Multi-Parameter Random Simplicial Complex Model

• The paper demonstrates that the multi-parameter random simplicial complex model generalizes classical models like Erdős–Rényi graphs through distinct simplex inclusion probabilities.
• It emphasizes recursive construction and sharp threshold phenomena, with Betti numbers and other invariants scaling predictably as functions of the probability vector.
• The model enables precise statistical inference and phase transition analyses, offering actionable insights into topological, geometric, and algebraic properties.

A multi-parameter random simplicial complex model is a stochastic framework for generating random abstract simplicial complexes, where the probability of including faces of given dimension is governed by a vector of probabilities or exponents for each dimension. This model strictly generalizes the classical Erdős–Rényi random graph, Linial–Meshulam complexes, and random clique complexes by assigning a distinct probability parameter to the inclusion of each $k$-simplex, conditional on its boundary being present. The topological, geometric, and algebraic features of complexes generated by this model exhibit sharp threshold phenomena and phase transitions, typically described by convex regions in the parameter space. Multi-dimensional algebraic-topological invariants such as Betti numbers, cohomology rings, fundamental group properties, and spectral laws depend intricately on the full multi-parameter vector.

1. Model Definition and Construction

Let $V=[n]=\{1,2,\dots,n\}$ be a finite vertex set and fix maximal dimension $r$. The model is specified by probability vector $\p=(p_0, p_1, \dots, p_r)\in [0,1]^{r+1}$, or, equivalently, by exponents $\alpha_i$ so that $p_i = n^{-\alpha_i}$.

The construction is recursive:

• Start with the 0-skeleton: all vertices are present with probability $p_0$.
• For each $k = 1$ to $r$, and for each $(k+1)$-subset $\sigma \subset V$ whose entire boundary $\partial\sigma$ lies in the existing complex, include $\sigma$ as a $k$-simplex independently with probability $p_k$.
• The resulting complex is downward-closed and denoted $Y_r(n;\p)$ (Costa et al., 2014).

The probability of a given complex $Y$ is

$$P_r(Y) = \prod_{i=0}^r p_i^{f_i(Y)} (1-p_i)^{e_i(Y)},$$

where $f_i(Y)$ is the number of $i$-faces and $e_i(Y)$ the number of external $i$-faces whose boundary lies in $Y$ but which are themselves absent (Costa et al., 2015).

Alternative formulations via hypergraph closures (upper model) and downward-closures (lower model) yield slightly different stochastic laws for face inclusion and generate dual large-scale topological phenomena (Kogan, 2024, Farber et al., 2022).

2. Relation to Classical Simplicial Complex Models

Special parameter settings recover classical models:

• Erdős–Rényi random graph $G(n,p_1)$: $r=1$, $p_0=1$, $p_1=p_1$, $p_i=0$ for $i>1$.
• Random clique complex: $r\geq 2$, $p_0=1$, $p_1=p_1$, $p_i=1$ for $i\geq 2$.
• Linial–Meshulam $d$-complex: $p_0=p_1=\dots=p_{d-1}=1$, $p_d=p$, $p_{d+1}=\dots=p_r=0$ (Costa et al., 2015, Fowler, 2015).

This interpolation allows for precise control over the local face-density at each dimension and supports a broader phase space for topological phenomena (Costa et al., 2015).

3. Thresholds, Critical Dimension, and Homological Phase Transitions

The multi-parameter model exhibits phase transitions governed by convex domains:

• In the "power-law" regime, $p_k = n^{-\alpha_k}$, thresholds for appearance of subcomplexes and topological features become convex sets in $(\alpha_0, \dots, \alpha_r)$-space.
• Critical dimension $d_c$ is defined by linear forms

$$\psi_k(\alpha) = \sum_{i=0}^r \binom{k}{i} \alpha_i,$$

and is the unique $k$ such that $\psi_k(\alpha_*) < 1 < \psi_{k+1}(\alpha_*)$ (Costa et al., 2015, Costa et al., 2015).

• The expected face counts and Betti numbers in dimension $k$ scale as $n^{\tau_k}/(k+1)!$ with

$\tau_k = \sum_{i=0}^k[1-\psi_i(\alpha)],$

and with $\ell_k(\alpha)$ and $\psi_k$ controlling vanishing/emergence thresholds.

• Betti numbers in the critical dimension dominate all others (the homological domination principle):

$$\mathbb{E}[b_k(Y)] \gg \mathbb{E}[b_j(Y)], \quad j \ne k,$$

for $$\alpha \in D_k = \{\psi_k(\alpha)<1<\psi_{k+1}(\alpha)\}$$ (Costa et al., 2015).

• Phase transitions for connectivity, simple connectivity, fundamental group properties, and appearance of higher torsion occur along explicit affine hyperplanes in parameter space (Costa et al., 2015).

