for temporal network sequences.
method Develops Forman--Ricci curvature on spatiotemporal prism complexes.
result Two curvature variants disagree on 56-67% of temporal edges.
Enhanced neural network framework improves constraint satisfaction with topological conditioning.
problem Maintaining semantic coherence while satisfying physical and logical constraints in neuro-symbolic reasoning.
method Integrates topological conditioning with gradient stabilization mechanisms using Forman-Ricci curvature, Deep Delta Learning, and Covariance Matrix Adaptation Evolution Strategy.
result Achieves mean energy reduction to 1.15 compared to baseline values of 11.68, with 95 percent success rate.
Hypernetworks are simplified simplicial complexes with curvature.
problem Representing hypernetworks geometrically for analysis.
method Hypernetworks are interpreted as posets, which are simplicial complexes with Forman Ricci curvature.
result Hypernetworks have intrinsic curvature that correlates with their Euler characteristic.
Geometric sampling of networks using curvature measures.
problem Sampling and analyzing complex network structures.
method Three types of discrete curvature (Forman-, full Forman-, Haantjes-Ricci) for edge-based and node-based sampling.
result Effective detection of networks' backbone and coarse structure.
New edge features improve GNN performance in biological datasets.
problem Inefficient use of edge features in GNNs.
method Self-supervised and unsupervised learning for new edge features, incorporating Forman-Ricci curvature.
result Improved node classification performance over baseline GNN models.
HLRC offers a new curvature metric for hypergraphs that balances interpretability and efficiency.
problem Challenges in geometric characterization of hypergraphs with higher-order interactions.
method Hypergraph lower Ricci curvature (HLRC) defined in closed form.
result HLRC consistently reveals meaningful higher-order organization in diverse hypergraph datasets.
Sandpile Economics: Theory, Identification, and Evidencephysics.soc-ph
Sandpile Economics explains how economies can be prone to large crises from small shocks.
problem Capitalist economies' recurrent crises disproportionate to shocks.
method Formal framework interpreting instability as geometric fragility of production networks.
result Curvature of production networks predicts medium-run output dynamics and resilience.
Automatic segmentation of auditory ossicles from CT images using Ricci curvature.
problem Automatic diagnosis of ossicles' diseases from 3D CT images of the head.
method Proposes a completely automatic method that locates and segments ossicles without manual labels or templates, using Ricci curvature in an energy function.
result Performance of the proposed method using discrete Forman-Ricci curvature is superior to state-of-the-art methods.
Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further s…
Network geometry measures predict market instability.
problem Predicting financial market instability using network geometry.
method Discrete Ricci curvatures to capture network fragility.
result Different geometric measures distinguish normal and crash periods.
We study different notions of Riemannian curvatures: The -curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the -curvatures, which incorporate …
Paper establishes a relation between Berwald scalar curvature and S-curvature.
problem Understanding the relationship between Finsler metrics' curvature properties.
method Proved conditions for isotropic Berwald scalar curvature and weakly isotropic S-curvature.
result Finsler metrics with isotropic Berwald scalar curvature have weakly isotropic S-curvature.
New scalar curvature defined from Ollivier-Ricci curvature for graphs.
problem Defining scalar curvature for graphs and point clouds.
method Defining a new scalar version of Ollivier-Ricci curvature and proving its convergence.
result The new scalar curvature converges to scalar curvature for sampled manifolds.
We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyper-pseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with co…
Study examines preservation of curvature-adaptedness during mean curvature flow.
problem Preservation of curvature-adaptedness during mean curvature flow.
method Investigates curvature-adaptedness in locally symmetric spaces.
result Curvature-adaptedness is preserved along mean curvature flow.
Paper explores entropic curvature in Markov chains, comparing it to other curvatures.
problem Comparing entropic curvature to other curvatures in Markov chains.
method Adapted Γ-calculus for θ-curvatures, explicit lower bounds, curvature perturbation.
result Entropic curvature differs significantly from other curvature notions.
The paper studies Berwald scalar curvature properties in Finsler geometry.
problem Characterizing Finsler manifolds based on Berwald scalar curvature.
method Analyzes properties of Berwald scalar curvature and its implications for Finsler manifolds.
result Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds.
The paper studies Finsler manifolds with a new curvature concept.
problem Understanding Finsler manifolds with positive weighted flag curvature.
method Introducing a new curvature concept based on the flag curvature and a non-Riemannian quantity, T-curvature.
result Positive weighted flag curvature implies the manifold is diffeomorphic to Euclidean space.
New proof shows holomorphic sectional curvature fully determines curvature tensor.
problem Determining the curvature tensor from holomorphic sectional curvature.
method Representation-theoretic means to calculate -norm of holomorphic sectional curvature.
result Holomorphic sectional curvature fully determines the curvature tensor.
Given a compact four dimensional smooth Riemannian manifold with smooth boundary, we consider the evolution equation by -curvature in the interior keeping the -curvature and the mean curvature to be zero and the evolution equation by -curvature at the boundary with the condition that the -curvature …
Compact shrinkers with curvature pinching conditions proven.
problem Ensuring shrinkers are compact under curvature pinching conditions.
method Various curvature pinching conditions applied to shrinkers with positive Ricci curvature and asymptotically nonnegative sectional curvature.
result Shrinkers with curvature pinching conditions are proven to be compact.
Study geodesic curvature of logarithmic spirals on curved surfaces.
problem Understanding geodesic curvature on curved surfaces.
method Computed geodesic curvature of logarithmic spirals on surfaces of constant Gaussian curvature.
result Asymptotic behavior of geodesic curvature is independent of the ambient surface's curvature.
