Extends knot concordance invariant to balanced spatial graphs using grid homology.
problem Defining a concordance invariant for balanced spatial graphs.
method Using grid homology to extend the invariant from knots to spatial graphs.
result The combinatorial Υ invariant is a concordance invariant for balanced spatial graphs. The article studies embeddings of edge-colored graphs related to balanced 3- and 4-manifolds.
problem Investigating embeddings of edge-colored dual graphs of balanced 3- and 4-manifolds.
method Introducing the concept of balanced genus and proving lower bounds for the genus of 3- and 4-manifolds.
result Established lower bounds for the balanced genus of 3- and 4-manifolds, and conditions for homeomorphism to spheres.
A {\em balanced} spatial graph has an integer weight on each edge, so that the directed sum of the weights at each vertex is zero. We describe the Alexander module and polynomial for balanced spatial graphs (originally due to Kinoshita \cite{ki}), and examine their behavior under some common operations on the graph. We…
It has been recently shown that a large class of balanced graph cuts allows for an exact relaxation into a nonlinear eigenproblem. We review briefly some of these results and propose a family of algorithms to compute nonlinear eigenvectors which encompasses previous work as special cases. We provide a detailed analysis…
New Ricci flow method for directed graphs with balancing factor.
problem Analyzing asymmetry in directed networks.
method Rigorous formulation of Ricci flow on directed weighted graphs with balancing factor.
result Existence and uniqueness of discrete Ricci flow solutions.
Extends knot polynomial to knotted 4-valent graphs.
problem Constructing an invariant for knotted 4-valent graphs.
method Graphical calculus and Reidemeister moves for 4-valent graphs.
result Extension of sl(n) polynomial to knotted 4-valent graphs. Balancing graph summarization and change detection in streaming data.
problem Balancing compression rate in graph summarization and accuracy in change detection.
method Introducing a probabilistic hierarchical latent variable model and optimizing parameters based on the minimum description length principle to balance the trade-off.
result Guaranteed suppression of Type I error probability (false alarms) in change detection.
New method predicts graph structure changes over time.
problem Existing graph prediction methods assume static vertices, limiting their applicability.
method Combines time series prediction with adapted FBA for growing graphs.
result Efficacy demonstrated on synthetic and real datasets.
New method finds balanced clusters in graphs using auxiliary information.
problem Finding balanced clusters in graphs with population-level constraints.
method Proposes individual-level balancing constraint and develops spectral clustering algorithms.
result Establishes first statistical consistency result for constrained spectral clustering.
The paper studies matrix normalization and graph balancing using a new functional and gradient descent.
problem Matrix normalization and graph balancing.
method A new functional called the non-normal energy, and gradient descent.
result Gradient descent of the non-normal energy converges to balanced graphs and preserves spectra and realness of weights.
This work improves policy-based training by proposing an evaluation balance objective for GFlowNets.
problem Reliable estimation of policy divergence under directed acyclic graphs remains challenging.
method Proposes an evaluation balance objective over partial episodes to measure policy divergence and improve policy-based training reliability.
result Evaluation balance strengthens policy-based training reliability and broadens its flexibility.
In this paper, we develop a novel weighted Laplacian method, which is partially inspired by the theory of graph Laplacian, to study recent popular graph problems, such as multilevel graph partitioning and balanced minimum cut problem, in a more convenient manner. Since the weighted Laplacian strategy inherits the virtu…
The study extends Tutte's conflict graph concept to nonplanar graphs.
problem Understanding the structure of nonplanar graphs through conflict graphs.
method Defining a signed conflict graph for maximally planar subgraphs and analyzing their balance.
result For graphs with a flat embedding, every maximal planar subgraph has unbalanced conflict graphs if and only if the graph is intrinsically linked.