4. Topological Invariants: Betti Numbers, Cohomology, Group Properties

Betti Numbers and Cohomology

• For lower model complexes, Betti numbers exhibit sharp vanishing and emergence thresholds, respectively governed by linear forms $S_1^k(\alpha)$ and $S_2^k(\alpha)$ (Kogan, 2024, Fowler, 2015).
• The cup-product cohomology ring is a.a.s. trivial for the lower model (cup-length $=1$), but nontrivial Steenrod operations generically appear for certain parameter ranges, reflecting embedded strongly-connected subcomplexes of arbitrary topology (Kogan, 2024).
• In upper models, cohomology is a.a.s. concentrated in a single dimension—the complex collapses onto its critical skeleton, confirming and strengthening earlier results (Farber et al., 2022).
• Simultaneous nontrivial homology in consecutive dimensions is achievable by parameter tuning (Fowler, 2015).

Fundamental Group and Asphericity

• The fundamental group $\pi_1$ transitions from trivial to hyperbolic with explicit multi-parameter threshold conditions, e.g., $\alpha_0+3\alpha_1+2\alpha_2=1$ (Costa et al., 2015).
• In hyperbolic regimes, only 2-torsion occurs; odd-prime torsion is excluded (Costa et al., 2015).
• In random complexes of dimension $>2$, the group has geometric and cohomological dimension at most $2$ in specified regimes.
• The probabilistic Whitehead Conjecture holds for 2-dimensional aspherical subcomplexes: all subcomplexes are also a.s. aspherical in relevant parameter domains (Costa et al., 2015).

5. Limit Theorems, Spectral Laws, and Large Deviations

Limit Theorems for Topological Invariants

• In regimes where face-counts of a particular dimension dominate, both Euler characteristic and Betti numbers obey strong laws of large numbers and functional central limit theorems (FCLT); limiting Gaussian processes for fluctuation depend only on dynamics in the smallest non-trivial dimension (Owada et al., 2020).
• For higher-dimensional Betti numbers (above the critical dimension), scaling constants in CLT and LDP lower-tail exponents exhibit phase transitions at explicit parameter boundaries (Owada et al., 2023).
• Large deviation results for subcomplex counts and Betti numbers at and below critical dimension indicate upper-tail probabilities decay at rates governed by combinatorially defined exponents $T_q$ and extremal parameters $M_{F,n}(p)$ (Samorodnitsky et al., 2022).

Spectral Properties

• For adjacency matrices (signed and unsigned) of $d$-dimensional multi-parameter random simplicial complexes, the empirical spectral distribution converges to Wigner's semicircle law under appropriate scaling conditions: dense lower-dimensional skeleton, sparse top-dimensional cells (Adhikari et al., 9 Jan 2026).
• Operator norm and empirical law convergence hold for both the principal block and the random-dimension submatrix, extending classical random graph spectral universality (Adhikari et al., 9 Jan 2026).

6. Statistical Methodology and Goodness-of-Fit

• Statistical inference (e.g., goodness-of-fit tests) based on critical simplex counts and subcomplex occurrence is feasible in the multi-parameter model:
  • Multivariate CLT applies to subcomplex counts; covariances become perfectly correlated in the dense regime.
  • Lexicographical acyclic matching yields distributional theory for critical simplex counts.
  • MLE estimators for $\p$ are asymptotically unbiased, consistent, normally distributed, and uncorrelated under standard conditions (Temčinas et al., 2023).
• Contrast with geometric models: tests exploiting higher-order counts outperform lower-order subgraph-based statistics for goodness-of-fit.

7. Open Problems and Advanced Phenomena

• Precise thresholds for integer homology remain partly unresolved, especially near critical parameter boundaries (Costa et al., 2015, Fowler, 2015).
• Embedding windows and containment criteria for arbitrary fixed subcomplexes are characterized by convex density domains and intersective reduced density domains in parameter space.
• The interplay between local attachment rules and global algebraic topology, including ring and Steenrod-algebra structures, is subtle and may yield new stochastic topological phenomena (Kogan, 2024).
• Phase diagrams in the multi-parameter space reveal regimes of simultaneous homological activity, collapse phenomena, and spectral transitions; these can be explicitly stratified via analytic invariants $\psi_k$, $T_j$, $B_i$ (Costa et al., 2014, Farber et al., 2022).

---

The multi-parameter random simplicial complex model thus provides a rigorous unification and generalization of classical stochastic topology, capturing rich threshold, collapse, and domination phenomena, and supporting precise limit theorems, probabilistic group theory, and statistical inference in high-dimensional random topologies (Costa et al., 2014, Costa et al., 2015, Costa et al., 2015, Kogan, 2024, Farber et al., 2022, Adhikari et al., 9 Jan 2026, Owada et al., 2020, Fowler, 2015, Owada et al., 2023, Samorodnitsky et al., 2022, Costa et al., 2015, Temčinas et al., 2023).