Related papers: Path homologies of deep feedforward networks

While standard persistent homology has been successful in extracting information from metric datasets, its applicability to more general data, e.g. directed networks, is hindered by its natural insensitivity to asymmetry. We study a…

Directed graphs can be studied by their associated directed flag complex. The homology of this complex has been successful in applications as a topological invariant for digraphs. Through comparison with path homology theory, we derive a…

Stratified digraphs are popular models for feedforward neural networks. However, computation of their path homologies has been limited to low dimensions due to high computational complexity. A recursive algorithm is proposed to compute…

A path graph is the intersection graph of paths in a tree. A directed path graph is the intersection graph of paths in a directed tree. Even if path graphs and directed path graphs are characterized very similarly, their recognition…

This paper focuses on developing an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique for such topological analysis involves computation of homology groups and their…

Neural networks are powerful function estimators, leading to their status as a paradigm of choice for modeling structured data. However, unlike other structured representations that emphasize the modularity of the problem -- e.g., factor…

In real-world systems, the relationships and connections between components are highly complex. Real systems are often described as networks, where nodes represent objects in the system and edges represent relationships or connections…

We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy…

We study random graph models for directed acyclic graphs, an important class of networks that includes citation networks, food webs, and feed-forward neural networks among others. We propose two specific models, roughly analogous to the…

This paper addresses the following question of neural network identifiability: Does the input-output map realized by a feed-forward neural network with respect to a given nonlinearity uniquely specify the network architecture, weights, and…

We present an algorithm to compute path homology for simple digraphs, and use it to topologically analyze various small digraphs en route to an analysis of complex temporal networks which exhibit such digraphs as underlying motifs. The…

Directed graphs are ubiquitous models for networks, and topological spaces they generate, such as the directed flag complex, have become useful objects in applied topology. The simplices are formed from directed cliques. We extend Atkin's…

The dynamics of large complex systems are predominately modeled through pairwise interactions, the principle underlying structure being a network of the form of a digraph or quiver. Significant success has been obtained in applying the…

Complex networks encountered in biology are often characterized by significant structural diversity. Whether it be differences in the three-dimensional structure of allosteric proteins, or the variation among the micro-scale structures of…

Directed acyclic graphs are a fundamental class of networks that includes citation networks, food webs, and family trees, among others. Here we define a random graph model for directed acyclic graphs and give solutions for a number of the…

The rapid progress of Artificial Intelligence research came with the development of increasingly complex deep learning models, leading to growing challenges in terms of computational complexity, energy efficiency and interpretability. In…

In weighted graphs the shortest path between two nodes is often reached through an indirect path, out of all possible connections, leading to structural redundancies which play key roles in the dynamics and evolution of complex networks. We…

Alignment, the tendency of adjacent weight matrices in deep networks to develop compatible subspace orientations, underlies gradient flow, Neural Collapse, and representation similarity across architectures. Despite extensive empirical…

We introduce new algebro-topological invariants of directed networks, based on the topological construction of the directed clique complex. The shape of the underlying directed graph is encoded in a way that can be studied mathematically to…

We explore pseudometrics for directed graphs in order to better understand their topological properties. The directed flag complex associated to a directed graph provides a useful bridge between network science and topology. Indeed, it has…