In 2003, Ozsváth and Szabó defined the concordance invariant τ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of τ for knots in S3 and a combinatorial proof that τ gives a lower bound for the slice genus of a knot. Recently, Har…
Let f:S2→S2 be an orientation-preserving branched covering map of degree d≥2, and let Σ be an oriented Jordan curve passing through the critical values of f. Then Γ:=f−1(Σ) is an oriented graph on the sphere. In a group email discussion in Fall 2010, W. Thurston introduced balanced planar graphs a…
Spectral Clustering as a relaxation of the normalized/ratio cut has become one of the standard graph-based clustering methods. Existing methods for the computation of multiple clusters, corresponding to a balanced k-cut of the graph, are either based on greedy techniques or heuristics which have weak connection to th…
We generalize the construction of the Heegaard Floer homology for a singular knot to that for a balanced bipartite graph. For a given graph, we provide a combinatorial description of the Euler characteristic of its Heegaard Floer homology by using the "Kauffman states" on a graph diagram.
Graph machine learning lacks a balanced theory, focusing on expressive power and optimization.
problem Insufficient theoretical understanding of GNNs' generalization behavior.
method Develop a balanced theory focusing on expressive power, generalization, and optimization.
result Theoretical advancements need to align with practical success in graph machine learning.
The paper shows conflict graphs of Petersen family graphs are mostly unbalanced.
problem Understanding the balance of conflict graphs in Petersen family graphs.
method Analyzing maximally planar subgraphs and their conflict graphs.
result All but three strong conflict graphs from Petersen Family Graphs are unbalanced.
A new GCN model detects cryptocurrency fraud by considering network evolution and balance theory.
problem Detecting fraud in evolving signed cryptocurrency trust networks.
method Motif-aware temporal GCN using balance theory and learnable weights.
result The model outperforms existing methods on bitcoin datasets.
Several structure learning algorithms have been proposed towards discovering causal or Bayesian Network (BN) graphs. The validity of these algorithms tends to be evaluated by assessing the relationship between the learnt and the ground truth graph. However, there is no agreed scoring metric to determine this relationsh…
In graph-based active learning, algorithms based on expected error minimization (EEM) have been popular and yield good empirical performance. The exact computation of EEM optimally balances exploration and exploitation. In practice, however, EEM-based algorithms employ various approximations due to the computational ha…
GHNet improves graph learning by balancing homogeneity and heterogeneity.
problem Over-smoothing in GCN leads to similar node representations.
method GHNet uses gating units to balance homogeneity and heterogeneity in feature propagation.
result GHNet achieves larger receptive fields without over-smoothing.
DeepSphere improves spherical CNNs by balancing efficiency and rotation equivariance.
problem Designing efficient and rotation-equivariant convolutional layers for spherical data.
method Graph-based approach to represent spherical data, focusing on the number of vertices and neighbors.
result DeepSphere achieves state-of-the-art performance and demonstrates efficiency and flexibility.
FairACE improves fairness in GNNs by balancing node performance across degree groups.
problem Degree biases in GNNs lead to unequal prediction performance among nodes with varying degrees.
method Integrates asymmetric contrastive learning with adversarial training to balance performance between high-degree and low-degree nodes.
result Significantly improves degree fairness metrics while maintaining competitive accuracy.
In practical machine learning systems, graph based data representation has been widely used in various learning paradigms, ranging from unsupervised clustering to supervised classification. Besides those applications with natural graph or network structure data, such as social network analysis and relational learning, …
A well-known theorem of Wolpert shows that the Weil-Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more gene…
The existence of a balanced vertex is proven for geodesic nets with three boundary vertices.
problem Existence of a balanced vertex in geodesic nets with specific boundary conditions.
method Proof of existence on a general two-dimensional Riemannian surface.
result Existence of a balanced vertex for geodesic nets with three unbalanced boundary vertices.
Improved GFlowNets learn more efficiently with trajectory balance.
problem Inefficient credit assignment in GFlowNets leads to suboptimal learning.
method Proposed trajectory balance as a new learning objective.
result Trajectory balance leads to more efficient and robust GFlowNet learning.
This article explores and analyzes the unsupervised clustering of large partially observed graphs. We propose a scalable and provable randomized framework for clustering graphs generated from the stochastic block model. The clustering is first applied to a sub-matrix of the graph's adjacency matrix associated with a re…
Alexander polynomial equals spanning tree count at t=1.
problem Alexander polynomial for spatial graphs.
method Combinatorial constructions generalized to weighted graphs.
result Value of Alexander polynomial at t=1 equals weighted spanning tree count.
