This thesis explores Ollivier-Ricci curvature in graphs and manifolds, with applications to graph neural networks.
problem Understanding curvature in metric spaces and graphs.
method Combines optimal transport theory, Riemannian manifolds, and graph theory to define and analyze Ollivier-Ricci curvature.
result Extensions of Ollivier-Ricci curvature to directed graphs and applications in network science.
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.
New methods for calculating curvature in graph theory.
problem Calculating curvature in graphs and random walks.
method Analyzing continuous and discrete-time Ollivier-Ricci curvatures of weighted graphs.
result Generalized existence and properties of Ollivier-Ricci curvature for various random walks.
This study examines how removing edges from complete graphs affects Ollivier Ricci curvature.
problem Conditions under which Ollivier Ricci curvature changes sign after edge removal.
method Defined and analyzed graphs obtained by removing matching, vertex incident, and cycle edges from complete graphs.
result Ollivier Ricci curvature remains positive or zero for graphs formed by removing edges from complete graphs.
Unified curvature for hypergraphs from Ollivier-Ricci.
problem Generalizing curvature to hypergraphs.
method Developed ORCHID framework to generalize Ollivier-Ricci curvature to hypergraphs.
result ORCHID curvatures have favorable theoretical properties and are scalable for hypergraph tasks.
Graphs with bounded degrees and non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk.
problem Understanding geometric properties of graphs with non-negative Ollivier-Ricci curvature.
method Analyzing the geometric properties of graphs with non-negative Ollivier-Ricci curvature, proving subexponential growth and diffusive random walk.
result For graphs with bounded degrees and non-negative Ollivier-Ricci curvature, the average log-volume growth and random walk displacement are subexponential.
Graphs can be smoothed or squashed too, study finds.
problem Graph Neural Networks struggle with over-smoothing and over-squashing issues.
method Unified framework using Ollivier-Ricci curvature to address both issues.
result Over-smoothing and over-squashing linked to positive and negative graph curvature respectively.
We simplify evaluation of Ollivier-Ricci curvature bounds in hypergraphs.
problem Computational challenges in evaluating Ollivier-Ricci curvature bounds in hypergraphs.
method Simplified approach with linear computational complexity.
result Significant improvements in evaluating Ollivier-Ricci curvature bounds.
Graphs with non-negative Ollivier-Ricci curvature cannot be expanders.
problem Understanding the relationship between graph curvature and expansion properties.
method Proving an inequality linking isoperimetric profiles to total variation decay of random walks.
result Graphs with non-negative Ollivier-Ricci curvature cannot be expanders.
In this paper, we compare Ollivier Ricci curvature and Bakry-Émery curvature notions on combinatorial graphs and discuss connections to various types of Ricci flatness. We show that non-negativity of Ollivier Ricci curvature implies non-negativity of Bakry-Émery curvature under triangle-freeness and an additional in-de…
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 3 linear parts, with at most 2 linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartes…
Curvature formulas on regular graphs identified bone idle edges and graphs.
problem Understanding curvature in regular graphs and identifying bone idle edges.
method Explicit formulas for Lin-Lu-Yau and Ollivier-Ricci curvatures derived from graph parameters.
result Equality condition on regular graphs for Ollivier-Ricci curvature and characterization of bone idle edges.
The paper refines Steinerberger curvature for block graphs and bridges.
problem Understanding curvature in graph theory.
method Formulas and relations for curvature in block graphs and graph bridges.
result Self-centered Bonnet-Myers sharp graphs are antipodal.
New curvature measure for causal sets derived from optimal transport.
problem Capturing Ricci curvature in causal sets.
method Using Lorentzian optimal transport, novel curvature defined along maximal chains.
result Recovery of timelike Ricci curvature from order-theoretic data.
The study proves inequalities and curvature properties for Markov chains.
problem Isoperimetric and concentration inequalities for Markov chains.
method Laplacian separation principle for eikonal equation; modified log-Sobolev constant; Ollivier curvature.
result Affirmative answers to open questions and new inequalities.
Characterizes graphs with Lin-Lu-Yau curvature at least one and explores bone-idle graphs.
problem Characterizing graphs with specific curvature properties.
method Study of Ollivier-Ricci curvature and Lin-Lu-Yau curvature, exploration of regular graphs, and exact formula derivation.
result Characterizes edges that are bone-idle in regular graphs and provides a complete characterization of 4-regular bone-idle graphs.
We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs J(2n,n), the Gosset graph and suitable Cartesian …
The study examines discrete curvature notions on Cayley graphs of certain groups.
problem Understanding curvature in discrete settings for various groups.
method Introduced Right Angled Artin-Coxeter Hybrids (RAACHs) and derived curvatures of Cayley graphs.
result Addition of relators does not decrease weighted curvatures of Cayley graphs.
We prove the following estimate for the spectrum of the normalized Laplace operator Δ on a finite graph G, \begin{equation*}1- (1- k[t])^{\frac{1}{t}}\leq λ_1 \leq \cdots \leq λ_{N-1}\leq 1+ (1- k[t])^{\frac{1}{t}}, \,\forall \,\,\text{integers}\,\, t\geq 1. \end{equation*} Here k[t] is a lower bound for the Olli…
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).
Edge subdivision affects the Perron eigenvalue of tree Ricci matrices.
problem Understanding how edge subdivision impacts the Perron eigenvalue of tree Ricci matrices.
method Compressing branches into scalar feedback functions via Schur complement, reducing the spectral problem to a one-dimensional Chebyshev equation.
result Edge subdivision can decrease, preserve, or increase the Perron eigenvalue of tree Ricci matrices.
