arXiv research

The study classifies graphs with specific curvature and maximum degree.

Study Kazdan-Warner equations on graphs using Brouwer degree theory.

Graph data widely exist in many high-impact applications. Inspired by the success of deep learning in grid-structured data, graph neural network models have been proposed to learn powerful node-level or graph-level representation. However, most of the existing graph neural networks suffer from the following limitations…

Extends graph degree theorem to simplicial closure of Auter space.

New graphs with maximum degree 4 found to be Ricci-flat.

Graph Lie algebras have infinite prolongation if they have a vertex of degree one.

Efficiently matches random graphs with inhomogeneous edge probabilities.

Paper proves edge-connectivity equals minimum degree for graphs with non-negative curvature.

Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.

The G-degree of colored graphs is a key concept in the approach to Quantum Gravity via tensor models. The present paper studies the properties of the G-degree for the large class of graphs representing singular manifolds (including closed PL manifolds). In particular, the complete topological classification up to G-deg…

We define and study the statistical models in exponential family form whose sufficient statistics are the degree distributions and the bi-degree distributions of undirected labelled simple graphs. Graphs that are constrained by the joint degree distributions are called $dK$-graphs in the computer science literature and…

A graph clustering method that moves nodes to highest-degree neighbors.

The sinh-Gordon equation is solved on finite, symmetric graphs.

For a graph representation of a dataset, a straightforward normality measure for a sample can be its graph degree. Considering a weighted graph, degree of a sample is the sum of the corresponding row's values in a similarity matrix. The measure is intuitive given the abnormal samples are usually rare and they are dissi…

Graphs with bounded degrees and non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk.

Lower bound on minimum vertex degree for non-negative Lin-Lu-Yau curvature on graphs.

New spectral clustering method for graphs with uneven node degrees.

We prove that every simple graph of order 12 which has minimum degree 6 contains a K_6 minor.

GCNs favor high-degree nodes, leading to biased performance; a new method mitigates this.

FairACE improves fairness in GNNs by balancing node performance across degree groups.

A strong interaction is known to exist between edge-colored graphs (which encode PL pseudo-manifolds of arbitrary dimension) and random tensor models (as a possible approach to the study of Quantum Gravity). The key tool is the {\it G-degree} of the involved graphs, which drives the {\it $1/N$ expansion} in the tensor …

The paper corrects for node degree in spectral clustering using random walk Laplacian.

Graph alignment in two correlated random graphs refers to the task of identifying the correspondence between vertex sets of the graphs. Recent results have characterized the exact information-theoretic threshold for graph alignment in correlated Erdős-Rényi graphs. However, very little is known about the existence of e…

The paper explores graphons of line graphs from sparse finite graphs.

Improved model for grouping nodes in bipartite networks.

Study shows neural ODEs generalize well on synthetic graphs but struggle with degree heterogeneity and clustering.

This paper tests the multivariate normality of node degrees in Erdős-Rényi graphs.

We prove that two links related by a surgery along a connected, strict graph clasper of degree n are C_n-equivalent, i.e, related by a sequence of surgeries along strict tree claspers of degree n.

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of nonorientable surfaces into which such graphs may be embedded. In a previous paper by the authors, the problem of calculating whether a given *-graph in which all vertices have degree 4 or 6 admi…

Random graph matching refers to recovering the underlying vertex correspondence between two random graphs with correlated edges; a prominent example is when the two random graphs are given by Erdős-Rényi graphs $G(n,\frac{d}{n})$. This can be viewed as an average-case and noisy version of the graph isomorphism problem.…

Graphs with nonnegative Bakry-Émery curvature have volume doubling and Poincaré inequalities.

New algorithms find half-optimal independent sets in sparse graphs.

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael, and, independently, Mattman showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the …

Graphs with maximum degree Δ have at most O(1) equiangular lines for λ < 3/sqrt(2).

A new topology improves decentralized learning efficiency and accuracy.

Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if all interior edge degrees are even. Using a refinement process which cuts the in…

The paper bounds the complexity of GCNs using Rademacher complexity.

We prove diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result using only curvature and maximal vertex degree is sharp in case of hypercubes. The other result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result i…

A family of Markov blankets in a faithful Bayesian network satisfies the symmetry and consistency properties. In this paper, we draw a bijection between families of consistent Markov blankets and moral graphs. We define the new concepts of weak recursive simpliciality and perfect elimination kits. We prove that they ar…

IFH models graph generation with adjustable sequentiality.

We present a model for random simple graphs with a degree distribution that obeys a power law (i.e., is heavy-tailed). To attain this behavior, the edge probabilities in the graph are constructed from Bertoin-Fujita-Roynette-Yor (BFRY) random variables, which have been recently utilized in Bayesian statistics for the c…

The paper defines surface area for graphs and derives spectral estimates.

For given closed orientable 3-manifolds $M$ and $N$ let $cD(M,N)$ be the set of mapping degrees from $M$ to $N$. We address the problem: For which $N$, $cD(M,N)$ is finite for all $M$? The answer is known in Thurston's picture of closed orientable irreducible 3-manifolds unless the target is a non-trivial graph manifol…

Generative model captures hubs and dense communities in social networks.

Degree-Quant improves GNN efficiency by quantizing them without losing accuracy.

A graph is called intrinsically knotted if every embedding of the graph contains a knotted cycle. Johnson, Kidwell and Michael showed that intrinsically knotted graphs have at least 21 edges. Recently Lee, Kim, Lee and Oh, and, independently, Barsotti and Mattman, showed that $K_7$ and the 13 graphs obtained from $K_7$…

Given a finite or infinite planar graph all of whose faces have degree 4, we study embeddings in the plane in which all edges have length 1, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such emb…

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A *-graph is a graph endowed with a formal adjacency structure on the half-edges around each vertex, and an embedding of a *-graph is…