New algorithm for multiway spectral clustering on Grassmann manifolds.
problem Efficiently computing multiple eigenvectors of a nonlinear graph Laplacian.
method Direct multiway spectral clustering in p-norm, reformulated as minimization on Grassmann manifold. result Monotonic decrease of balanced graph cuts leads to optimal solutions.
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).
NodeSig efficiently computes binary node embeddings for scalable graph analysis.
problem Scalability issues in graph representation learning models.
method NodeSig uses random walk diffusion probabilities and stable random projections to compute binary node embeddings efficiently.
result NodeSig achieves a good balance between accuracy and efficiency on node classification and link prediction tasks.
Study on harmonic maps between cones, linking degrees to graph Laplacian eigenvalues.
problem Understanding harmonic maps between singular spaces.
method Analyzing homogeneous harmonic maps between simplicial cones and their degrees.
result Degrees of homogeneous harmonic maps are related to eigenvalues of discrete graph Laplacians.
Generalized Thurston's characterization for branched coverings of the 2-sphere.
problem Characterize branched coverings of the 2-sphere.
method Introduced local balance and operations against balanced graphs.
result New proof of a theorem by Eremenko-Gabrielov-Mukhin-Tarasov-Varchenko.
In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation φ_y+ [φ^{2}/2]_t=w, where w is a bounded function depending o…
Paper predicts future graph structures using time series methods.
problem Forecasting dynamic graph structures with unseen nodes and edges.
method Time series forecasting for node degree prediction combined with flux balance analysis.
result Demonstrated utility and applicability of the approach on synthetic and real-world datasets.
FairDTD improves fairness in GNNs by distilling dual teacher knowledge, balancing utility and bias.
problem Bias in GNN predictions due to sensitive attributes.
method Dual-Teacher Distillation with a causal graph model, feature and structure teachers, and graph-level distillation.
result Achieves optimal fairness while preserving high model utility.
Study exact community recovery in noisy SBM with limited queries.
problem Community recovery in noisy stochastic block models with limited queries.
method Balanced uniform querying, two-stage adaptive strategy, sublinear queries, subsampled graph.
result Adaptive querying can improve exact recovery limits in noisy SBM.
New geometric analysis of PWSPDs balances density and geometry in high-dimensional data.
problem Balancing density and geometry in high-dimensional data.
method Power-weighted shortest-path distances (PWSPDs) and their geometric and computational analyses.
result High probability guarantees on the equivalence of PWSPDs on complete and nearest neighbor graphs.
Energy savings for DNN inference on resource-constrained devices.
problem Energy efficiency in deep learning inference for constrained devices.
method Efficiently searches through equivalent DNN graphs to find the one with the least execution cost.
result Achieves 24% energy savings with minimal performance impact.
New method improves graph neural networks by considering different types of relations in sampling.
problem Current graph neural networks ignore relation types in biomedical graphs, leading to suboptimal performance.
method Proposes relation-dependent sampling for multi-relational graphs to balance relation frequency and importance.
result State-of-the-art graph neural networks achieve better accuracy and efficiency with relation-dependent sampling.
This paper studies the large sample asymptotics of data analysis procedures based on the optimization of functionals defined on k-NN graphs on point clouds. The paper is framed in the context of minimization of balanced cut functionals, but our techniques, ideas and results can be adapted to other functionals of rele…
Most state-of-the-art graph kernels only take local graph properties into account, i.e., the kernel is computed with regard to properties of the neighborhood of vertices or other small substructures. On the other hand, kernels that do take global graph propertiesinto account may not scale well to large graph databases.…
We study the problem of partitioning a small sample of n individuals from a mixture of k product distributions over a Boolean cube {0,1}K according to their distributions. Each distribution is described by a vector of allele frequencies in RK. Given two distributions, we use γ to denote the average $\el…
We consider the problem of learning the weighted edges of a balanced mixture of two undirected graphs from epidemic cascades. While mixture models are popular modeling tools, algorithmic development with rigorous guarantees has lagged. Graph mixtures are apparently no exception: until now, very little is known about wh…
BLISS optimizes GNN training by adaptively sampling nodes.
problem High computational costs in training GNNs on large graphs.
method Uses Bandit Layer Importance Sampling to dynamically select nodes.
result Improves GNN performance with reduced computational cost.