CoLaS: Copula-Seeded Local Latent-Space Graphs

• The paper introduces CoLaS, a modular latent-variable graph model that unifies degree heterogeneity, persistent clustering, and assortativity through a copula-based framework.
• The model separates marginal specifications from dependence structure by coupling node popularity and latent spatial locations, enabling explicit parameter control.
• The CoLaS-HT extension further refines the approach to produce power-law degree distributions while preserving network sparsity and locality.

CoLaS (Copula-Seeded Local Latent-Space Graphs) is a modular latent-variable random graph model that unifies degree heterogeneity, persistent clustering, and systematic degree mixing within sparse regimes. CoLaS separates marginal specifications from dependence structure by utilizing a copula to couple node "popularity" and latent geometric location, introducing explicit and interpretable parameter control for assortativity. The framework is supported by sparse-limit theory for degree distributions, transitivity, and assortativity, and includes a minimal extension—CoLaS-HT—that enables power-law degree tails while preserving sparsity and locality (Papamichalis et al., 23 Dec 2025).

1. Latent Variable Construction and Edge Formation

Each node $i$ is assigned latent marks:

• Popularity $W_i \sim F_W$ controlling degree heterogeneity.
• Location $X_i \sim \mathrm{Unif}(\mathbb{T}^d)$, where $\mathbb{T}^d$ is the $d$-dimensional torus, specifying spatial locality.

The joint distribution of $(W_i, X_i)$ is constructed using Sklar's theorem with a copula $C_\theta$ on $[0,1]^{d+1}$. Writing $U_i = F_W(W_i)$ and $V_i = X_i$, this yields $(U_i, V_i) \sim C_\theta$, so marginals are preserved: $F_{W,X}(w,x) = C_\theta(F_W(w), F_X(x))$, with $F_X(x)$ uniform.

Given the latent marks, edges are conditionally independent: for $\{A_{ij}\}_{i<j}$, the edge indicators are independent Bernoulli variables conditional on $\{(W_i, X_i)\}_{i=1}^n$.

Edge probabilities under the fixed-range rule are:

$$p^{(n)}_{ij} = 1 - \exp\left\{ -\frac{\lambda}{\rho_n} W_iW_j\, k\left(\frac{X_i - X_j}{\varepsilon_n}\right) \right\}$$

where $\lambda > 0$ is global intensity, $k: \mathbb{R}^d \to [0, \infty)$ a local kernel with compact support, $\varepsilon_n \downarrow 0$ a shrinking-range sequence, and $\rho_n = n\varepsilon_n^d \to \rho \in (0, \infty)$.

2. Control of Degree Mixing and Separation of Marginals

Degree mixing (assortativity) is engineered through the copula parameter $\theta$, which modulates the degree to which higher popularity ($W$) aligns with favorable spatial locations ($X$), while keeping $F_W$ and the law of $X$ fixed.

For conditional moments:

$$m_{p,\theta}(x) = \mathbb{E}\left[ W^p \mid X = x \right], \quad p=1,2,3,\dots$$

which underlie large-$n$ limiting behavior.

Endpoint assortativity in the sparse regime converges to:

$$r(\theta) = \frac{ \mathrm{Cov}_{\nu_\theta}(\Lambda_\theta(Z), \Lambda_\theta(Z')) + \mathbb{E}_{\nu_\theta}[\Gamma_\theta(Z, Z')] } { \mathrm{Var}_{\nu_\theta}(\Lambda_\theta(Z)) + \mathbb{E}_{\nu_\theta}[\Lambda_\theta(Z)] }$$

where $\nu_\theta$ is the limiting edge–Palm law, $\Lambda_\theta(z)$ is the limiting degree intensity, and $\Gamma_\theta(z, z')$ is the limiting common-neighbor intensity.

The copula construction ensures that all changes to mixing properties arise through its concordance, with marginals unchanged for all $\theta$.

3. Sparse-Limit Degree Distributions and Tail Dichotomy

In the fixed-range CoLaS regime with a compact kernel, degree distributions obey a mixed-Poisson limit:

$$D_i \xrightarrow{d} \mathrm{Poisson}(\Lambda_\theta(W_i, X_i)),$$

with

$$\Lambda_\theta(w, x) = \rho \int_{\mathbb{R}^d} \mathbb{E}\left[ 1 - \exp\left( -\frac{\lambda}{\rho} w W' k(u) \right) \bigg| X = x \right] du$$

This yields a universal "degree-tail dichotomy":

• For bounded, fixed-range kernels, $\Lambda_\theta(W, X) \leq M < \infty$ a.s., and thus $D$ is stochastically dominated by $\mathrm{Poisson}(M)$, forcing degree distributions to have exponentially light tails, regardless of the popularity marginal.

