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 $W_i \sim F_W$0, so marginals are preserved: $W_i \sim F_W$1, with $W_i \sim F_W$2 uniform.

Given the latent marks, edges are conditionally independent: for $W_i \sim F_W$3, the edge indicators are independent Bernoulli variables conditional on $W_i \sim F_W$4.

Edge probabilities under the fixed-range rule are:

$W_i \sim F_W$5

where $W_i \sim F_W$6 is global intensity, $W_i \sim F_W$7 a local kernel with compact support, $W_i \sim F_W$8 a shrinking-range sequence, and $W_i \sim F_W$9.

2. Control of Degree Mixing and Separation of Marginals

Degree mixing (assortativity) is engineered through the copula parameter $X_i \sim \mathrm{Unif}(\mathbb{T}^d)$0, which modulates the degree to which higher popularity ($X_i \sim \mathrm{Unif}(\mathbb{T}^d)$1) aligns with favorable spatial locations ($X_i \sim \mathrm{Unif}(\mathbb{T}^d)$2), while keeping $X_i \sim \mathrm{Unif}(\mathbb{T}^d)$3 and the law of $X_i \sim \mathrm{Unif}(\mathbb{T}^d)$4 fixed.

For conditional moments:

$X_i \sim \mathrm{Unif}(\mathbb{T}^d)$5

which underlie large-$X_i \sim \mathrm{Unif}(\mathbb{T}^d)$6 limiting behavior.

Endpoint assortativity in the sparse regime converges to:

$X_i \sim \mathrm{Unif}(\mathbb{T}^d)$7

where $X_i \sim \mathrm{Unif}(\mathbb{T}^d)$8 is the limiting edge–Palm law, $X_i \sim \mathrm{Unif}(\mathbb{T}^d)$9 is the limiting degree intensity, and $\mathbb{T}^d$0 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 $\mathbb{T}^d$1.

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:

$\mathbb{T}^d$2

with

$\mathbb{T}^d$3

This yields a universal "degree-tail dichotomy":

• For bounded, fixed-range kernels, $\mathbb{T}^d$4 a.s., and thus $\mathbb{T}^d$5 is stochastically dominated by $\mathbb{T}^d$6, 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:

$\mathbb{T}^d$7

where triangle counts $\mathbb{T}^d$8 and the function

$\mathbb{T}^d$9

with $d$0.

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:

$d$1

The limiting degree intensity is now

$d$2

with $d$3. If $d$4 is regularly varying of index $d$5, 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 $d$6 and the density parameter $d$7 is feasible from a single observed graph under injectivity of the map $d$8:

• Fix $d$9, spatial dimension $(W_i, X_i)$0, and kernel $(W_i, X_i)$1.
• Estimate $(W_i, X_i)$2 by matching the empirical mean degree $(W_i, X_i)$3 with $(W_i, X_i)$4.
• Compute empirical global transitivity $(W_i, X_i)$5 and endpoint assortativity $(W_i, X_i)$6.
• Obtain $(W_i, X_i)$7 via minimum-distance moment matching:

$(W_i, X_i)$8

A joint $(W_i, X_i)$9-CLT for $C_\theta$0 ensures consistency and asymptotic normality:

$C_\theta$1

where $C_\theta$2 is the Jacobian of the moment map, and $C_\theta$3 is the covariance of $C_\theta$4.

7. Conceptual Synthesis and Model Characteristics

CoLaS achieves explicit, modular separation of mechanisms:

• Marginal degree heterogeneity is governed by $C_\theta$5.
• Clustering is controlled by the geometric kernel $C_\theta$6 under shrinking range, preventing vanishing transitivity.
• Degree mixing (assortativity) is tuned solely by the copula parameter $C_\theta$7, without secondary rewiring.

This structure enables explicit large-$C_\theta$8 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).