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We examine the generic local and global rigidity of various graphs in R^d. Bruce Hendrickson showed that some necessary conditions for generic global rigidity are (d+1)-connectedness and generic redundant rigidity and hypothesized that they…
Text classification aims to assign labels to textual units by making use of global information. Recent studies have applied graph neural network (GNN) to capture the global word co-occurrence in a corpus. Existing approaches require that…
We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link L there is a link containing L for which the volume…
Despite the widespread application of recurrent neural networks (RNNs) across a variety of tasks, a unified understanding of how RNNs solve these tasks remains elusive. In particular, it is unclear what dynamical patterns arise in trained…
Every graph G can be embedded in a Euclidean space as a two-distance set. This allows us to reformulate the analogue of Borsuk's conjecture for two-distance sets in terms of graphs. This conjecture remains open for dimensions from 4 to 63.…
Starting from the working hypothesis that both physics and the corresponding mathematics have to be described by means of discrete concepts on the Planck-scale, one of the many problems one has to face in this enterprise is to find the…
In this paper we study the dimension spectrum of general conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We perform a comprehensive study of the dimension spectrum addressing questions…
A simple graph more often than not contains adjacent vertices with equal degrees. This in particular holds for all pairs of neighbours in regular graphs, while a lot such pairs can be expected e.g. in many random models. Is there a…
Regular colored graphs are dual representations of pure colored D-dimensional complexes. These graphs can be classified with respect to an integer, their degree, much like maps are characterized by the genus. We analyse the structure of…
We study weighted residual dynamics associated with a rank-one projection in finite dimension. The iteration reduces, after finitely many steps, to a nonlinear recursion on a stabilized active subspace. We prove that this recursion can be…
We consider multiple and set-indexed sums of random vectors taking values in Euclidean space of growing dimension. It is shown that, when viewed as finite metric spaces, the sets of values of such sums converge in probability. The limit is…
What is a good vector representation of an object? We believe that it should be generative in 3D, in the sense that it can produce new 3D objects; as well as be predictable from 2D, in the sense that it can be perceived from 2D images. We…
We consider the question of whether the injective modules generate the unbounded derived category of a ring as a triangulated category with arbitrary coproducts. We give an example of a non-Noetherian commutative ring where they don't, but…
Recent research on graph neural network (GNN) models successfully applied GNNs to classical graph algorithms and combinatorial optimisation problems. This has numerous benefits, such as allowing applications of algorithms when preconditions…
The trust in the predictions of Graph Neural Networks is limited by their opaque reasoning process. Prior methods have tried to explain graph networks via concept-based explanations extracted from the latent representations obtained after…
High dimensional data can have a surprising property: pairs of data points may be easily separated from each other, or even from arbitrary subsets, with high probability using just simple linear classifiers. However, this is more of a rule…
Recent research in machine teaching has explored the instruction of any concept expressed in a universal language. In this compositional context, new experimental results have shown that there exist data teaching sets surprisingly shorter…
Threshold graphs are recursive deterministic network models that have been proposed for describing certain economic and social interactions. One drawback of this graph family is that it has limited generative attachment rules. To mitigate…
The graph reconstruction conjecture asserts that every simple graph on at least three vertices is uniquely determined by its deck of vertex-deleted subgraphs. In this expository article we survey the conjecture and present an…
Guarded recursion is a powerful modal approach to recursion that can be seen as an abstract form of step-indexing. It is currently used extensively in separation logic to model programming languages with advanced features by solving domain…