Two-Layer Random Hypergraph Model

• The two-layer random hypergraph model is a mathematical framework that couples deterministic household structures with random workplace affiliations to capture higher-order network interactions.
• It employs discrete-time Markov chain dynamics to simulate opinion spread, adaptive rewiring, and polarization by leveraging local agreement measures and threshold-based updates.
• Statistical inference techniques and simulation studies reveal regime-dependent behaviors in homogeneity, degree distribution, and clustering that enhance understanding of complex social systems.

A two-layer random hypergraph model is a mathematical framework for representing networks with two overlapping sets of higher-order relationships among a fixed set of vertices, typically designed to analyze phenomena such as opinion spread, clustering, and degree heterogeneity in multiway systems. Each layer consists of hyperedges with distinct construction rules: one deterministic (e.g., households) and one random (e.g., workplaces), enabling the study of adaptive, multi-layer interactions and their effect on global network properties, polarization dynamics, and statistical inference.

1. Formal Construction of Two-Layer Random Hypergraph Models

Let $V$ be a collection of $n$ labeled vertices representing individuals. The model consists of two hypergraph layers:

• Layer 1 (Households): A deterministic partition $E_1 = \{H_1, \dots, H_{n/k_h}\}$ of $V$ into disjoint blocks of fixed size $k_h$, where each $H_i = \{(i-1)k_h + 1, \dots, i k_h\}$ and $|H_i| = k_h$ for all $i$.
• Layer 2 (Workplaces): A random covering $E_2 = \{W_1, \dots, W_W\}$ where each vertex $v \in V$ is independently assigned to a workplace via a random mapping $\varphi: V \to \{1, \ldots, W\}$, $P[\varphi(v) = j] = 1/W$. Thus, $W_j = \{v \in V: \varphi(v) = j\}$, with initial sizes following a multinomial law subject to $\sum_j |W_j| = n$.

Households are static, while workplace memberships may adapt through dynamics driven by local and global conditions.

2. Dynamical Processes: Opinion Spread and Adaptive Rewiring

The system's state at time $t$ is $$S_t = (\sigma_t, \varphi_t) \in \{A, B\}^V \times \{1,\dots,W\}^V$$, where $\sigma_t$ encodes each vertex's opinion. The Markovian dynamics proceed in discrete time steps:

1. Vertex Selection: $v \in V$ is sampled uniformly.
2. Local Agreement Calculations:
   • Household agreement: $$h = \frac{1}{|H_v|} \#\{u \in H_v : \sigma_t(u) = \sigma_t(v)\}$$
   • Workplace agreement: $$w = \frac{1}{|W_v(t)|} \#\{u \in W_v(t): \sigma_t(u) = \sigma_t(v)\}$$
   • Combined agreement: $a = \frac{h + \lambda w}{1 + \lambda}$, where $\lambda > 0$ weighs workplace influence.
3. Opinion Change: If $a < r_1$, $v$ flips attitude with probability $p_{\rm change}(a) = \beta(r_1 - a)$, where $\beta \in (0, 1]$ measures opinion-change strength and $r_1$ is the threshold.
4. Workplace Switching: If no flip occurs and $w < r_2$, with probability $p_{\rm quit}(w) = q(r_2 - w)$, vertex $v$ switches to a different workplace $W_j$ where its opinion holds strict majority, or selects randomly if none exist. Here, $q \in [0,1]$ measures workplace-change strength and $r_2$ is the threshold.

These updates induce a time-homogeneous Markov chain on the finite state space.

3. Parameter Regimes and Model Classification

Key parameters:

Parameter	Range	Interpretation
$\beta$	$(0, 1]$	Opinion-change strength
$q$	$[0, 1]$	Workplace-change strength
$r_1, r_2$	$(0, 1]$	Thresholds for change
$\lambda$	$(0, \infty)$	Workplace vs household weight

Two analytically distinct regimes are notable:

• Linear Model ($r_1 = r_2 = 1$): Changes always possible unless full local agreement. $p_{\rm change}(a) = \beta(1 - a)$, $p_{\rm quit}(w) = q(1 - w)$.
• Nonlinear Model ($r_1 = r_2 = 1/2$): Actions only when in minority in both layers, resulting in strong peer-pressure and possible trapping in mixed configurations.

4. Absorbing States, Homophily, and Polarization Dynamics

Absorbing States:

• In the linear regime, all absorbing states are globally homogeneous: every household and workplace is monochromatic. Absorption is inevitable and reached in finite time.
• The nonlinear regime admits additional absorbing states: some hyperedges remain mixed if no individual is in the minority in both groups, allowing for persistent local heterogeneity.

