WoRSCH 2025
Workshop on Random Simplicial
Complexes and Hypergraphs
1-2nd September, 2025 | Nottingham Trent University
Omer Bobrowski:
Universality in Random Geometric Complexes
Random geometric complexes are simplicial complexes whose vertices are generated by a random point process in a metric space. In this talk we will focus on the homology (cycles/holes in various dimensions) of these complexes. Our main results show that the lifetime distribution of homological cycles obeys a universal law, that depends on neither the support nor the original distribution of the point process. This phenomenon was first discovered in an experimental study, and the theory has been developed over the past few years. We will present the main theoretical proven results and open conjectures, as well as some potential applications for this line of research.
Yanna Kraakman:
Sampling random hypergraphs with fixed degrees
Comparing a network to number of random networks can reveal structures of the network that are non-trivial, i.e., not caused by randomness. In this talk, we study two algorithms to sample random hypergraphs, both suited for undirected and directed hypergraphs. The algorithms are Markov chain Monte Carlo algorithms, so we consider the underlying Markov chain and experimentally compare the mixing times of the two algorithms, which is related to the speed of the algorithms. I will briefly show an application of the algorithms to a chemical reaction network, which we model as a directed hypergraph. This talk is based on the article https://doi.org/10.1093/comnet/cnaf007, which is joined work with Clara Stegehuis.
Guillermo Restrepo:
Chemical hypergraphs
Graphs have long found applications in chemistry as a par excellence model of molecular structures. However, not every molecular structure can be represented as a graph; in fact, only covalent molecules can. Although these constitute a large part of known compounds, other important classes—such as organometallic compounds and more complex molecular structures—still await an appropriate mathematical model. The challenge with non-covalent molecules is that they exhibit higher-order relations among atoms, rather than the typical binary relationships of covalent bonds. In this talk, I will present several cases of such higher-order atomic relationships and discuss how hypergraphs provide a suitable and comprehensive framework to capture the diversity of the molecular world.
Chemistry, however, is more than its substances: it also encompasses the methods by which these chemicals are produced or transformed. Reactions therefore constitute a central part of the discipline. From a mathematical perspective, a chemical reaction involves relationships among substances, which have traditionally been modelled using (directed) graphs. Yet graphs cannot capture the essential set-theoretical nature of reactions, namely the relationship between the set of reactants and the set of products. This is naturally represented by directed hypergraphs, which will be discussed in the talk.
Substances and their reactions together constitute the chemical space, which can be modelled as a gigantic network of directed hypergraphs. Recently, some features of this network and its temporal expansion have been analysed, raising questions about its nearness to random structures. This has motivated the development of an Erdős–Rényi model for chemical hypergraphs, whose properties will be presented.
The talk concludes with results on the temporal unfolding of the chemical space, which not only shed light on the practice of chemistry but also raise intriguing mathematical questions.
Hanlin Sun:
Higher-order percolation processes on multiplex hypergraphs
Abstract: Higher-order interactions are increasingly recognized as a fundamental aspect of complex systems ranging from the brain to social contact networks. Hypergraphs as well as simplicial complexes capture the higher-order interactions of complex systems and allow us to investigate the relation between their higher-order structure and their function. Here we establish a general framework for assessing hypergraph robustness and we characterize the critical properties of simple and higher-order percolation processes. This general framework builds on the formulation of the random multiplex hypergraph ensemble where each layer is characterized by hyperedges of given cardinality. We observe that in presence of the structural cutoff the ensemble of multiplex hypergraphs can be mapped to an ensemble of multiplex bipartite networks. We reveal the relation between higher-order percolation processes in random multiplex hypergraphs, interdependent percolation of multiplex networks, and 𝐾-core percolation. The structural correlations of the random multiplex hypergraphs are shown to have a significant effect on their percolation properties. The wide range of critical behaviors observed for higher-order percolation processes on multiplex hypergraphs elucidates the mechanisms responsible for the emergence of discontinuous transition and uncovers interesting critical properties which can be applied to the study of epidemic spreading and contagion processes on higher-order networks.
Giulia Preti:
Beyond the Configuration Model: Descriptive Null Models for (Hyper)graphs
Null models are essential tools for understanding complex networks: they generate randomized versions of a system that preserve chosen structural properties, allowing us to test whether observed patterns are statistically significant or simply due to chance. Classical null models have mostly been developed for simple graphs; however, many real-world systems are better represented by higher-order structures such as hypergraphs. At the same time, even in simple graphs, preserving additional constraints such as color assortativity can dramatically change the insights we obtain.
In this talk, I will present a family of null models we developed for three settings: bipartite graphs (linked to undirected hypergraphs), directed hypergraphs, and colored multigraphs. All of these models rely on Markov Chain Monte Carlo (MCMC) algorithms, with specialized swap operations — including restricted and parity swaps — that ensure uniform and efficient sampling of the corresponding ensembles.
Through case studies on purchase networks, endorsement data, and legislative co-sponsorship, I will show how selecting the appropriate null model and sampling it correctly yields qualitatively different insights into structural patterns. Finally, I will highlight ongoing work extending the idea of preserving color assortativity from graphs to hypergraphs, further expanding the scope of descriptive null models.
Michael Farber:
Models of random simplicial complexes and their topological properties
In my talk I will discuss the constructions of two major classes of probability measures on the set of simplicial complexes. I will present results on properties of random simplicial complexes under various assumptions on the probability parameters. The properties of random simplicial complexes which will appear in my talk include the Betti numbers, the fundamental groups, the Lusternik-Schnirelmann category, the topological complexity and some others.
