Random topology refers to the study of topological properties of randomly generated spaces. This area of research dates back to the seminal result by Erdős and Rényi on the connectivity of random graphs (Erdős and Rényi 1959). Over the past two decades this field has gained increasing attention, driven by two primary developments. The first is the elegant generalization of the Erdős-Rényi result, by Linial and Meshulam (2006), from connectivity in random graphs to homological-connectivity in random simplicial complexes. This result has facilitated a new area of study in combinatorial topology. The second development is the rise of Topological Data Analysis (TDA). This field focuses on utilizing methods from algebraic topology in the analysis of data and networks, based on their global shape. As data is random, probabilistic analysis is essential for developing solid statistical tests. In this line of research, the focus is on objects of geometric nature (i.e., geometric complexes, random fields). Though these two lines of research have evolved largely in parallel, they share numerous concepts and methods.

In this special issue, we tried to gather topics that cover the wide spectrum of studies that one might consider as “random topology". The types of models and phenomena that fall under this category are quite varied. The most fruitful line of study has been the topology of random simplicial complexes. The result in Linial and Meshulam (2006) has sparked a whole line of research on combinatorial models. There is a variety of models including the random k-complex, clique complex, multi-parameter complex, Rado complex, and more. There are also various phenomena studied, including phase transitions for homology and the fundamental group, spectral properties, central limit theorems, and more. The study of random geometric complexes was initiated by Kahle (2011), extending the notion of random geometric graphs (Penrose 2003). Here, the most well studied models are the Čech and the Vietoris-Rips complexes, which are also the ones often used in TDA. The range of phenomena studied in the geometric context is largely similar to the combinatorial ones, covering various phase transitions and limit theorems. In addition, there is a significant emphasis on analyzing persistent homology  (Hiraoka et al. 2018), the main workhorse in TDA. In addition to the study of random simplicial complexes, random topology covers many other areas including the study of random fields (Bobrowski and Borman 2012; Sarnak and Wigman 2019), random knots (Even-Zohar et al. 2016), random walks (Parzanchevski and Rosenthal 2016), growth models (Manin et al. 2023), and more.

We now give more context and background to the papers in the special issue. The entire collection of articles can be accessed online at https://springerlink.bibliotecabuap.elogim.com/collections/bdgddefcgg.

Determining connectivity threshold for random geometric graphs on the torus (Penrose 2003) was an important first step towards the study of homological connectivity thresholds for random geometric complexes (Bobrowski 2022) and it is expected that the behaviour will hold for random geometric complexes on compact manifolds without a boundary. The case of manifolds with a boundary is more delicate and in our special issue, (Penrose et al. 2023) has tried to address the same by considering largest nearest-neighbour link and connectivity threshold for random samples from a polytope. More importantly they identify the dependence on the geometry of polytopal domains. The thresholds for homological connectivity are now understood for many random complex models, built on undirected graphs (Bobrowski 2022; Kahle 2014; Linial and Meshulam 2006). As a new direction, in this special issue we see an attempt towards the study of homological connectivity thresholds in directed random graphs (Chaplin 2022) using path homology, a topological invariant which accounts for the asymmetry in directed graphs.

Another active line of investigation in random topology is limit theorems for various topological invariants of different random complexes and unsurprisingly, we find in the special issue, articles extending the state-of-art in this direction. In Botnan and Hirsch (2022), asymptotic normality is proved for multiparameter persistent Betti numbers complementing earlier works on single-parameter persistent Betti number. These results are used to derive goodness-of-fit tests, yielding an illuminating comparison with classical statistical tests. In the case of random cubical complexes, (Kanazawa et al. 2024) proves a large deviation principle for persistent diagrams and en route establishing a general method to lift large deviation principles from Betti numbers to persistence diagrams. In Temčinas et al. (2023) a multivariate normal approximations for Erdős -Rényi random clique complexes was studied. Extending Stein’s method for dissociated sums into the multivariate setting, enabled the derivation for central limit theorems for both the vector of simplex counts and the vector of Morse critical simplex counts. Also related to the Erdős -Rényi random clique complex, but in a different direction, (Ababneh and Kahle 2023) determines the order of maximal persistence. More precise asymptotics and distributional limits for maximal persistence of random complexes remain a challenge.

The special issue also covers other interesting properties of random complexes. The survey (Farber 2023) reviews recent progress in large random simplicial complexes, with particular focus on ampleness – a property characterizing stability and resilience of connectivity properties under small alterations of the complex. In Meshulam (2023); Hiraoka and Shirai (2024) the focus is on spectral properties of random complexes. The former article proves a spectral expansion for random Cayley complexes. This is a higher-dimensional analogue of Alon-Roichman theorem, which investigates spectral expansion for Cayley graphs with respect to random sets of generators. The latter article computes eigenvalues of the up-Laplacian on cubical lattices, which is used to derive torsion-weighted count of spanning acycles using Matrix-tree theorem for cell complexes. Acycles are a higher-dimensional analogue of trees and have recently attracted attention in random topology.

Moving forward, dynamical models can be expected to play an important role in random topology as in random graph theory. In Rosenthal (2023) branching random walks on simplicial complexes are introduced, and as in the classical case of graphs, their connection with spectral and topological properties of the simplicial complex are studied. In Hua et al. (2023) the Eden model (a classical growth model) is studied, on vertex-transitive graphs, examining the topology of its boundary profile. These results are applied to understand the behaviour of the Eden model on non-Euclidean spaces.

As an extremely useful companion to anyone wishing to understand excursion sets of random fields, (Wigman 2023) surveys progress in this direction with emphasis on ideas and techniques that have shaped the study of nodal domains.

With topological data analysis being a prime motivation for research in random topology, a special issue like this would be incomplete without a nod to statistical aspects of the subject. Already, we have seen in Botnan and Hirsch (2022), a more classical goodness-of-fit tests via multi-parameter persistence, and a comparison to other classical tests. Further motivated by TDA of connectomes, (Unger and Krebs 2024) studies an MCMC-based algorithm to sample directed graphs with a fixed number of cliques or equivalently simplices in the associated flag complex. The conclusion that observed high Betti numbers in the Blue Brain project are statistical outliers, is a very promising news for TDA overall. This, along with (Chaplin 2022) makes a case for more TDA-motivated research into directed random graphs. Siu et al. (2024) proposes a new robust method to distinguish small holes from noise, by investigating persistence diagrams with respect to an alternative filtration called the Robust Density-Aware Distance filtration.

To conclude, we wish to express our gratitude to all the authors who have enriched the special issue with a diverse collection of articles, encompassing the various topics within random topology. A special thank to the invited authors – Michael Farber, Matthew Kahle, Roy Meshulam, Mathew D. Penrose, Ron Rosenthal, Tomoyuki Shirai and Igor Wigman, who agreed to contribute to this special issue at its early stages. We also wish to thank the many referees for their thorough work helping to shape the contents of this special issue.