Related papers: $\ell^{p}$ improving estimates for multilinear for…
We present three results which support the conjecture that a graph is minimally rigid in $d$-dimensional $\ell_p$-space, where $p\in (1,\infty)$ and $p\not=2$, if and only if it is $(d,d)$-tight. Firstly, we introduce a graph bracing…
Consider the twin problems of estimating the connection probability matrix of an inhomogeneous random graph and the graphon of a W-random graph. We establish the minimax estimation rates with respect to the cut metric for classes of block…
Classes of graphs with bounded expansion are a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r,…
Let $L$ be a sequence $(\ell_1,\ell_2,\ldots,\ell_n)$ of $n$ lines in $\mathbb{C}^3$. We define the {\it intersection graph} $G_L=([n],E)$ of $L$, where $[n]:=\{1,\ldots, n\}$, and with $\{i,j\}\in E$ if and only if $i\neq j$ and the…
Graph Neural Networks (GNNs) have received increasing attention for representation learning in various machine learning tasks. However, most existing GNNs applying neighborhood aggregation usually perform poorly on the graph with…
Mapper graphs are widely used tools in topological data analysis and visualization. They can be understood as discrete approximations of Reeb graphs, providing insight into the shape and connectivity of complex data. Given a…
Random geometric graphs are random graph models defined on metric spaces. Such a model is defined by first sampling points from a metric space and then connecting each pair of sampled points with probability that depends on their distance,…
Although Graph Neural Networks (GNNs) have become the dominant approach for graph representation learning, their performance on link prediction tasks does not always surpass that of traditional heuristic methods such as Common Neighbors and…
This paper introduces Polynomial Graphical Lasso (PGL), a new approach to learning graph structures from nodal signals. Our key contribution lies in modeling the signals as Gaussian and stationary on the graph, enabling the development of a…
Graph embedding seeks to build a low-dimensional representation of a graph G. This low-dimensional representation is then used for various downstream tasks. One popular approach is Laplacian Eigenmaps, which constructs a graph embedding…
Motivated by the shape of transportation networks such as subways, we consider a distribution of points in the plane and ask for the network $G$ of given length $L$ that is optimal in a certain sense. In the general model, the optimality…
We prove an expanded range of $\ell ^{p}(\mathbb{Z}^d)$-improving properties and sparse bounds for discrete spherical maximal means in every dimension $d\geq 6$. Essential elements of the proofs are bounds for high exponent averages of…
We study the expected adjacency matrix of a uniformly random multigraph with fixed degree sequence $\mathbf{d} \in \mathbb{Z}_+^n$. This matrix arises in a variety of analyses of networked data sets, including modularity-maximization and…
We present a new, systematic approach for analyzing network topologies. We first introduce the dK-series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Increasing values of d…
Due to their flexibility to represent almost any kind of relational data, graph-based models have enjoyed a tremendous success over the past decades. While graphs are inherently only combinatorial objects, however, many prominent analysis…
The size sz(G) of an l_1-graph G=(V,E) is the minimum of n_f/t_f over all its possible l_1-embeddings f into n_f-dimensional hypercube with scale t_f. In terms of v=|V|, the sum of distances between all the pairs of vertices of G is at most…
Optimal transport and its related problems, including optimal partial transport, have proven to be valuable tools in machine learning for computing meaningful distances between probability or positive measures. This success has led to a…
Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with…
The distance matrix of a connected graph is the symmetric matrix with columns and rows indexed by the vertices and entries that are the pairwise distances between the corresponding vertices. We give a construction for graphs which differ in…
It is known from the algebraic graph theory that if $L$ is the Laplacian matrix of some tree $G$ with a vertex degree sequence $\mathbf{d}=(d_1, ..., d_n)^\top$ and $D$ is its distance matrix, then…