Related papers: Information and dimensionality of anisotropic rand…
Complex systems, ranging from soft materials to wireless communication, are often organised as random geometric networks in which nodes and edges evenly fill up the volume of some space. Studying such networks is difficult because they…
Characterizing anisotropic correlations in quantum and statistical systems requires a coordinate-invariant framework. We introduce a geometric map based on the local informational line element, calibrated by the Euclidean benchmark scale…
We consider 15 properties of labeled random graphs that are of interest in the graph-theoretical and the graph mining literature, such as clustering coefficients, centrality measures, spectral radius, degree assortativity, treedepth,…
In high-dimensional graph learning problems, some topological properties of the graph, such as bounded node degree or tree structure, are typically assumed to hold so that the sample complexity of recovering the graph structure can be…
We classify isomorphism-invariant random digraphs according to where randomness resides, namely, arcs, vertices, and vertices and arcs together which in turn yield arc random digraphs (ARD), vertex random digraphs (VRD) and vertex-arc…
In this paper, we revisit the problem of sampling edges in an unknown graph $G = (V, E)$ from a distribution that is (pointwise) almost uniform over $E$. We consider the case where there is some a priori upper bound on the arboriciy of $G$.…
Various different random graph models have been proposed in which the vertices of the graph are seen as members of a metric space, and edges between vertices are determined as a function of the distance between the corresponding metric…
A random geometric graph $G(\mathcal{X}_n, r_n)$ is formed by taking a binomial process $\mathcal{X}_n$ as the set of vertices and joining any two distinct points with an edge if they lie within distance $r_n$ of each other. We investigate…
We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of…
In a random graph, counts for the number of vertices with given degrees will typically be dependent. We show via a multivariate normal and a Poisson process approximation that, for graphs which have independent edges, with a possibly…
It is well known that 3--regular graphs with arbitrarily large girth exist. Three constructions are given that use the former to produce non-Hamiltonian 3--regular graphs without reducing the girth, thereby proving that such graphs with…
In this work, we study a family of random geometric graphs on hyperbolic spaces. In this setting, N points are chosen randomly on a hyperbolic space and any two of them are joined by an edge with probability that depends on their hyperbolic…
We extend the latent position random graph model to the line graph of a random graph, which is formed by creating a vertex for each edge in the original random graph, and connecting each pair of edges incident to a common vertex in the…
In this paper we investigate the geometry of the likelihood of the unknown parameters in a simple class of Bayesian directed graphs with hidden variables. This enables us, before any numerical algorithms are employed, to obtain certain…
Gaussian graphical models emerge in a wide range of fields. They model the statistical relationships between variables as a graph, where an edge between two variables indicates conditional dependence. Unfortunately, well-established…
We investigate distributional properties of a class of spectral spatial statistics under irregular sampling of a random field that is defined on $\mathbb{R}^d$, and use this to obtain a test for isotropy. Within this context, edge effects…
We consider the following two algorithmic problems: given a graph $G$ and a subgraph $H\subseteq G$, decide whether $H$ is an isometric or a geodesically convex subgraph of $G$. It is relatively easy to see that the problems can be solved…
Random geometric graphs are widely used in modeling geometry and dependence structure in networks. In a random geometric graph, nodes are independently generated from some probability distribution $F$ over a metric space, and edges link…
A large driver of the complexity of graph learning is the interplay between structure and features. When analyzing the expressivity of graph neural networks, however, existing approaches ignore features in favor of structure, making it…
We study geometric random graphs defined on the points of a Poisson process in $d$-dimensional space, which additionally carry independent random marks. Edges are established at random using the marks of the endpoints and the distance…