Related papers: A Simple Differential Geometry for Networks and it…
The threshold network model is a type of finite random graphs. In this paper, we introduce a generalized threshold network model. A pair of vertices with random weights is connected by an edge when real-valued functions of the pair of…
We define the Ricci curvature on simplicial complexes by modifying the definition of the Ricci curvature on graphs, and we prove the upper and lower bounds of the Ricci curvature. These properties are generalizations of previous studies.…
Mostly acyclic directed networks, treated mathematically as directed graphs, arise in machine learning, biology, social science, physics, and other applications. Newman [1] has noted the mathematical challenges of such networks. In this…
We propose a data-driven mean-curvature solver for the level-set method. This work is the natural extension to $\mathbb{R}^3$ of our two-dimensional strategy in [DOI: 10.1007/s10915-022-01952-2][1] and the hybrid inference system of [DOI:…
Magnitude pruning is one of the mainstream methods in lightweight architecture design whose goal is to extract subnetworks with the largest weight connections. This method is known to be successful, but under very high pruning regimes, it…
While Graph Neural Networks (GNNs) have been successfully leveraged for learning on graph-structured data across domains, several potential pitfalls have been described recently. Those include the inability to accurately leverage…
Spin networks are at the core of quantum gravity. Our aim is to plug the mathematical community at large into the procedures turn to create a finite quantum theory of general relativity. For this, because of the different cultural…
Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric…
In this paper, we introduce a unified framework for defining Lin-Lu-Yau (LLY) Ricci curvature on both undirected and directed hypergraphs. By establishing upper bounds and monotonicity properties for the parameterized curvature…
We develop a comprehensive geometric framework for defining spaces $\mathcal{G}(M,E)$ of nonlinear generalized sections of vector bundles $E \to M$ containing spaces of distributional sections $\mathcal{D}'(M, E)$. Our theory incorporates…
We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for…
In this paper, we propose a novel formulation to extend CNNs to two-dimensional (2D) manifolds using orthogonal basis functions, called Zernike polynomials. In many areas, geometric features play a key role in understanding scientific…
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…
We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong…
We review and extend the recently proposed model of combinatorial quantum gravity. Contrary to previous discrete approaches, this model is defined on (regular) random graphs and is driven by a purely combinatorial version of Ricci…
Statistical ensembles of networks, i.e., probability spaces of all networks that are consistent with given aggregate statistics, have become instrumental in the analysis of complex networks. Their numerical and analytical study provides the…
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 is to find the discrete protoforms of…
Bridging geometry and topology, curvature is a powerful and expressive invariant. While the utility of curvature has been theoretically and empirically confirmed in the context of manifolds and graphs, its generalization to the emerging…
Real networks are finite metric spaces. Yet the geometry induced by shortest path distances in a network is definitely not its only geometry. Other forms of network geometry are the geometry of latent spaces underlying many networks, and…
We present a framework to define a large class of neural networks for which, by construction, training by gradient flow provably reaches arbitrarily low loss when the number of parameters grows. Distinct from the fixed-space global…