The inability to produce power-law degrees in this sparse, fixed-range context motivates the extension described in the next section.

4. Persistent Clustering and Transitivity

Clustering, quantified by global transitivity, remains nonvanishing in the sparse local regime:

$$C_n = \frac{3 T_n}{\sum_i \binom{D_i}{2}} \xrightarrow{\mathbb{P}} C(\theta) := \frac{2 \mathbb{E}\left[ \tau_\theta(W, X) \right]}{\mathbb{E}[\Lambda_\theta(W, X)^2]}$$

where triangle counts $T_n$ and the function

$$\tau_\theta(w, x) = \frac{\rho^2}{2} \iint_{\mathbb{R}^d \times \mathbb{R}^d} \mathbb{E}\bigl[ q_{w, x}(u; W_1) q_{w, x}(v; W_2) q_{W_1, W_2}(u-v) \mid X = x \bigr] du \, dv$$

with $$q_{a, b}(u) = 1 - \exp\{ -\frac{\lambda}{\rho} ab k(u)\}$$.

The numerator counts rooted triangles, while the squared intensity in the denominator counts wedges; their ratio determines asymptotic clustering.

5. Tail Inheritance: The CoLaS-HT Extension

To achieve genuine power-law degree distributions while retaining locality and sparsity, CoLaS introduces a tail-inheriting extension ("CoLaS-HT"). The key modification is to replace the fixed interaction range with a weight-dependent range:

$$\P(A_{ij}=1 \mid W_i, W_j, X_i, X_j) = 1 - \exp \left\{ -\frac{\lambda}{\rho_n} k\left( \frac{X_i - X_j}{\varepsilon_n (W_i W_j)^{1/d}} \right) \right\}$$

The limiting degree intensity is now

$$\Lambda_\theta^{\mathrm{HT}}(w, x) = \rho \kappa_2^{(\lambda)} w \, m_{1,\theta}(x)$$

with $$\kappa_2^{(\lambda)} = \int \left(1 - e^{-(\lambda/\rho)k(u)} \right) du$$. If $W$ is regularly varying of index $\alpha$, the limiting intensity and thus the mixed-Poisson degree also inherit this tail—a property absent from the fixed-range regime.

6. Model Calibration via One-Graph Estimation

Identification of the copula parameter $\theta$ and the density parameter $\lambda$ is feasible from a single observed graph under injectivity of the map $\theta \mapsto (C(\theta), r(\theta))$:

• Fix $F_W$, spatial dimension $d$, and kernel $k$.
• Estimate $\lambda$ by matching the empirical mean degree $\bar{d}$ with $\rho_n \mathbb{E}_W[\Lambda_\theta(W, X)]$.
• Compute empirical global transitivity $\widehat{C}_n$ and endpoint assortativity $\widehat{r}_n$.
• Obtain $\widehat{\theta}_n$ via minimum-distance moment matching:

$$\widehat{\theta}_n \in \arg \min_{\theta \in \Theta} \left\| (\widehat{r}_n, \widehat{C}_n) - (r(\theta), C(\theta)) \right\|_2^2.$$

A joint $\sqrt{n}$-CLT for $(\widehat{C}_n, \widehat{r}_n)$ ensures consistency and asymptotic normality:

$$\sqrt{n} (\widehat{\theta}_n - \theta_0) \Rightarrow \mathcal{N}\left( 0,\, (G^\top G)^{-1} G^\top \Sigma G (G^\top G)^{-1} \right)$$

where $G = D_\theta(r, C)|_{\theta_0}$ is the Jacobian of the moment map, and $\Sigma$ is the covariance of $(\widehat{r}_n, \widehat{C}_n)$.

7. Conceptual Synthesis and Model Characteristics

CoLaS achieves explicit, modular separation of mechanisms:

• Marginal degree heterogeneity is governed by $F_W$.
• Clustering is controlled by the geometric kernel $k$ under shrinking range, preventing vanishing transitivity.
• Degree mixing (assortativity) is tuned solely by the copula parameter $\theta$, without secondary rewiring.

This structure enables explicit large-$n$ limit theorems for degree distribution (mixed-Poisson laws), consistent estimation via one-graph calibration, and, with the CoLaS-HT extension, sharply distinguishes between light-tailed and heavy-tailed degree behaviors contingent on kernel and node-weight interactions. This provides a unified analytic framework for sparse empirical networks exhibiting heterogeneity, clustering, and assortativity, with direct parameter-to-mechanism correspondence (Papamichalis et al., 23 Dec 2025).