Generating Networks to Target Assortativity via Archimedean Copula Graphons

TL;DR

The paper addresses generating random graphs with a prescribed assortativity without edge rewiring by embedding Archimedean copulas into graphons. It develops three models—Copula Graphon, Copula Density Graphon, and Tensor Copula Graphon—whose assortativity is tied to graphon homomorphism densities, enabling control over subgraph frequencies alongside $r$. Theoretical results show how $r$ can be expressed via $t(F,W)$ and proven convergence of empirical assortativity to the graphon-level target, with algorithms for estimating the copula parameter $\theta$ and constructing networks. Numerically, the authors demonstrate the ability to realize no, weak, and strong assortativity as well as disassortativity by varying copula families, parameters, and tensor combinations. The approach offers a principled, transparent alternative to rewiring for benchmarking graph-learning algorithms and studying motif-assortativity interactions in complex networks, with avenues for extending to non-Archimedean copulas in future work.

Abstract

We develop an approach to generate random graphs to a target level of assortativity by using copula structures in graphons. Unlike existing random graph generators, we do not use rewiring or binning approaches to generate the desired random graph. Instead, we connect Archimedean bivariate copulas to graphons in order to produce flexible models that can generate random graphs to target assortativity. We propose three models that use the copula distribution function, copula density function and their mixed tensor product to produce networks. We express the assortativity coefficient in terms of homomorphism densities. Establishing this relationship forges a connection between the parameter of the copula and the frequency of subgraphs in the generated network. Therefore, our method attains a desired the subgraph distribution as well as the target assortativity. We establish the homomorphism densities and assortativity coefficient for each of the models. Numerical examples demonstrate the ability of the proposed models to produce graphs with different levels of assortativity.

Figures (5)

• Figure 1: Star and path networks in the combinatorial expression of Newman's assortativity coefficient
• Figure 2: Assortativity coefficient range by $W_G, W_J$ for $\theta \in [0,10]$. Blue, red line is the average assortativity coefficient of $W_G$, $W_J$ respectively, over 10 repetitions, the minimum and maximum within the range is the shaded region
• Figure 3: Assortativity coefficient range by $W_G$ for $\theta \in [0,10]$. Blue line is the average assortativity coefficient over 10 repetitions, the minimum and maximum within the range is the shaded region
• Figure 4: Assortativity coefficient range through $\Tilde{W}_J \otimes \Tilde{W}_G$ for $\theta_J = 2$, $\theta_G \in[1,10]$. Blue line is the average assortativity coefficient over 10 repetitions, the minimum and maximum within the range is the shaded region
• Figure 5: Assortativity coefficient range through $\Tilde{W}_F \otimes \Tilde{W}_F$ for $\theta_F \in[1,5]$ at different lags. Assortativity coefficient is averaged over 10 repetitions. Line is the average assortativity coefficient over 10 repetitions, the minimum and maximum within the range is the shaded region

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: Degree Assortativity Coefficient with Graphon Homomorphism Densities
• proof
• Corollary 2.2
• Theorem 2.3: Convergence to the Homomorphism Density Degree Assortativity Coefficient
• Lemma 2.1
• proof
• Theorem 3.1
• proof