Related papers: Random Graphons and a Weak Positivstellensatz for …
A large body of work has investigated the properties of graph neural networks and identified several limitations, particularly pertaining to their expressive power. Their inability to count certain patterns (e.g., cycles) in a graph lies at…
In this paper we consider a random graph on which topological restrictions are imposed, such as constraints on the total number of edges, wedges, and triangles. We work in the dense regime, in which the number of edges per vertex scales…
In this work, we study the properties of sampling sets on families of large graphs by leveraging the theory of graphons and graph limits. To this end, we extend to graphon signals the notion of removable and uniqueness sets, which was…
Graph neural networks (GNNs) have become powerful tools for processing graph-based information in various domains. A desirable property of GNNs is transferability, where a trained network can swap in information from a different graph…
Using the theory of negative association for measures and the notion of random weak limits of sparse graphs, we establish the validity of the cavity method for counting spanning subgraphs subject to local constraints in asymptotically…
These notes concern aspects of various graphs whose vertex set is a group $G$ and whose edges reflect group structure in some way (so that they are invariant under the action of the automorphism group of $G$). The graphs I will discuss are…
We study spectral graph convolutional neural networks (GCNNs), where filters are defined as continuous functions of the graph shift operator (GSO) through functional calculus. A spectral GCNN is not tailored to one specific graph and can be…
For any fixed integer $R \geq 2$ we characterise the typical structure of undirected graphs with vertices $1, ..., n$ and maximum degree $R$, as $n$ tends to infinity. The information is used to prove that such graphs satisfy a labelled…
In a 1977 paper, Steffens identified an elegant criterion for determining when a countable graph has a perfect matching. In this paper, we will investigate the proof-theoretic strength of this result and related theorems. We show that a…
Introduced the quantitative measure of the structural complexity of the graph (complex network, etc.) based on a procedure similar to the renormalization process, considering the difference between actual and averaged graph structures on…
A sequence of graphs is FO-convergent if the probability of satisfaction of every first-order formula converges. A graph modeling is a graph, whose domain is a standard probability space, with the property that every definable set is Borel.…
We consider uniform random cographs (either labeled or unlabeled) of large size. Our first main result is the convergence towards a Brownian limiting object in the space of graphons. We then show that the degree of a uniform random vertex…
Graphon is a nonparametric model that generates graphs with arbitrary sizes and can be induced from graphs easily. Based on this model, we propose a novel algorithmic framework called \textit{graphon autoencoder} to build an interpretable…
In the short note, we describe a sampling construction that yields a sequence of graphons converging to a prescribed limit graphon in 1-norm. This convergence is stronger than the convergence in the cut norm, usually used to study graphon…
A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on…
The functionality of a graph $G$ is the minimum number $k$ such that in every induced subgraph of $G$ there exists a vertex whose neighbourhood is uniquely determined by the neighborhoods of at most $k$ other vertices in the subgraph. The…
We give a survey of basic results on the cut norm and cut metric for graphons (and sometimes more general kernels), with emphasis on the equivalence problem. The main results are not new, but we add various technical complements, and a new…
We establish a central limit theorem for the sum of $\epsilon$-independent random variables, extending both the classical and free probability setting. Central to our approach is the use of graphon limits to characterize the limiting…
Large-scale graph machine learning is challenging as the complexity of learning models scales with the graph size. Subsampling the graph is a viable alternative, but sampling on graphs is nontrivial as graphs are non-Euclidean. Existing…
Signal analysis on graphs relies heavily on the graph Fourier transform, which is defined as the projection of a signal onto an eigenbasis of the associated shift operator. Large graphs of similar structure may be represented by a graphon.…