Bernoulli Connectivity Distributions

• Bernoulli connectivity distributions are defined by superposing independent Bernoulli subgraphs, each characterized by a community size and an edge probability.
• The framework establishes connectivity and k-connectivity thresholds using mixed moments and empirical measures, leading to sharp zero-one laws.
• Practical insights include unified criteria for network connectivity that generalize classical Erdős–Rényi models to multilayer and spatial contexts.

A Bernoulli connectivity distribution characterizes the probabilistic structure and connectivity transitions in random graphs formed by superposing independent Bernoulli random subgraphs—each defined on variable-sized vertex subsets and equipped with a per-edge Bernoulli connection probability. This framework generalizes classical models, such as @@@@1@@@@ and random connection models, to multilayer or community-based constructions, where connectivity properties depend on both local graph parameters (community sizes, edge densities) and their empirical distributions. Threshold results, zero-one laws, and asymptotic behaviors in such models have been rigorously established, providing unified criteria for connectivity and k-connectivity phenomena.

1. Formal Model Definition and Construction

Let $V = [n]$ be a set of $n$ base vertices. One constructs $m$ independent layers (or communities), denoted $G_{n,i} = (V_{n,i}, E_{n,i})$, as Bernoulli subgraphs via the following mechanism:

• For each $i$ ($1\leq i\leq m$), draw a pair $(X_{n,i}, Q_{n,i})$ with $X_{n,i} \in \{0,1,\dots,n\}$ (community size) and $Q_{n,i} \in [0,1]$ (Bernoulli edge probability) according to a specified law.
• Select a uniform random subset $V_{n,i}\subset V$ of size $X_{n,i}$.
• On $V_{n,i}$, form a standard Bernoulli graph: each possible edge in $\binom{X_{n,i}}{2}$ is included independently with probability $Q_{n,i}$.

Layers are mutually independent in both their size–connectivity draws and edge formation. The superposed union graph is

$$G[n,m] = (V, E_{n,1} \cup E_{n,2} \cup \cdots \cup E_{n,m}).$$

The construction is fully characterized by the empirical measure $$P_{n,m} = \frac{1}{m} \sum_{i=1}^m \delta_{(X_{n,i}, Q_{n,i})}$$ in the general case, allowing for both i.i.d. and non-identically distributed communities (Ardickas et al., 2023, Bloznelis et al., 2023, Ardickas et al., 21 Mar 2025).

2. Mixed Moments and the Connectivity Threshold Parameter

Central to the analysis is the mixed moment defined by

$$\kappa_n = \int x \left[1 - (1-q)^{(x-1)_+} \right] P_{n,m}(dx, dq),$$

with $h(x, q) = 1 - (1-q)^{(x-1)_+}$ quantifying the expected probability that a vertex in a Bernoulli subgraph is non-isolated. For the i.i.d. case, this reduces to $\alpha = E[X h(X, Q)]$.

The connectivity transition for $G[n, m]$ is governed by the parameter

$A_{n, m} = \ln n - \frac{m}{n}\, \kappa_n$

in the general setting, or $A_{m, n} = \ln n - \frac{m}{n}\, \alpha$ in the i.i.d. case. This parameter encodes the competition between the logarithmic vertex set size and the cumulative edge contribution stemming from the communities, and admits an asymptotic zero-one law for connectivity (Ardickas et al., 2023, Bloznelis et al., 2023).

3. Zero–One Laws and Asymptotic Double-Exponential Limit

Provided the additional moment condition

$\int x\, h(x, q)\, \ln(1+x)\, P(dx, dq) < \infty$

holds (ensuring concentration of isolated vertices), the following dichotomy is established:

• If $A_{n, m} \to -\infty$, then $\Pr\{ G[n, m]\;\text{is connected} \} \to 1$.
• If $A_{n, m} \to +\infty$, then $\Pr\{ G[n, m]\;\text{is connected} \} \to 0$.

