Uniform Random Hypergraphs

Updated 11 December 2025

Uniform random hypergraphs are hypergraphs on a fixed vertex set where each k-subset is chosen uniformly at random, generalizing the Erdős–Rényi model.
They exhibit sharp threshold phenomena in areas such as colorability, Hamiltonicity, and bootstrap percolation, providing insights into phase transitions in high-dimensional structures.
Their study includes detailed spectral analysis, advanced enumeration, and efficient sampling methods that are critical for applications in CSPs, network science, and data structures.

A uniform random hypergraph is a fundamental probabilistic object in discrete mathematics and theoretical computer science, defined as a hypergraph on a fixed vertex set where each possible hyperedge of a specified cardinality (the uniformity) is chosen independently and with uniform probability. Its study unifies and generalizes classic random graph theory, providing a combinatorial framework essential to understanding high-dimensional random structures, threshold phenomena, spectral properties, and random constraint satisfaction problems.

1. Formal Models of Uniform Random Hypergraphs

There are two principal formulations for uniform random hypergraphs:

Binomial model $H_k(n,p)$ : Given a vertex set $V = [n]$ and fixed %%%%2%%%%, each $k$ -subset of $V$ appears as a hyperedge independently with probability $p \in [0, 1]$ . This is the direct hypergraph generalization of the classical Erdős–Rényi $G(n,p)$ model for graphs ( $k=2$ ).
Uniform model $H_k(n,m)$ : The hypergraph has $n$ vertices and exactly $m$ $k$ -edges, sampled uniformly at random (possibly with replacement, depending on the formulation).

The two models are tightly connected for sufficiently large $m$ and appropriate $p \approx m / \binom{n}{k}$ , but differ in probability space, particularly for sparse edge regimes (Achlioptas et al., 2020, Cooper, 2015).

2. Threshold Phenomena and Phase Transitions

Uniform random hypergraphs exhibit sharp threshold phenomena, analogous to those in random graphs but with richer complexity due to higher uniformity. Key examples include:

2-colorability: For $H_k(n, m = rn)$ with $k \geq 3$ , the sharp threshold for 2-colorability lies at

$r_c(k) = 2^{k-1} \ln 2 - \frac12 \ln 2,$

so that for $r \geq r_c(k)$ , the probability of 2-colorability tends to zero, and for $r \leq r_c(k)-1$ , it tends to one. The proof utilizes first and second moment methods, large deviations, and connections to the NAE $k$ -SAT threshold (Achlioptas et al., 2020).

Hamiltonicity: The threshold for the appearance of loose or tight Hamilton cycles in $k$ -uniform hypergraphs is determined sharply; for tight cycles ( $\ell = k-1$ ), the threshold for $H_k(n,p)$ is $p \sim e/n$ for all $k \geq 4$ (Dudek et al., 2011).
Bootstrap percolation: In $H_k(n,p)$ , for fixed $k, r > 1$ , the threshold for percolative spreading (infection) is at an initial infected set size $b_{k,r}(n,p)$ , with sharp phase transition between subcritical and supercritical regimes dictated by the initial seed size relative to $b_{k,r}(n,p)$ , itself depending critically on $n$ , $k$ , $r$ , and $p$ (Kang et al., 2017).
Zero-One Laws: First-order properties of $G^s(n,p = n^{-\alpha})$ exhibit a complete characterization: there is a countable set $A_s$ of exceptional exponents $\alpha$ for which the zero-one law fails, closely tracking phase transitions in the emergence of strictly balanced subhypergraphs (Matushkin, 2016).
Packing and isomorphism thresholds: The threshold for packing two independent $k$ -uniform random hypergraphs edge-disjointly is $p_1p_2 \sim (\log n)/n^{k-1}$ , paralleling perfect matching thresholds (Bollobás et al., 2014). Polynomial-time canonical labeling is possible for both binomial and $r$ -regular models in supercritical regimes (Chakraborti et al., 2020, Lenoir, 7 May 2024).

