Characterizes Forman curvature bounds and proves curvature equivalence.
problem Characterize Forman curvature bounds and prove curvature equivalence.
method Contractivity of the Hodge Laplacian semigroup, translation between 2-cells and transport plans.
result Ollivier and Forman curvature coincide on edges when maximizing Forman curvature.
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…
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.
The study classifies graphs on surfaces with positive curvature properties.
problem Classifying graphs on surfaces with specific curvature properties.
method Using medial graphs and classification techniques.
result Complete classification of graphs on surfaces with positive Forman curvature and corner curvature.
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.
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.
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…
The paper introduces a new type of Ricci flow on graphs to study their curvature.
problem Understanding the curvature of graphs and their convergence properties.
method Proposes a weighted Forman and Lin-Lu-Yau Ricci flow on graphs and proves the existence and uniqueness of solutions.
result The normalized curvature flow on trees converges to a constant curvature metric.
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.
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.
We develop a computationally efficient method to estimate Ollivier-Ricci curvature.
problem Computational infeasibility of evaluating Ollivier-Ricci curvature on large graphs.
method Derive explicit transfer moduli between OR and BF curvatures, construct lazy transport envelopes, and use cross-edge matching.
result Deterministic bounds for OR curvature parameterized by local graph combinatorics, reducing complexity to worst-case O(max_v deg(v)^1.5).
The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching μ on a finite regular CW complex X, Forman introduced a discrete analogue of gradient flows. Although Forman's gradient flow has been proved to be useful in practical computations of homology groups, i…
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.
Extends Morse-Forman theory to vector-valued functions for multiparameter persistence.
problem Computing multiparameter persistence with new tools and methods.
method Adapting Forman's theory to vectorial setting and using combinatorial topological dynamics.
result Established more general result for sublevel sets and found a way to induce Morse decomposition.
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.
New method calculates discrete curvature using effective resistances.
problem Calculating discrete curvature on graphs.
method Effective resistances to calculate curvature on graph nodes and links.
result Relation to established discrete curvatures and convergence to continuous curvature.
Enhanced Markov chain sampler learns network statistics faster.
problem Learning network statistics efficiently.
method Integrates graph Forman curvature into Markov chain transition probabilities and stationary distribution.
result Curved Markov chain Monte Carlo achieves faster convergence.
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…
This paper shows that when the Riemannian metric on a contact manifold is blown up along the direction orthogonal to the contact distribution, the corresponding harmonic forms rescaled and normalized in the L2-norms will converge to Rumin's harmonic forms. This proves a conjecture in Gromov `` Carnot-Caratheodory sp…
Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.
problem Computing homology in discrete and smooth dynamical systems.
method Counting flow lines between orbits and critical points.
result Directly recovers Z2 homology from flow lines. 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.
A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical properties, e.g. the principal curvature directions dynamics on surfaces or the stress…
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.
We present a detailed description of a fundamental group algorithm based on Forman's combinatorial version of Morse theory. We use this algorithm in a classification problem of prime knots up to 14 crossings.
We derive a discrete analogue of Morse-Bott theory on CW complexes and use this discrete Morse-Bott function to do some Conley theory analysis. It turns out that our discrete Morse-Bott theory is indeed a generalization of Forman's discrete Morse theory.
Unified piecewise-linear Ricci flows improve community detection.
problem Improving community detection in graph neural networks.
method Proposed piecewise-linear Ricci curvature flows with surgeries.
result Flow consistently outperforms baseline models on real-world datasets.
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.
In the case of smooth manifolds, we use Forman's discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a couple triangulation-discrete Morse function. As an application, we prove that any Euler structure on a smooth oriented closed 3-manifold has a particular rea…
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.
For general Riemannian foliations, spectral asymptotics of the Laplacian is studied when the metric on the ambient manifold is blown up in directions normal to the leaves (adiabatic limit). The number of ``small'' eigenvalues is given in terms of the differentiable spectral sequence of the foliation. The asymptotics of…
After surveying classical notions of PL topology of the Seventies, we clarify the relation between Morse theory and its discretization by Forman. We show that PL handles theory and discrete Morse theory are equivalent, in the sense that every discrete Morse vector on some PL triangulation is also a PL handle vector, an…
It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical point…
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.
New spectral sequences derived from shellable tilings.
problem Discrete Morse theory and shellable complexes.
method Introduced tilings and quivers to support spectral sequences.
result Spectral sequences converge to relative (co)homology.
In his PhD thesis, Abrams proved that, for a natural number n and a graph G with at least n vertices, the n-strand configuration space of G deformation retracts to a compact subspace, the discretized n-strand configuration space, provided G satisfies two conditions: each path between distinct essential vertices (vertic…
In bounding the homology of a manifold, Forman's Discrete Morse theory recovers the full precision of classical Morse theory: Given a PL triangulation of a manifold that admits a Morse function with c_i critical points of index i, we show that some subdivision of the triangulation admits a boundary-critical discrete Mo…
The paper uses MDM theory to analyze multifiltering functions on simplicial complexes.
problem Understanding multifiltering functions through discrete Morse theory.
method Applying multiparameter discrete Morse theory to vector-valued multifiltering functions.
result Any multifiltering function can be approximated by a compatible MDM function.
The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow Φ:R×X→X on pfaffian set X is tame if the graph of Φ is a pfaffian subset of R×X×X. Any compact tame set admits plenty tame flows. We prove …
New technique connects graph matching complexes to Morse theory for better topology understanding.
problem Understanding the topology of matching complexes of complete graphs.
method Developed discrete Morse theory technique to analyze Mn. result Showed Mn is geometrically (νn−1)-connected, improving on previous homotopical results. Researchers use discrete Morse theory to improve the topology of matching complexes of complete graphs.
problem Understanding the topology of matching complexes of complete graphs, especially for small n.
method Developed gradient vector fields to simplify the computation of homology groups.
result Computed the homology groups of M7 efficiently and conjectured an optimal gradient vector field. We study different notions of Riemannian curvatures: The p-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 (p,q)-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.