At critical scaling ($A_{n, m} \to c$ for $c\in\R$):

• The expected number of isolated vertices is $n \exp(-A_{n, m})(1+o(1))$.
• The count of isolated vertices converges in distribution to Poisson with mean $e^{-c}$, and the connectivity probability converges to $e^{-e^{-c}}$, mirroring the classical transition in Erdős–Rényi $G(n, p)$ (Ardickas et al., 2023, Bloznelis et al., 2023).

4. Generalizations: Non-identical Distributions and k-Connectivity

The main connectivity law generalizes to non-identically distributed pairs $(X_{n,i}, Q_{n,i})$, using the empirical average

$$K_n = \frac{1}{m} \sum_{i=1}^m E[X_{n,i} h(X_{n,i}, Q_{n,i})],$$

and corresponding threshold parameter $A_n = \ln n - \frac{m}{n} K_n$. The connectivity transition is preserved under analogous integrability and non-degeneracy conditions (Bloznelis et al., 2023).

For $k$-connectivity ($k \geq 2$), the threshold becomes

$$\Delta_{n, m, k} = \ln n + (k-1) \ln \ln n - \frac{m}{n} K^*$$

where $K^* = E[ X h(X, Q) ]$. If $\Delta_{n, m, k} \to +\infty$, vertex-$k$-connectivity occurs with probability tending to $1$; if $\Delta_{n, m, k} \to -\infty$, edge-$k$-connectivity vanishes (Ardickas et al., 21 Mar 2025).

5. Intuitive Interpretation and Comparison to Classical Models

The mixed moment $E[X h(X, Q)]$ serves as the effective "edge mass" per vertex contributed by the union of communities. Consequently, $(m / n) E[X h(X, Q)]$ quantifies the mean number of useful edges per base vertex. This structure reproduces the celebrated $np$ criterion for $G(n, p)$, with the threshold for connectivity corresponding to $(m / n) E[X h(X, Q)] \approx \ln n$.

Variations in community size and edge density, captured by the empirical distribution $P_{n, m}$, directly impact connectivity. The additional integrability condition involving $\ln(1+x)$ is essential to rule out heavy-tailed community-size effects that could disrupt sharp threshold behavior (Ardickas et al., 2023).

6. Extensions and Related Bernoulli Connectivity Distributions

Bernoulli connectivity distributions encompass models beyond the complete-graph superposition, including the anisotropic Bernoulli bond percolation framework (Lima et al., 2015) and random connection models where connections depend on geometric or spatial distances with Bernoulli probabilities (Iyer, 2015). In these, connectivity transitions are similarly mediated by edge probabilities, geometric parameters, and associated mixed moments or integral criteria.

For example, in the random connection model on $\mathbb{R}^d$, the connectivity regime is governed by the parameter $\alpha = \int_{\mathbb{R}^d} g(|x|)dx$, resulting in the threshold

$$\lim_{n\to\infty} \frac{\alpha n d_n^d}{\log n} = 1,$$

where $d_n$ is the smallest radius ensuring absence of isolated nodes. Analogous moment-based expressions for connectivity probabilities and degree distributions arise (Iyer, 2015).

7. Practical Consequences and Open Problems

The zero–one laws for Bernoulli connectivity distributions ensure practical predictability for large-scale network design and analysis: knowing the empirical distribution of community sizes and edge probabilities allows direct calculation of connectivity transitions. This framework accommodates models with heterogeneous mixtures, spatially localized connectivity, and overlapping community structures.

A remaining open direction is the extension of these limit laws for $k$-connectivity to sharp nondegenerate distributions (e.g., Gumbel-type limit in the vein of Erdős–Rényi), especially in the presence of more complex community-level dependencies or heavy-tailed regimes (Ardickas et al., 21 Mar 2025). This suggests ongoing research into the finer structure of degree and component distributions within Bernoulli connectivity models.