Related papers: Uniform linear embeddings of graphons

The $2$-cell embeddings of graphs on closed surfaces have been widely studied. It is well known that ($2$-cell) embedding a given graph $G$ on a closed orientable surface is equivalent to cyclically ordering the edges incident to each…

Random geometric graphs (RGGs) are commonly used to model networked systems that depend on the underlying spatial embedding. We concern ourselves with the probability distribution of an RGG, which is crucial for studying its random…

A graph embedding is a representation of graph vertices in a low-dimensional space, which approximately preserves properties such as distances between nodes. Vertex sequence-based embedding procedures use features extracted from linear…

When analyzing weighted networks using spectral embedding, a judicious transformation of the edge weights may produce better results. To formalize this idea, we consider the asymptotic behavior of spectral embedding for different…

A graph drawing in the plane is called an almost embedding if the images of any two non-adjacent simplices (i.e. vertices or edges) are disjoint. Almost embeddings (more precisely, their higher-dimensional analogues) naturally appear in…

Embedding graphs in a geographical or latent space, i.e.\ inferring locations for vertices in Euclidean space or on a smooth manifold or submanifold, is a common task in network analysis, statistical inference, and graph visualization. We…

In this paper, we study orthogonal colourings of random geometric graphs. Two colourings of a graph are orthogonal if they have the property that when two vertices receive the same colour in one colouring, then those vertices receive…

A random algebraic graph is defined by a group $G$ with a uniform distribution over it and a connection $\sigma:G\longrightarrow[0,1]$ with expectation $p,$ satisfying $\sigma(g)=\sigma(g^{-1}).$ The random graph…

Consider an ergodic unimodular random one-ended planar graph $\G$ of finite expected degree. We prove that it has an isometry-invariant locally finite embedding in the Euclidean plane if and only if it is invariantly amenable. By "locally…

Feature extraction and dimension reduction for networks is critical in a wide variety of domains. Efficiently and accurately learning features for multiple graphs has important applications in statistical inference on graphs. We propose a…

Given a graph $G$ and $p\in [0,1]$, the random subgraph $G_p$ is obtained by retaining each edge of $G$ independently with probability $p$. We show that for every $\epsilon>0$, there exists a constant $C>0$ such that the following holds.…

Let $F_G(P)$ be a functional defined on the set of all the probability distributions on the vertex set of a graph $G$. We say that $G$ is \emph{symmetric with respect to $F_G(P)$} if the uniform distribution on $V(G)$ maximizes $F_G(P)$.…

We study random graphs with latent geometric structure, where the probability of each edge depends on the underlying random positions corresponding to the two endpoints. We focus on the setting where this conditional probability is a…

This paper provides a necessary and sufficient condition for a random network with nodes Poissonly distributed on a unit square and a pair of nodes directly connected following a generic random connection model to be asymptotically almost…

Joint spectral embeddings facilitate analysis of multiple network data by simultaneously mapping vertices in each network to points in Euclidean space where statistical inference is then performed. In this work, we consider one such joint…

In this paper, we study the task of detecting the edge dependency between two weighted random graphs. We formulate this task as a simple hypothesis testing problem, where under the null hypothesis, the two observed graphs are statistically…

Let $X_1,..., X_n$ be independent, uniformly random points from $[0,1]^2$. We prove that if we add edges between these points one by one by order of increasing edge length then, with probability tending to 1 as the number of points $n$…

Let $G$ be a uniformly chosen simple (labelled) random graph with given degree sequence $\boldsymbol{d}$ and let $X,Y,L$ be edge-disjoint graphs on the same vertex set as $G$. We investigate the probability that $X \subseteq G$ and that $G…

There has been much recent interest in random graphs sampled uniformly from the n-vertex graphs in a suitable structured class, such as the class of all planar graphs. Here we consider a general 'bridge-addable' class of graphs - if a graph…

We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem---to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph---and…