Related papers: On nonlocal minimal graphs
Graph classification benchmarks, vital for assessing and developing graph neural networks (GNNs), have recently been scrutinized, as simple methods like MLPs have demonstrated comparable performance. This leads to an important question: Do…
In this paper, we give definitions of three kinds of minimal charts, and we investigate properties of minimal charts and establish fundamental theorems characterizing minimal charts. To classify charts with two or three crossings we use the…
In this paper, local monomialization theorems are proven for analytic morphisms of complex and real analytic spaces. This gives the generalization of the local monomialization theorem for morphisms of algebraic varieties over a field of…
We present the first work-optimal polylogarithmic-depth parallel algorithm for the minimum cut problem on non-sparse graphs. For $m\geq n^{1+\epsilon}$ for any constant $\epsilon>0$, our algorithm requires $O(m \log n)$ work and $O(\log^3…
In this note, we use eigenvalue interlacing to derive an inequality between the maximum degree of a graph and its maximum and minimum adjacency eigenvalues. The case of equality is fully characterized.
We present an approach to reduced-order modelling that builds off recent graph-theoretic work for representation, exploration, and analysis of computed states of physical systems (Banerjee et al., Comp. Meth. App. Mech. Eng., 351, 501-530,…
Computing subgraph frequencies is a fundamental task that lies at the core of several network analysis methodologies, such as network motifs and graphlet-based metrics, which have been widely used to categorize and compare networks from…
We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under…
We present a parallel version of the cut-pursuit algorithm for minimizing functionals involving the graph total variation. We show that the decomposition of the iterate into constant connected components, which is at the center of this…
The purpose of this paper is to prove some results on the absence of bound states for certain nonlinear Schr\"odinger equations on noncompact metric graphs with localized nonlinearity. In particular, we show how the topological and metric…
Graph learning has become essential in various domains, including recommendation systems and social network analysis. Graph Neural Networks (GNNs) have emerged as promising techniques for encoding structural information and improving…
For real application and theoretical investigation of ordinary hypergraphs and non-ordinary hypergraphs, researchers need to establish standard rules and feasible operating methods. We propose a visualization tool for investigating…
This thesis presents a local-to-global perspective on graph neural networks (GNN), the leading architecture to process graph-structured data. After categorizing GNN into local Message Passing Neural Networks (MPNN) and global Graph…
This survey describes some useful properties of the local homology of abstract simplicial complexes. Although the existing literature on local homology is somewhat dispersed, it is largely dedicated to the study of manifolds, submanifolds,…
We propose a new step-wise approach to proving observational equivalence, and in particular reasoning about fragility of observational equivalence. Our approach is based on what we call local reasoning. The local reasoning exploits the…
We prove existence and uniqueness of minimizers for a family of energy functionals that arises in Elasticity and involves polyconvex integrands over a certain subset of displacement maps. This work extends previous results by Awi and Gangbo…
Problems in econometrics, insurance, reliability engineering, and statistics quite often rely on the assumption that certain functions are non-decreasing. To satisfy this requirement, researchers frequently model the underlying phenomena…
We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth…
Graph clustering is a fundamental problem that has been extensively studied both in theory and practice. The problem has been defined in several ways in literature and most of them have been proven to be NP-Hard. Due to their high practical…
We prove local inequalities for analytic functions defined on a convex body in $\Re^{n}$ which generalize well-known classical inequalities for polynomials.