Study on singularity behavior of mean curvature flow with bounded curvature and index.
problem Understanding singularity formation in mean curvature flow with constraints.
method Analyzing flow with bounded mean curvature and Morse index.
result Either mean curvature or Morse index blows up at first singular time.
Study on curvature in finitely generated groups, showing positive curvature in specific cases.
problem Understanding curvature in finitely generated groups.
method Analyzing dead-end elements and related elements to find curvature, studying effect of radius.
result Examples of positive curvature for arbitrary radius in lamplighter and Houghton's group.
Introduces new curvature concept for Kähler manifolds.
problem Optimizing curvature constraints for projective Kähler manifolds.
method Introduces weighted orthogonal Ricci curvature and proves vanishing theorems.
result Proves optimal curvature constraints for projective Kähler manifolds.
The paper studies Kropina metrics with a specific curvature property.
problem Characterizing Kropina metrics with isotropic scalar curvature.
method Tensor analysis to derive expressions and characterize metrics.
result Characterization of Kropina metrics with isotropic scalar curvature.
We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the -curvatures. They are a generalization of the -curvat…
The paper examines geometric properties of a unique spacetime model.
problem Investigating the geometric properties of a point-like global monopole spacetime.
method Analyzing the spacetime's pseudosymmetry structures, energy-momentum tensor, and curvature properties.
result The point-like global monopole spacetime exhibits various pseudosymmetry structures and properties.
Quantitative estimate for curvature in mean curvature flow.
problem Estimating curvature in mean curvature flow.
method Proving a curvature estimate for smooth convex ancient flows.
result Curvature grows at most quadratically in terms of rescaled extrinsic distance.
Lower bounds on curvature integral for manifolds with curvature constraints.
problem Bounding curvature integrals under curvature constraints.
method Proving a lower bound for the curvature integral using dimension, upper curvature bounds, and injectivity radius.
result Uniformly bounded below integral of scalar curvature.
New insights into SGD and generalization via shift-curvature and bias-curvature mechanisms.
problem Understanding the role of curvature in generalization and how SGD affects it.
method Derivation of new SGD steady-state distribution and analysis of shift-curvature and bias-curvature mechanisms.
result Shift-curvature is a significant factor in test performance, especially for small SGD noise.
Solves curvature problems on manifolds with negative curvature.
problem Prescribed curvature problems on closed manifolds with negative curvature.
method Investigates fully nonlinear prescribed curvature problems for modified Schouten tensor on closed Riemannian manifolds with negative curvature.
result Proves solvability of curvature problems under certain conditions.
The paper finds and analyzes the Funk-Finsler structure in constant curvature spaces.
problem Investigating the Funk-Finsler metric in spaces of constant curvature.
method Explicitly computed -curvature, Riemann curvature, Ricci curvature, and flag curvature.
result The -curvature and flag curvature of the Funk-Finsler metric in hyperbolic, spherical, and Euclidean spaces are bounded.
The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) non-Riemannian Finsler metrics on an open subset in R^n with negative flag curvature and constant S-curvature. In th…
Paper estimates curvature of convex hypersurfaces with prescribed curvature.
problem Estimating curvature of -convex hypersurfaces with prescribed curvature.
method Establishes curvature estimates for -convex hypersurfaces in with .
result Proves existence of star-shaped hypersurface of prescribed curvature and interior estimates.
A complete surface of constant mean curvature 1 (CMC-1) in hyperbolic 3-space with constant curvature -1 has two natural notions of "total curvature"-- one is the total absolute curvature which is the integral over the surface of the absolute value of the Gaussian curvature, and the other is the dual total absolute cur…
Curvature estimates prove existence of smooth hypersurfaces in hyperbolic space.
problem Existence of smooth complete hypersurfaces with constant curvature in hyperbolic space.
method Deriving curvature estimates to prove existence for all curvature values.
result Existence of smooth hypersurfaces for all possible curvature values.
Study shows no large mean curvature fill-ins for nonnegative scalar curvature.
problem Existence of fill-ins with nonnegative scalar curvature and large mean curvature.
method Analyzes closed Riemannian manifolds and scalar curvature properties.
result No closed Riemannian manifold admits a fill-in with nonnegative scalar curvature and large mean curvature.
The abstract finds conditions for creating curves of constant curvature.
problem Finding conditions for closed embedded curves of constant curvature.
method Using Almgren-Pitts min-max method for geodesic curvature.
result Closed embedded curves of any prescribed constant curvature found in .
New manifolds with negative curvature limit to one with negative curvature.
problem Preserving nonnegative scalar curvature under intrinsic flat convergence.
method Constructing sequences of manifolds with positive scalar curvature.
result Intrinsic flat limit of manifolds with negative scalar curvature.
Estimates mean curvature, scalar curvature, shape operator in warped products.
problem Estimating geometric properties in warped product spaces.
method Local and global upper estimates for curvature and shape operator.
result Results on pseudo-hyperbolic spaces and space forms.
Study proves surfaces with constant curvature are simple shapes.
problem Characterizing singular minimal surfaces with constant curvature.
method Proved geometric properties of surfaces with constant curvature.
result Singular minimal surfaces with constant curvature are planes, spheres, and cylindrical surfaces.
Calculates S-curvature and mean Berwald curvature in homogeneous Finsler spaces.
problem Calculating curvatures in homogeneous Finsler spaces.
method Explicit formula for S-curvature and mean Berwald curvature in homogeneous generalized m-Kropina metric.
result Explicit formula for S-curvature and mean Berwald curvature in homogeneous generalized m-Kropina metric.