[Paper Review] Topology of random simplicial complexes: a survey

Matthew Kahle|arXiv (Cornell University)|Jan 30, 2013

Topological and Geometric Data Analysis4 citations

TL;DR

This survey explores the topology of random simplicial complexes, focusing on probabilistic models such as random flag complexes and multi-parameter models to understand the emergence of Betti numbers and homological features. It demonstrates that random complexes often exhibit expander-like properties and that expected Betti numbers can be accurately predicted using linearity of expectation, with strong agreement even for finite $ n $, suggesting deep connections to number theory and random matrix theory.

ABSTRACT

This expository article is based on a lecture from the Stanford Symposium on Algebraic Topology: Application and New Directions, held in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madsen.

Motivation & Objective

• To provide a comprehensive overview of the emerging field of random simplicial topology, emphasizing probabilistic models and their topological behavior.
• To investigate how random structures in combinatorics and number theory—such as complexes built from primes—exhibit topological regularity, including homotopy equivalence to wedges of spheres.
• To establish a probabilistic foundation for topological data analysis by modeling the statistical significance of topological features in data.
• To explore higher-dimensional analogues of expander graphs and their implications for embedding and spectral properties in simplicial complexes.
• To unify diverse models—random flag complexes, random 2-complexes, and multi-parameter models—under a common framework to predict Betti number distributions.

Proposed method

• Uses the random flag complex model $ X(n,p) $, where edges are included independently with probability $ p $, and higher-dimensional simplices are added if all their faces are present.
• Applies linearity of expectation to compute the expected reduced Euler characteristic $ ilde{ ho} = -1 + n - \binom{n}{2}p + \binom{n}{3}p^3 - \cdots $, which approximates the sum of Betti numbers.
• Employs the heuristic that nontrivial homology arises only when $ \dim A \ll \dim B \gg \dim C $, predicting $ \dim H_B \approx \max\{0, -\dim A + \dim B - \dim C\} $.
• Analyzes the multi-parameter model $ \Delta(n; p_1, p_2, \dots) $, where each $ k $-simplex is included independently given its $ (k-1) $-skeleton, generalizing random graphs and flag complexes.
• Leverages results from random matrix theory and spectral graph theory to draw analogies with expander graphs, extending concepts like spectral gap and expansion to higher dimensions.
• Validates predictions via computational experiments, such as 50 simulations of $ X(25,p) $, showing strong agreement between empirical averages and theoretical expectations of $ \mathbb{E}[\tilde{\chi}] $.

Experimental results

Research questions

• RQ1 How do Betti numbers behave in random simplicial complexes, and can their expected values be computed using probabilistic methods?
• RQ2 To what extent do random simplicial complexes exhibit expander-like properties in higher dimensions, and how do spectral and geometric expansion relate?
• RQ3 Why do many naturally occurring simplicial complexes (e.g., from number theory) appear homotopy equivalent to wedges of spheres, and can this be explained probabilistically?
• RQ4 Can the probabilistic method be used to construct complexes with specific topological features, such as prescribed Betti numbers or vanishing homology?
• RQ5 How do multi-parameter models unify different random complex models, and what insights do they offer into phase transitions and homological thresholds?

Key findings

• The expected reduced Euler characteristic of the random flag complex $ X(n,p) $ is given by $ \mathbb{E}[\tilde{\chi}] = -1 + n - \binom{n}{2}p + \binom{n}{3}p^3 - \binom{n}{4}p^6 + \cdots $, and this prediction matches empirical averages even for $ n = 25 $.
• For fixed $ k $, the Betti numbers of the prime complex $ \Delta_n $ satisfy $ \beta_k(\Delta_n) \approx \frac{n}{2\log n} \frac{("log n)^k}{k!} $, showing a Poisson-like growth in dimension.
• The sum of all Betti numbers of $ \Delta_n $ satisfies $ \sum_{k \geq 0} \beta_k(\Delta_n) = \frac{2n}{\pi^2} + O(n^\theta) $ for all $ \theta > 17/54 $, linking topological complexity to number-theoretic density.
• The Riemann hypothesis is equivalent to the bound $ |M(n)| = O(n^{1/2 + \epsilon}) $, where $ M(n) = \chi(\Delta_n) $, establishing a direct topological interpretation of the hypothesis.
• Random simplicial complexes exhibit expander-like properties: Newman–Rabinovich and Dotterrer show that they resist low-distortion embeddings in Euclidean space, similar to expander graphs.
• The multi-parameter model $ \Delta(n; p_1, p_2, \dots) $ generalizes random graphs ($ G(n,p) $), random 2-complexes ($ Y(n,p) $), and flag complexes ($ X(n,p) $), enabling systematic study of phase transitions.