arXiv subjects

Marzieh Eidi

Publications and source records attributed to Marzieh Eidi.

8 recordsLinked to original sources

arXiv2026

From the discrete to the continuous, from simplicial complexes to Riemannian manifolds. Approximating flows and cuts on manifolds by discrete versions

Many fundamental structures of Riemannian geometry have found discrete counterparts for graphs or combinatorial ones for simplicial complexes. These include those discussed in this survey, Hodge theory, Morse theory, the spectral theory of Laplace type operators and Cheeger inequalities, and their interconnections. This raises the question of the relation between them, abstractly as structural analogies and concretely what happens when a graph constructed from random sampling of a Riemannian manifold or a simplicial complex triangulating such a manifold converge to that manifold. We survey the current state of research, highlighting some recent developments like Cheeger type inequalities for the higher dimensional geometry of simplicial complexes, Floer type constructions in the presence of periodic or homoclinic orbits of dynamical systems or the disorientability of simplicial complexes.

math.DG↗

arXiv2025

Geometric characterisation of structural and regular equivalences in undirected (hyper)graphs

Similarity notions between vertices in a graph, such as structural and regular equivalence, are one of the main ingredients in clustering tools in complex network science. We generalise structural and regular equivalences for undirected hypergraphs and provide a characterisation of structural and regular equivalences of undirected graphs and hypergraphs through neighbourhood graphs and Ollivier-Ricci curvature. Our characterisation sheds new light on these similarity notions opening a new avenue for their exploration. These characterisations also enable the construction of a possibly wide family of regular partitions, thereby offering a new route to a task that has so far been computationally challenging.

math.CO↗

arXiv2025

Demystifying Topological Message-Passing with Relational Structures: A Case Study on Oversquashing in Simplicial Message-Passing

Topological deep learning (TDL) has emerged as a powerful tool for modeling higher-order interactions in relational data. However, phenomena such as oversquashing in topological message-passing remain understudied and lack theoretical analysis. We propose a unifying axiomatic framework that bridges graph and topological message-passing by viewing simplicial and cellular complexes and their message-passing schemes through the lens of relational structures. This approach extends graph-theoretic results and algorithms to higher-order structures, facilitating the analysis and mitigation of oversquashing in topological message-passing networks. Through theoretical analysis and empirical studies on simplicial networks, we demonstrate the potential of this framework to advance TDL.

cs.LG↗

arXiv2024

Higher Order Bipartiteness vs Bi-Partitioning in Simplicial Complexes

Bipartite graphs are a fundamental concept in graph theory with diverse applications. A graph is bipartite iff it contains no odd cycles, a characteristic that has many implications in diverse fields ranging from matching problems to the construction of complex networks. Another key identifying feature is their Laplacian spectrum as bipartite graphs achieve the maximum possible eigenvalue of graph Laplacian. However, for modeling higher-order connections in complex systems, hypergraphs and simplicial complexes are required due to the limitations of graphs in representing pairwise interactions. In this article, using simple tools from graph theory, we extend the cycle-based characterization from bipartite graphs to those simplicial complexes that achieve the maximum Hodge Laplacian eigenvalue, known as disorientable simplicial complexes. We show that a $N$-dimensional simplicial complex is disorientable if its down dual graph contains no simple odd cycle of distinct edges and no twisted even cycle of distinct edges. Furthermore, we see that in a $N$-simplicial complex without twisting cycles, the fewer the number of (non-branching) simple odd cycles in its down dual graph, the closer is its maximum eigenvalue to the possible maximum eigenvalue of Hodge Laplacian. Similar to the graph case, the absence of odd cycles plays a crucial role in solving the bi-partitioning problem of simplexes in higher dimensions.

math.CO↗

arXiv2024

Floer Homology: From Generalized Morse-Smale Dynamical Systems to Forman's Combinatorial Vector Fields

We construct a Floer type boundary operator for generalised Morse-Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the $\mathbb{Z}_2$ homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.

math.DS↗

arXiv2023

Irreducibility of Markov Chains on simplicial complexes, the Spectrum of the Discrete Hodge Laplacian and Homology

Random walks on graphs are a fundamental concept in graph theory and play a crucial role in solving a wide range of theoretical and applied problems in discrete math, probability, theoretical computer science, network science, and machine learning. The connection between Markov chains on graphs and their geometric and topological structures is the main reason why such a wide range of theoretical and practical applications exist. Graph connectedness ensures irreducibility of a Markov chain. The convergence rate to the stationary distribution is determined by the spectrum of the graph Laplacian which is associated with lower bounds on graph curvature. Furthermore, walks on graphs are used to infer structural properties of underlying manifolds in data analysis and manifold learning. However, an important question remains: can similar connections be established between Markov chains on simplicial complexes and the topology, geometry, and spectral properties of complexes? Additionally, can we gain topological, geometric, or analytic information about a manifold by defining appropriate Markov chains on its triangulations? These questions are not only theoretically important but answers to them provide powerful tools for the analysis of complex networks that go beyond the analysis of pairwise interactions. In this paper, we provide an integrated overview of the existing results on random walks on simplicial complexes, using the novel perspective of signed graphs. This perspective sheds light on previously unknown aspects such as irreducibility conditions. We show that while up-walks on higher dimensional simplexes can never be irreducible, the down walks become irreducible if and only if the complex is orientable. We believe that this new integrated perspective can be extended beyond discrete structures and enables exploration of classical problems for triangulable manifolds.

math.SP↗

arXiv2020

Edge-based analysis of networks: Curvatures of graphs and hypergraphs

The relations, rather than the elements, constitute the structure of networks. We therefore develop a systematic approach to the analysis of networks, modelled as graphs or hypergraphs, that is based on structural properties of (hyper)edges, instead of vertices. For that purpose, we utilize so-called network curvatures. These curvatures quantify the local structural properties of (hyper)edges, that is, how, and how well, they are connected to others. In the case of directed networks, they assess the input they receive and the output they produce, and relations between them. With those tools, we can investigate biological networks. As examples, we apply our methods here to protein-protein interaction, transcriptional regulatory and metabolic networks.

cs.DM↗

arXiv2019

Ollivier Ricci Curvature of Directed Hypergraphs

We develop a definition of Ricci curvature on directed hypergraphs and explore the consequences of that definition. The definition generalizes Ollivier's definition for graphs. It involves a carefully designed optimal transport problem between sets of vertices.

cs.DM↗