Small-scale analysis ($n=10$): Two absorbing-state isomorphism classes in the linear case; eight in the nonlinear case. In simulation, linear dynamics favor full homogeneity as $\beta$ increases and $q$ decreases; nonlinear dynamics are less sensitive, leaving mixed households prevalent.

Large-scale simulations ($n=1000$):

• Variance of $N_A$ (number of $A$ opinions): Increases with $\beta$ (opinion flips) and decreases with $q$ (rewiring), reflecting consensus vs. segregation dynamics. Nonlinear variants exhibit lower variance.
• Homophily Indices: For layer $\ell$ ($=1$ for households, $=2$ for workplaces), $$H_{\ell}(t) = \frac{1}{n} \sum_{e \in E_{\ell}(t)} |e| \mathbb{1}_{\{\sigma_t|_e \text{ is constant}\}}$$. In the linear regime, both $H_1$ and $H_2$ tend to unity; in the nonlinear case, $H_2 \rightarrow 1$ (workplace homogenization), but $H_1 < 1$ persists (incomplete household segregation). Households homogenize rapidly for high $q$ and low $\beta$.
• Component-size behavior: The network splits into two near-equal monochromatic components for large $q$; for large $\beta$ and small $q$, a giant component remains until consensus.

Polarization Speed: Hitting times $\tau_{hh}$ and $\tau_{wp}$ for homophily thresholds $>0.4$ decrease with $q$, increase with $\beta$. Workplaces segregate faster than households.

5. Estimation, Inference, and Statistical Properties

Parameter inference is integral to multi-layer hypergraph analysis. For degree heterogeneity, the two-layer hypergraph $\boldsymbol{\beta}$-model (Nandy et al., 2023) allows each layer (edge-size $k_1$, $k_2$) to have independent edge probabilities parameterized as:

$$p_e^{(k_1)} = \frac{\exp(\sum_{i \in e} \beta^{(k_1)}_i)}{1 + \exp(\sum_{i \in e} \beta^{(k_1)}_i)}, \qquad p_f^{(k_2)} = \frac{\exp(\sum_{j \in f} \beta^{(k_2)}_j)}{1 + \exp(\sum_{j \in f} \beta^{(k_2)}_j)}$$

MLE for each layer $\beta^{(s)}$ possesses optimal convergence properties (L$_2$-rate $O_p(n^{-(s-2)/2})$, L$_\infty$-rate $O_p(\sqrt{\log n / n^{s-1}})$), with corresponding CLTs and asymptotic confidence intervals. Likelihood-ratio tests exhibit minimax optimality; the detection threshold for layer $s$ is $n^{-(2s-3)/4}$ in L$_2$-norm.

6. Structural Insights, Topological Characteristics, and Limit Theorems

Workplace-size distributions are sharply centered about $k_w$, as moves only reassign individuals without growth or shrinkage of hyperedges. Over long times, the workplace layer forms a two-block structure, favoring same-opinion clustering. Analytical tractability is restricted; large-$n$ behavior is characterized via simulation.

For related models such as random connection hypergraphs (Brun et al., 2024), key analytic results include higher-order degree distributions with power-law tails:

$$\lim_{k \to \infty} \frac{\log \mathbb{P}(D_m \geq k)}{\log k} = -\left(m - \frac{m+1}{\alpha}\right)$$

and central/stable limit theorems for simplex counts and Betti numbers under varying mark distributions. Gaussian limits require light-tailed marks; heavy-tailed marks ($\alpha > 1/2$) yield stable laws for counts and topology, with empirical application underlining underestimation of higher-order clustering in collaboration networks.

7. Applications and Empirical Observations

Multi-layer random hypergraph models substantiate mechanistic analyses of opinion formation, polarization, and clustering in social systems. Simulation studies with parameters tuned to empirical network datasets—such as arXiv author collaboration networks—replicate marginal degree distributions but underfit higher Betti numbers and simplex-counts, indicating the necessity for more nuanced mechanisms or additional layers to capture real-world clustering.

In summary, the two-layer random hypergraph model provides a rigorous framework for dissecting the emergence of consensus, segregation, and degree heterogeneity in adaptive multiway systems. The interplay between local opinion dynamics, adaptive group membership, and higher-order structure underpins both theoretical advances in statistical inference and practical modeling capability for complex networks.