Institute Output
A Cosine Rule-Based Discrete Sectional Curvature for Graphs
Xerxes D. Arsiwalla, J.F. Du Plessis
How does one generalize differential geometric constructs such as curvature of a manifold to the discrete world of graphs and other combinatorial structures? This problem carries significant importance for analyzing models of discrete spacetime in quantum gravity; inferring network geometry in network science; and manifold learning in data science. The key contribution of this paper is to introduce and validate a new estimator of discrete sectional curvature for random graphs with low metric-distortion.
Heaps of Fish: arrays, generalized associativity and heapoids
Carlos Zapata-Carratala, Xerxes D. Arsiwalla, Taliesin Beynon
In this paper we investigate a ternary generalization of associativity by defining a diagrammatic calculus of hypergraphs that extends the usual notions of tensor networks, categories and relational algebras. In doing so we rediscover the ternary structures known as heaps and are able to give a more comprehensive treatment of their mergence in the context of dagger categories and their generalizations.
Some Relativistic and Gravitational Properties of the Wolfram Model
Jonathan Gorard
The article shows that causal invariance in the Wolfram Model leads to discrete general and Lorentz covariance, introducing curvature concepts for hypergraphs related to the Ricci tensor and Einstein field equations