Daniela Egas Santander:
How neuron physicality shapes the structure in biological neural networks
A strong hypothesis in neuroscience is that many aspects of brain function are determined by the ‘’map of the brain’’ and that its computational power relies on its connectivity architecture. Impressive scientific and engineering advances in recent years generated a plethora of large brain networks of incredibly complex architectures.
A central feature of the architecture is its inherent directionality, which reflects the flow of information. Evidence shows that reciprocal connections and higher order motifs, such as directed cliques, emerge selectively rather than at random in biological neural networks. This raises fundamental questions in both mathematics and computational neuroscience. In this talk, we explore how such structure arises from the physicality of the neurons themselves and propose a framework to control and quantify the over or under representation of higher order motifs.
Florian Unger:
Sampling Directed Flag Complexes
As the more constrained cousin of hypergraphs and simplicial complexes, flag complexes (the maximal simplicial complex over a given graph) are notoriously hard to sample from.
We will first have a look at an older (somewhat successful) MCMC-based attempt of sampling directed flag complexes, which in order to work further had to restrain connectivity between vertices, left to only modify directionality of the edges.
Then we discuss its shortcomings, both practical and theoretical. The latter takes the form of the refuting the following Conjecture, whose correctness was (strictly speaking) necessary for the samplers universal correctness: "Any finite simple oriented graph exhibit a sequence of edge-flips monotonically decreasing the number of oriented 3-cycles to zero". We present the construction of a counterexample to this Conjecture.
Lastly, we discuss various approaches to generalizing this approach to be no longer restricted to only modifying directionality, both from the perspective of small-steps iterative processes and global reshuffling procedures.
Chunyin Siu:
Homological and Homotopical Properties of Preferential Attachment Clique Complexes
The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ``hubs''. We study the higher-order connectivity of such a network by investigating the algebraic-topological properties of its clique complex. By determining the asymptotic growth rates of the Betti numbers, we discover that the graph undergoes higher-order phase transitions within the infinite-variance regime. Homotopically, we found similar phase transitions in the homotopy-connectedness of the infinite preferential attachment clique complex. This is a joint work with Gennady Samorodnitsky, Christina Lee Yu and Rongyi He. This talk is based on the articles https://doi.org/10.1017/apr.2024.66 and https://doi.org/10.1137/24M1693179Jan Spaliński:
From Erdős–Rényi graphs to Linial-Meshulam complexes via the multineighbor construction
Neighborhood complexes, introduced by Lovász in his proof of the Kneser conjecture, have long served as a bridge between combinatorics and algebraic topology. The m-neighbor complex Nm(G) generalizes this construction: its simplices are vertex sets in a graph G that share at least m common neighbors. In this talk, we investigate Nm(G(n, p)) for an Erdős–Rényi random graph G(n, p) and identify parameter regimes in which the complex almost surely exhibits a sharply concentrated dimension. Under explicit growth conditions on m and p, the random complex has a complete (t−2)-skeleton, no t-faces, and (t−1)-faces whose marginal probabilities match those of a Linial– Meshulam (t−1)-complex Yn,t−1,q. We give precise formulas for t and q, prove concentration bounds that force the dimension lock, and compare the resulting distribution with the independent-face Linial–Meshulam model. Our proofs rely on binomial tail estimates calibrated to the m-neighbor threshold and union bounds over candidate faces. We also interpret Nm(G) as a Dowker complex of the vertex–neighbor incidence relation, suggesting potential applications to topological data analysis. The presentation will outline the motivation from both topological combinatorics and random complex theory, state the main theorems, sketch the probabilistic arguments, and highlight examples illustrating the sharp transition in dimension.
Caterina De Bacco:
Broad Spectrum Structure Discovery in Large-Scale Higher-Order Networks
Like traditional networks, real-world hypergraphs exhibit mesoscale structure, patterns of interaction among groups of nodes.
Broadly, modeling such structure reduces to clustering nodes into groups and characterizing how those groups interact, if at all. In doing so, one reduces the conceptual complexity of the system and the dimensionality of the data, potentially revealing real functional components and underlying mechanisms that drive observed interactions.
Mesoscale structure comes in many different forms, not all of which are tractably modeled. Perhaps the most commonly modeled structure is assortative structure wherein nodes form “communities” and interact mostly with other similar nodes within the same community. Recent work develops methods for efficient detection of these communities in hypergraph data, which will be discussed in the talk.
However, many complex systems also exhibit some degree of disassortative structure, wherein similar nodes form “classes”, but may not interact within these classes. Within these classes, nodes have similar characteristics, but may (exclusively) interact with nodes in other classes.
Even in traditional network settings, any degree of disassortativity is generally difficult to model. This challenge stems from the baseline complexity required to model both how nodes form groups and how these combinations of groups interact.
For hypergraphs, this complexity compounds: higher-order interactions among nodes introduce higher-order interactions within and between groups, resulting in a combinatoric explosion in the number of model parameters that makes estimation impossible.
In this talk I will present a family of probabilistic generative models to tractably capture a wide range of mesoscale structure underlying
large-scale hypergraphs, ranging from strictly assortative to highly disassortative.
The key insight enabling this approach is to treat classes of similar units as themselves nodes in a latent hypergraph.
By modeling observed node interactions through latent interactions among classes using low-rank representations, this approach tractably
captures rich structural patterns while ensuring model identifiability.
I will present empirical results on real hypergraph data that shows how this approach improves link prediction tasks and discovers interpretable structures in various systems, including a case study on drug interactions data.
Finally, I will discuss some open challenges and questions.