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Generating graph structures is a challenging problem due to the diverse representations and complex dependencies among nodes. In this paper, we introduce Graph Variational Recurrent Neural Network (GraphVRNN), a probabilistic autoregressive…
We report on work in progress on 'nested term graphs' for formalizing higher-order terms (e.g. finite or infinite lambda-terms), including those expressing recursion (e.g. terms in the lambda-calculus with letrec). The idea is to represent…
In this paper we consider aspects of geometric observability for hypergraphs, extending our earlier work from the uniform to the nonuniform case. Hypergraphs, a generalization of graphs, allow hyperedges to connect multiple nodes and…
An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a hierarchical embedding. Such hierarchical structure can be global in the data set, or…
Comparing networks is essential for a number of downstream tasks, from clustering to anomaly detection. Despite higher-order interactions being critical for understanding the dynamics of complex systems, traditional approaches for network…
Higher-order network analysis uses the ideas of hypergraphs, simplicial complexes, multilinear and tensor algebra, and more, to study complex systems. These are by now well established mathematical abstractions. What's new is that the ideas…
A hypergraph as a generalization of graphs records higher-order interactions among nodes, yields a more flexible network model, and allows non-linear features for a group of nodes. In this article, we propose a hypergraph echo state network…
We introduce a new concept called uncertainty spaces which is an extended concept of probability spaces. Then, we express n-layer uncertainty which we call hierarchical uncertainty by a hierarchically constructed sequence of uncertainty…
We propose a general multi-class visual recognition model, termed the Classifier Graph, which aims to generalize and integrate ideas from many of today's successful hierarchical recognition approaches. Our graph-based model has the…
We generalize the tree-confluent graphs to a broader class of graphs called Delta-confluent graphs. This class of graphs and distance-hereditary graphs, a well-known class of graphs, coincide. Some results about the visualization of…
Inside the discipline of graph theory exists an extension known as the hypergraph. This generalization of graphs includes vertices along with hyperedges consisting of collections of two or more vertices. One well-studied application of this…
We introduce a hypergraph matrix, named the unified matrix, and use it to represent the hypergraph as a graph. We show that the unified matrix of a hypergraph is identical to the adjacency matrix of the associated graph. This enables us to…
A conjecture regarding the structure of expander graphs is discussed.
We introduce a new operad-like structure that we call a reconnectad; the ``input'' of an element of a reconnectad is a finite simple graph, rather than a finite set, and ``compositions'' of elements are performed according to the notion of…
Random recursive hypergraphs grow by adding, at each step, a vertex and an edge formed by joining the new vertex to a randomly chosen existing edge. The model is parameter-free, and several characteristics of emerging hypergraphs admit neat…
Knowledge graph (KG) embedding is well-known in learning representations of KGs. Many models have been proposed to learn the interactions between entities and relations of the triplets. However, long-term information among multiple triplets…
Hypergraphs play a pivotal role in the modelling of data featuring higher-order relations involving more than two entities. Hypergraph neural networks emerge as a powerful tool for processing hypergraph-structured data, delivering…
We study the implications of the modeling choice to use a graph, instead of a hypergraph, to represent real-world interconnected systems whose constituent relationships are of higher order by nature. Such a modeling choice typically…
In this paper we will relate hyperstructures and the general $\mathscr{H}$-principle to known mathematical structures, and also discuss how they may give rise to new mathematical structures. The main purpose is to point out new ideas and…
The past few years have witnessed the great success of a new family of paradigms, so-called folksonomy, which allows users to freely associate tags to resources and efficiently manage them. In order to uncover the underlying structures and…