Related papers: Limits of Embedded Graphs, and Universality Conjec…

Graph embedding has recently gained momentum in the research community, in particular after the introduction of random walk and neural network based approaches. However, most of the embedding approaches focus on representing the local…

We introduce a general class of algorithms and supply a number of general results useful for analysing these algorithms when applied to regular graphs of large girth. As a result, we can transfer a number of results proved for random…

Schramm's Locality Conjecture asserts that the value of the critical percolation parameter $p_c$ of a graph satisfying $p_c<1$ depends only on its local structure. In this note, we prove this conjecture in the particular case of transitive…

Amply regular graphs are graphs with local distance-regularity constraints. In this paper, we prove a weaker version of a conjecture proposed by Qiao, Park, and Koolen on diameter bounds of amply regular graphs and make new progress on…

Numerous important problems can be framed as learning from graph data. We propose a framework for learning convolutional neural networks for arbitrary graphs. These graphs may be undirected, directed, and with both discrete and continuous…

A graph $G$ is said to be ubiquitous, if every graph $\Gamma$ that contains arbitrarily many disjoint $G$-minors automatically contains infinitely many disjoint $G$-minors. The well-known Ubiquity conjecture of Andreae says that every…

We introduce an unsupervised graph embedding that trades off local node similarity and connectivity, and global structure. The embedding is based on a generalized graph Laplacian, whose eigenvectors compactly capture both network structure…

We establish that for q>=1, the class of convex combinations of q translates of a smooth probability density has local doubling dimension proportional to q. The key difficulty in the proof is to control the local geometric structure of…

Motivated in part by understanding average case analysis of fundamental algorithms in computer science, and in part by the wide array of network data available over the last decade, a variety of random graph models, with corresponding…

In this article, we study relations between the local geometry of planar graphs (combinatorial curvature) and em global geometric invariants, namely the Cheeger constants and the exponential growth. We also discuss spectral applications.

Consider a set of $n$ vertices, where each vertex has a location in $\mathbb{R}^d$ that is sampled uniformly from the unit cube in $\mathbb{R}^d$, and a weight associated to it. Construct a random graph by placing edges independently for…

Stable topological invariants are a cornerstone of persistence theory and applied topology, but their discriminative properties are often poorly-understood. In this paper we study a rich homology-based invariant first defined by Dey, Shi,…

Subgraph counts - in particular the number of occurrences of small shapes such as triangles - characterize properties of random networks, and as a result have seen wide use as network summary statistics. However, subgraphs are typically…

Let $\mathcal{P}_G$ be the family of all topologically mixing, but not exact self-maps of a topological graph $G$. It is proved that the infimum of topological entropies of maps from $\mathcal{P}_G$ is bounded from below by $(\log 3/…

Graph embedding aims at learning a vector-based representation of vertices that incorporates the structure of the graph. This representation then enables inference of graph properties. Existing graph embedding techniques, however, do not…

Embedding graph nodes into a vector space can allow the use of machine learning to e.g. predict node classes, but the study of node embedding algorithms is immature compared to the natural language processing field because of a diverse…

Graph embedding approaches attempt to project graphs into geometric entities, i.e, manifolds. The idea is that the geometric properties of the projected manifolds are helpful in the inference of graph properties. However, if the choice of…

In this article, we derive bounds for values of the global geometry of locally tessellating planar graphs, namely, the Cheeger constant and exponential growth, in terms of combinatorial curvatures. We also discuss spectral implications for…

Using topological circles in the Freudenthal compactification of a graph as infinite cycles, we extend to locally finite graphs a result of Oberly and Sumner on the Hamiltonicity of finite graphs. This answers a question of Stein, and gives…

In this paper, we investigate how local rules enforced at every node can influence the topology of a network. More precisely, we establish several results on the diameter of trees as a function of the number of nodes, as listed below. These…