Abstract simplicial complex
View article
Generalized network structures: The configuration model and the canonical ensemble of simplicial complexes Open
Simplicial complexes are generalized network structures able to encode interactions occurring between more than two nodes. Simplicial complexes describe a large variety of complex interacting systems ranging from brain networks to social a…
View article
Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes Open
Higher-order networks have emerged as a powerful framework to model complex systems and their collective behavior. Going beyond pairwise interactions, they encode structured relations among arbitrary numbers of units through representation…
View article
Full reconstruction of simplicial complexes from binary contagion and Ising data Open
Previous efforts on data-based reconstruction focused on complex networks with pairwise or two-body interactions. There is a growing interest in networks with higher-order or many-body interactions, raising the need to reconstruct such net…
View article
Weighted growing simplicial complexes Open
Simplicial complexes describe collaboration networks, protein interaction networks, and brain networks and in general network structures in which the interactions can include more than two nodes. In real applications, often simplicial comp…
View article
Weighted simplicial complexes and their representation power of higher-order network data and topology Open
Hypergraphs and simplical complexes both capture the higher-order interactions of complex systems, ranging from higher-order collaboration networks to brain networks. One open problem in the field is what should drive the choice of the ado…
View article
Random walks on simplicial complexes and harmonicsâ \n Open
In this paper, we introduce a class of random walks with absorbing states on simplicial complexes. Given a simplicial complex of dimension d, a random walk with an absorbing state is defined which relates to the spectrum of the kâdimensi…
View article
Homological percolation transitions in growing simplicial complexes Open
Simplicial complex (SC) representation is an elegant mathematical framework for representing the effect of complexes or groups with higher-order interactions in a variety of complex systems ranging from brain networks to social relationshi…
View article
Combinatorial Lefschetz theorems beyond positivity Open
Consider a simplicial complex that allows for an embedding into $\mathbb{R}^d$. How many faces of dimension $\frac{d}{2}$ or higher can it have? How dense can they be? This basic question goes back to Descartes' "Lost Theorem" and Euler's …
View article
Balanced Hodge Laplacians optimize consensus dynamics over simplicial complexes Open
Despite the vast literature on network dynamics, we still lack basic insights into dynamics on higher-order structures (e.g., edges, triangles, and more generally, k-dimensional “simplices”) and how they are influenced through higher-order…
View article
Homotopy theory of complete Lie algebras and Lie models of simplicial sets Open
In a previous work, by extending the classical Quillen construction to the non‐simply connected case, we have built a pair of adjoint functors, model and realization, between the categories of simplicial sets and complete differential grad…
View article
Yoneda Lemma for Simplicial Spaces Open
We study the Yoneda lemma for arbitrary simplicial spaces. We do that by introducing left fibrations of simplicial spaces and studying their associated model structure, the covariant model structure . In particular, we prove a recognition …
View article
Discrete topological complexity Open
We introduce a notion of discrete topological complexity in the setting of simplicial complexes, using only the combinatorial structure of the complex and replacing the concept of homotopy by that of contiguous simplicial maps. We study th…
View article
Semi-simplicial spaces Open
This is an exposition of homotopical results on the geometric realization of semi-simplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The top…
View article
On Fredholm determinants in topology Open
Given an abstract simplicial complex G, the connection graph G' of G has as vertex set the faces of the complex and connects two if they intersect. If A is the adjacency matrix of that connection graph, we prove that the Fredholm character…
View article
Creating Semiflows on Simplicial Complexes from Combinatorial Vector\n Fields Open
Combinatorial vector fields on simplicial complexes as introduced by Robin\nForman have found numerous and varied applications in recent years. Yet, their\nrelationship to classical dynamical systems has been less clear. In recent work\nit…
View article
Hodgelets: Localized Spectral Representations of Flows On Simplicial Complexes Open
We develop wavelet representations for edge-flows on simplicial complexes, using ideas rooted in combinatorial Hodge theory and spectral graph wavelets. We first show that the Hodge Laplacian can be used in lieu of the graph Laplacian to c…
View article
Strong Collapse for Persistence Open
We introduce a fast and memory efficient approach to compute the persistent homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by …
View article
Simplicial Neural Networks Open
Poster presented at the workshop Geometry of Complex Web (Les Diablaretes, February 2-5 2020 https://sites.google.com/view/geocow2020).
View article
A Notion of Harmonic Clustering in Simplicial Complexes Open
We outline a novel clustering scheme for simplicial complexes that produces\nclusters of simplices in a way that is sensitive to the homology of the\ncomplex. The method is inspired by, and can be seen as a higher-dimensional\nversion of, …
View article
Simplicial Complexity: piecewise linear motion planning in robotics Open
Using the notion of contiguity of simplicial maps, we adapt Farber's topological complexity to the realm of simplicial complexes. We show that, for a finite simplicial complex $K$, our discretized concept recovers the topological complexit…
View article
The amazing world of simplicial complexes Open
Defined by a single axiom, finite abstract simplicial complexes belong to the simplest constructs of mathematics. We look at a a few theorems.
View article
Simplicial Attention Networks Open
Graph representation learning methods have mostly been limited to the modelling of node-wise interactions. Recently, there has been an increased interest in understanding how higher-order structures can be utilised to further enhance the l…
View article
Stable integral simplicial volume of 3‐manifolds Open
We show that non-elliptic prime 3-manifolds satisfy integral approximation for the simplicial volume, that is, that their simplicial volume equals the stable integral simplicial volume. The proof makes use of integral foliated simplicial v…
View article
Principled Simplicial Neural Networks for Trajectory Prediction Open
We consider the construction of neural network architectures for data on simplicial complexes. In studying maps on the chain complex of a simplicial complex, we define three desirable properties of a simplicial neural network architecture:…
View article
Systolic geometry and simplicial complexity for groups Open
Twenty years ago Gromov asked about how large is the set of isomorphism classes of groups whose systolic area is bounded from above. This article introduces a new combinatorial invariant for finitely presentable groups called simplicial co…
View article
One can hear the Euler characteristic of a simplicial complex Open
We prove that that the number p of positive eigenvalues of the connection Laplacian L of a finite abstract simplicial complex G matches the number b of even dimensional simplices in G and that the number n of negative eigenvalues matches t…
View article
Extending Homotopy Type Theory with Strict Equality Open
In homotopy type theory (HoTT), all constructions are necessarily stable under homotopy equivalence. This has shortcomings: for example, it is believed that it is impossible to define a type of semi-simplicial types. More generally, it is …
View article
Higher-order connection Laplacians for directed simplicial complexes Open
Higher-order networks encode the many-body interactions existing in complex systems, such as the brain, protein complexes, and social interactions. Simplicial complexes are higher-order networks that allow a comprehensive investigation of …
View article
Simplicial Complex Representation Learning Open
Simplicial complexes form an important class of topological spaces that are frequently used in many application areas such as computer-aided design, computer graphics, and simulation. Representation learning on graphs, which are just 1-d s…
View article
Spikes and spines in 4D Lorentzian simplicial quantum gravity Open
A bstract Simplicial approaches to quantum gravity such as quantum Regge calculus and spin foams include configurations where bulk edges can become arbitrarily large while the boundary edges are kept small. Spikes and spines are prime exam…