Related papers: Finite graphs and amenability
Graph Neural Networks (GNNs) are a large class of relational models for graph processing. Recent theoretical studies on the expressive power of GNNs have focused on two issues. On the one hand, it has been proven that GNNs are as powerful…
The problem of node-similarity in networks has motivated a plethora of such measures between node-pairs, which make use of the underlying graph structure. However, higher-order relations cannot be losslessly captured by mere graphs and…
Machine learning on graphs, especially using graph neural networks (GNNs), has seen a surge in interest due to the wide availability of graph data across a broad spectrum of disciplines, from life to social and engineering sciences. Despite…
Consider a simple locally finite hypergraph on a countable vertex set, where each edge represents one unit of load which should be distributed among the vertices defining the edge. An allocation of load is called balanced if load cannot be…
Let $G$ be a finite group. The solubility graph associated with the finite group $G$, denoted by $\Gamma_{\cal S}(G)$, is a simple graph whose vertices are the non-trivial elements of $G$, and there is an edge between two distinct elements…
We define and study notions of amenability and skew-amenability of continuous actions of topological groups on compact topological spaces. Our main motivation is the question under what conditions amenability of a topological group passes…
Consider a sequence of finite regular graphs (GN) converging, in the sense of Benjamini-Schramm, to the infinite regular tree. We study the induced quantum graphs with equilateral edge lengths, Kirchhoff conditions (possibly with a non-zero…
Despite the exploding interest in graph neural networks there has been little effort to verify and improve their robustness. This is even more alarming given recent findings showing that they are extremely vulnerable to adversarial attacks…
We identify a sufficient condition, treewidth-pliability, that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint…
The purpose of this paper is to study the notion of relative extreme amenability for pairs of topological groups. We give a characterization by a fixed point property on universal spaces. In addition we introduce the concepts of an…
Hypergraphs are a generalization of graphs in which edges can connect any number of vertices. They allow the modeling of complex networks with higher-order interactions, and their spectral theory studies the qualitative properties that can…
We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of `relative compactness' for such graphs and study sufficient and necessary conditions for…
This survey describes some recent rigidity results obtained by the authors for the prescribed mean curvature problem on graphs $u : M \rightarrow \mathbb{R}$. Emphasis is put on minimal, CMC and capillary graphs, as well as on graphical…
A Riemann-Roch theorem on graph was initiated by M. Baker and S. Norine. In their article [2], a Riemann-Roch theorem on a finite graph with uniform vertex-weight and uniform edge-weight was established and it was suggested a Riemann-Roch…
The rankability of data is a recently proposed problem that considers the ability of a dataset, represented as a graph, to produce a meaningful ranking of the items it contains. To study this concept, a number of rankability measures have…
We establish a characterization of amenability for general Hausdorff topological groups in terms of matchings with respect to finite uniform coverings. Furthermore, we prove that it suffices to just consider two-element uniform coverings.…
Despite the celebrated popularity of Graph Neural Networks (GNNs) across numerous applications, the ability of GNNs to generalize remains less explored. In this work, we propose to study the generalization of GNNs through a novel…
Graph theoretical approach has proved an effective tool to understand, characterize and quantify the complex brain network. However, much less attention has been paid to methods that quantitatively compare two graphs, a crucial issue in the…
Graph neural networks have been shown to be very effective in utilizing pairwise relationships across samples. Recently, there have been several successful proposals to generalize graph neural networks to hypergraph neural networks to…
Complex networks are universal, arising in fields as disparate as sociology, physics, and biology. In the past decade, extensive research into the properties and behaviors of complex systems has uncovered surprising commonalities among the…