arXiv research

The random graph is an infinite graph with the universal property that any embedding of $G-v$ extends to an embedding of $G$, for any finite graph. In this paper we show that this graph embeds in the curve graph of a surface $Σ$ if and only if $Σ$ has infinite genus, showing that the curve system on an infinite genus s…

Study geodesics on graphs with random lengths, proving bi-infinite paths exist.

The article studies random infinite ideal hyperbolic polyhedra and their dual graphs, establishing new boundary theories.

Drawing on some recent results that provide the formalism necessary to definite stationarity for infinite random graphs, this paper initiates the study of statistical and learning questions pertaining to these objects. Specifically, a criterion for the existence of a consistent test for complex hypotheses is presented,…

Graph convolutional networks (GCNs) are a widely used method for graph representation learning. We investigate the power of GCNs, as a function of their number of layers, to distinguish between different random graph models on the basis of the embeddings of their sample graphs. In particular, the graph models that we c…

We present a nonparametric prior over reversible Markov chains. We use completely random measures, specifically gamma processes, to construct a countably infinite graph with weighted edges. By enforcing symmetry to make the edges undirected we define a prior over random walks on graphs that results in a reversible Mark…

A known failing of many popular random graph models is that the Aldous-Hoover Theorem guarantees these graphs are dense with probability one; that is, the number of edges grows quadratically with the number of nodes. This behavior is considered unrealistic in observed graphs. We define a notion of edge exchangeability …

We construct an infinitely exchangeable process on the set $\cate$ of subsets of the power set of the natural numbers $\mathbb{N}$ via a Poisson point process with mean measure $Λ$ on the power set of $\mathbb{N}$. Each $E\in\cate$ has a least monotone cover in $\catf$, the collection of monotone subsets of $\cate$, an…

Graphs with bounded degrees and non-negative Ollivier-Ricci curvature have subexponential growth and diffusive random walk.

Graph convolutional networks (GCNs) are a widely used method for graph representation learning. To elucidate the capabilities and limitations of GCNs, we investigate their power, as a function of their number of layers, to distinguish between different random graph models (corresponding to different class-conditional d…

Researchers explore statistical perspectives to understand GNN generalization.

First-passage percolation affects graph properties like curvature and geodesics.

The paper proposes a Gaussian mixture model for Hilbert-space-valued data.

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eige…

DGP learns speech recognition by modeling complex relationships between utterances.

We introduce a class of generative network models that insert edges by connecting the starting and terminal vertices of a random walk on the network graph. Within the taxonomy of statistical network models, this class is distinguished by permitting the location of a new edge to explicitly depend on the structure of the…

GGP models multivariate time series with latent sub-sequences for diverse behaviors.

Statistical physics approaches can be used to derive accurate predictions for the performance of inference methods learning from potentially noisy data, as quantified by the learning curve defined as the average error versus number of training examples. We analyse a challenging problem in the area of non-parametric inf…

Data-driven methods link graphon limits to random walks and spectral clustering.

Every infinitely edge-connected graph has a minor of Farey graph or $T_{\aleph_0}\ast t$.

The study examines convergence of stochastic processes on large graphs and adjacency matrices.

We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense. We show that the graphs we construct are generally connected, infinite diameter a…

The paper explores non-amenability in infinite-type surfaces and graphs.

The study connects geodesic flows on Riemann surfaces to random walks on their dual graphs.

Study flip graphs for surfaces of infinite type, finding uncountably many connected components.

Parabolic mapping class acts on curve graphs of infinite type surfaces.

Project infinite time series graphs to finite marginal models using number theory.

Study of harmonic functions on infinite penny graphs.

Study polynomial growth harmonic functions on infinite penny graphs.

Study of mapping class groups on infinite graphs, focusing on their large-scale geometry.

We study arc graphs and curve graphs for surfaces of infinite topological type. First, we define an arc graph relative to a finite number of (isolated) punctures and prove that it is a connected, uniformly hyperbolic graph of infinite diameter; this extends a recent result of J. Bavard to a large class of punctured sur…

Generates special homeomorphisms for complex surfaces.

Study of flip graphs and their automorphism groups for infinite-type surfaces.

Graph Lie algebras have infinite prolongation if they have a vertex of degree one.

Numerous networks in the real world change over time, in the sense that nodes and edges enter and leave the networks. Various dynamic random graph models have been proposed to explain the macroscopic properties of these systems and to provide a foundation for statistical inferences and predictions. It is of interest to…

Study of mapping class groups of infinite graphs, focusing on their finiteness and commensurability.

We show that the compression body graph has infinite diameter.

Study shows surfaces without certain curves have infinite orbit graph.

Study reconstructs hidden perfect matchings in random graphs with specific edge weights.

New simplicial complex for infinite-type surfaces shows graph properties.

A nonparametric Bayesian sparse graph linear dynamical system (SGLDS) is proposed to model sequentially observed multivariate data. SGLDS uses the Bernoulli-Poisson link together with a gamma process to generate an infinite dimensional sparse random graph to model state transitions. Depending on the sparsity pattern of…

The grand arc graph's asymptotic dimension is shown to be infinite.

New rays on infinite type surfaces help understand their boundaries.

Study of pure mapping class groups on infinite graphs.

The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric …

The paper defines when surfaces are homotopy equivalent to graphs and explores their mapping class groups.

New insights into ends of quotient spaces and graphs.

We investigate the geometry of the graphs of nonseparating curves for surfaces of finite positive genus with potentially infinitely many punctures. This graph has infinite diameter and is known to be Gromov hyperbolic by work of the author. We study finite covers between such surfaces and show that lifts of nonseparati…