The study classifies graphs with specific curvature and maximum degree.
problem Graphs with nonnegative Ricci curvature and maximum degree constraints.
method Classification of graphs with Lin-Lu-Yau-Ollivier Ricci curvature, maximum degree ≤ 3, and diameter ≥ 6.
result Classification of graphs meeting the specified criteria.
The Ollivier Ricci flow with prescribed curvature on infinite graphs.
problem Ricci flow with prescribed curvature on infinite graphs.
method Existence and uniqueness of the solution to the Ricci flow.
result Convergence of the Ricci flow for graphs with girth at least 6.
Every connected, weighted graph with non-negative curvature has exactly two ends.
problem Characterizing the structure of connected, weighted graphs with non-negative curvature.
method Extremal Lipschitz extensions, variational principle, study of harmonic functions.
result Every salami has exactly two ends and no vertices with positive curvature.
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…
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.
JORC-UMAP improves UMAP by incorporating geometric and topological priors.
problem UMAP's local Euclidean distance assumption fails to capture intrinsic manifold geometry, leading to topological tearing and structural collapse.
method JORC-UMAP introduces Ollivier-Ricci curvature as a geometric prior and Jaccard similarity as a topological prior to reinforce edges and reduce redundant links.
result JORC-UMAP reduces tearing and collapse more effectively than standard UMAP and other DR methods, as measured by SVM accuracy and triplet preservation scores.
Graphs with stronger curvature grow faster.
problem Understanding volume growth on graphs with various curvatures.
method Examined inner-outer and Ricci-Ollivier curvatures to relate them to volume growth.
result Graphs with stronger inner-outer curvature growth have faster volume growth.
Lower bound on minimum vertex degree for non-negative Lin-Lu-Yau curvature on graphs.
problem Determining the minimum vertex degree for non-negative Lin-Lu-Yau curvature.
method Investigation of Ollivier-Ricci curvature and Lin-Lu-Yau modification on locally finite graphs.
result Lower bound on minimum vertex degree ensuring non-negative Lin-Lu-Yau curvature.
We propose a new graph kernel for graph classification and comparison using Ollivier Ricci curvature. The Ricci curvature of an edge in a graph describes the connectivity in the local neighborhood. An edge in a densely connected neighborhood has positive curvature and an edge serving as a local bridge has negative curv…
Study of Ricci flow on trees, focusing on edge weights and curvatures.
problem Understanding the evolution of metrics on trees under Ricci flow.
method Continuous-time Ricci flow based on Lin-Lu-Yau Ollivier Ricci curvature.
result Ricci flow converges to zero curvature on edge weights of positive normalized values in caterpillar trees.
Existence and uniqueness theorem for Ricci flow on weighted graphs proved.
problem Existence and uniqueness of solutions to Ricci flow equations on weighted graphs.
method Continuous time normalized Ricci flow approach.
result Existence and uniqueness theorem for solutions to Ricci flow on weighted graphs.
Graph Ricci flow reveals hidden hierarchies in stock market correlations.
problem Detecting hidden structures in the complex stock market graph.
method Using graph Ricci curvature and flow techniques to analyze the NASDAQ 100 index.
result Algorithm detects hidden hierarchies, community behavior, and clustering in financial markets.
Discrete time random walks on a finite set naturally translate via a one-to-one correspondence to discrete Laplace operators. Typically, Ollivier curvature has been investigated via random walks. We first extend the definition of Ollivier curvature to general weighted graphs and then give a strikingly simple representa…
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.
The monitoring of large dynamic networks is a major chal- lenge for a wide range of application. The complexity stems from properties of the underlying graphs, in which slight local changes can lead to sizable variations of global prop- erties, e.g., under certain conditions, a single link cut that may be overlooked du…
New framework uses geometry of embeddings to predict robustness.
problem Monitoring robustness in models without OOD labels.
method Constructs graphs from embeddings, measures spectral complexity and curvature.
result Representation geometry predicts robustness reliably.
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…
TopoGeoScore selects robust checkpoints using only source-domain representations.
problem Selecting robust checkpoints without target-domain labels or samples.
method Constructs class-conditional mutual k-nearest-neighbour graphs and extracts three interpretable signals.
result Source representations contain measurable global-local-topological evidence of robustness.
Study reveals decurve flows in graph propagation models.
problem Limitations of traditional graph analysis and propagation mechanisms.
method Introduces Generalized Propagation Neural Networks (GPNNs) and Continuous Unified Ricci Curvature (CURC).
result Observation of decurve flow during training of graph neural networks, revealing propagation dynamics.
Spectro-Riemannian Graph Neural Networks integrate spectral and curvature signals for better graph representation learning.
problem Enhance graph representation learning by leveraging spectral and curvature signals.
method Proposes Spectro-Riemannian Graph Neural Networks (CUSP) that combines spectral and curvature insights.
result Empirical evaluation shows CUSP outperforms state-of-the-art models by up to 5.3%.
GeomHerd predicts herding behavior before market prices move, using Ricci curvature of agent interaction graphs.
problem Quantifying herding behavior in markets that lags behind actual price movements.
method Develops a geometric framework to track coordination on agent interaction graphs, bypassing lag in price-correlation statistics.
result GeomHerd anticipates herding long before market baselines, with significant lead times in predictions.
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.