3. Spectral Properties and Local Weak Limits

There is increasing understanding of the spectral theory and local weak structure of uniform random hypergraphs:

Adjacency and Laplacian spectra: The adjacency eigenvalues and Laplacian spectra are connected to those of the all-ones hypermatrix and exhibit semicircle-type laws under certain scalings. For $k=2,3$ , the spectrum of the complete uniform hypergraph approximates that of a scaled all-ones tensor; the spectral radius of the symmetric Bernoulli hyperensemble grows as $O(n^{(k-1)/2}\sqrt{\log n})$ (Cooper, 2015, Lu et al., 2011).
Loose Laplacian spectra: For $1 \leq s \leq r/2$ , the $s$ -Laplacian eigenvalues of $H^r(n,p)$ exhibit a sharp spectral gap and, in the appropriate regime, their empirical distribution converges to a shifted semicircle law, with implications for high-order random walks and expansion (Lu et al., 2011).
Local weak convergence: In the sparse regime, the local neighborhood structure around a random $r$ -set in the $r$ -set weighted line graph $G_r(H_n)$ converges to a $(\binom{k}{r}-1)$ -block Galton–Watson tree with Poisson offspring. The limiting empirical spectral distribution of the adjacency matrix converges to the corresponding spectral measure of the Galton–Watson tree (Adhikari et al., 5 Sep 2025).

4. Degree Sequences, Enumeration, and Sampling

The enumeration and sampling of uniform random hypergraphs with a prescribed degree sequence is governed by intricate constraint systems:

Asymptotic enumeration: For dense, nearly regular sequences, the number of $r$ -uniform hypergraphs with a given degree sequence $d$ can be estimated asymptotically by solving a system involving auxiliary parameters $(\beta_1, ..., \beta_n)$ and evaluating determinant-based corrections, closely paralleling classical graph enumeration (Greenhill et al., 2021).
Sampling algorithms: The Hypercurveball Markov chain samples uniformly from the space $\mathcal{H}_d$ of simple hypergraphs with prescribed degrees, outperforming edge-swap based chains for large hyperedge counts. Its stationarity, irreducibility, and mixing time are controlled explicitly; under regularity assumptions, its mixing is $O(n^2\log n)$ , which is polynomially faster than edge-swap for $m=\Omega(n)$ (Kraakman et al., 6 Dec 2024).
Model emulation and hash-based constructions: Simple hash-based constructions can closely emulate uniform random hypergraphs in the context of hash-based data structures, with provably low error probabilities via deficiency and conditioning arguments (Aumüller et al., 2016).

5. Extremal and Universality Properties

Uniform random hypergraphs furnish "generic" counterexamples and extremal behaviors intrinsic to combinatorics:

Existential closure: For $H^{(k)}(m,p)$ with $p\gg m^{-(k-1)/n}$ , the property of being $n$ -existentially closed (n-e.c.) holds with high probability, generalizing Erdős–Rényi results for graphs. Explicit infinite families are constructed using combinatorial designs (MOLS, BIBDs, $t$ -designs) (Burgess et al., 8 Jul 2024).
Sparse extremal properties: Natural extremal theorems (e.g., Mantel’s theorem, Turán-type results) extend to the random $k$ -uniform context. For instance, in $G^{(4)}(n,p)$ with $p > K \log n / n$ , every maximum $T_4$ -free subhypergraph is 4-partite with high probability, for a fixed threshold $K$ and appropriate forbidden configuration $T_4$ (Gu et al., 2014).
Peelability and orientability: The phase transition for the existence of nontrivial $k$ -cores (minimum degree $\geq2$ ) and related properties is sharply determined by the edge density threshold $c_k$ . Alternative models with segment structure can raise these thresholds well beyond the classical Erdős–Rényi value, with implications for hashing-based data structures (Dietzfelbinger et al., 2019).

6. Applications and Broader Implications

Random uniform hypergraphs serve as testbeds for algorithm design and as reference models for real-world networks. Their applications include:

Random CSPs: The study of thresholds and solution space geometry for $k$ -SAT, coloring, and Not-All-Equal $k$ -SAT is grounded in random $k$ -uniform hypergraphs (Achlioptas et al., 2020).
Network motif randomization: Uniform hypergraph models underlie null models in network science, giving the baseline for statistical significance of observed topological features (Kraakman et al., 6 Dec 2024).
Data structures: Their peelability properties govern the performance and efficiency of hash-based structures such as cuckoo hashing, Invertible Bloom Lookup Tables, and retrieval data structures (Dietzfelbinger et al., 2019, Aumüller et al., 2016).

Open problems include refining threshold windows, understanding the precise limiting spectral distributions for $k>3$ , and extending efficient sampling to sparser, more irregular degree sequences.

References