Weighted Random Dot Product Graphs

TL;DR

This work addresses the limitation of the classical RDPG in handling weighted graphs by introducing WRDPG, a nonparametric model where each node possesses a sequence of latent positions whose inner products define the moments of edge weight distributions. The approach links edge weights to a moment sequence via a moment-generating framework and estimates latent positions using the adjacency spectral embedding of powered weight matrices, backed by consistency and asymptotic normality results. A rich generative framework is developed to sample graphs matching prescribed or data-fitted moment sequences, with discrete, continuous, and mixed weight distributions handled through Vandermonde, Chebyshev, and maximum-entropy methodologies. Through illustrative examples and theoretical guarantees, WRDPG demonstrates discriminative power from higher-order moments, enabling more nuanced community detection and realistic graph generation, with practical implications for hypothesis testing and reference distribution construction in weighted networks.

Abstract

Modeling of intricate relational patterns has become a cornerstone of contemporary statistical research and related data science fields. Networks, represented as graphs, offer a natural framework for this analysis. This paper extends the Random Dot Product Graph (RDPG) model to accommodate weighted graphs, markedly broadening the model's scope to scenarios where edges exhibit heterogeneous weight distributions. We propose a nonparametric weighted (W)RDPG model that assigns a sequence of latent positions to each node. Inner products of these nodal vectors specify the moments of their incident edge weights' distribution via moment-generating functions. In this way, and unlike prior art, the WRDPG can discriminate between weight distributions that share the same mean but differ in other higher-order moments. We derive statistical guarantees for an estimator of the nodal's latent positions adapted from the workhorse adjacency spectral embedding, establishing its consistency and asymptotic normality. We also contribute a generative framework that enables sampling of graphs that adhere to a (prescribed or data-fitted) WRDPG, facilitating, e.g., the analysis and testing of observed graph metrics using judicious reference distributions. The paper is organized to formalize the model's definition, the estimation (or nodal embedding) process and its guarantees, as well as the methodologies for generating weighted graphs, all complemented by illustrative and reproducible examples showcasing the WRDPG's effectiveness in various network analytic applications.

Figures (12)

• Figure 1: Latent position estimation. Given an adjacency matrix ${\mathbf A}$ we compute its $k$-th entry-wise power ${\mathbf A}^{(k)}$. The ASE of ${\mathbf A}^{(k)}$ yields the estimates ${\hat{\mathbf X} }[k]$; see also Section \ref{['ssec:estimation']}.
• Figure 2: Graph generation. Given the latent positions of each vertex $\{{\mathbf X}[k]\}_{k \geq 0}$, we estimate a weight distribution whose sequence of moments is given by the corresponding dot products. Edge weights are then sampled from this estimated distribution; see also Section \ref{['sec:generative']}.
• Figure 3: True (dashed vertical line) and estimated (histograms) latent positions for an Erdös-Rényi model with $\mathcal{N}(1,0.01)$ weights.
• Figure 4: Estimated (blue and red circles) and true latent positions (black crosses) for a two-block SBM with $\mathcal{N}(1,0.01)$ weights.
• Figure 5: Theoretical latent positions (black crosses) and ASE embeddings of ${\mathbf W}^{(k)}$ for Gaussian ($\mu=5$ and $\sigma=0.1$; in blue) and Poisson ($\lambda=5$; in red) distributed weights for $d=2$ and $k=1$ (left), $k=2$ (center), and $k=3$ (right). Nodes with different weight distributions are clearly revealed for $k=3$, but they overlap for $k=1$.

Theorems & Definitions (33)

• Definition 2.1: Admissible moment sequence
• Remark 2.1
• Definition 2.2: Weighted inner-product distribution
• Definition 2.3: WRDPG
• Remark 2.2: Nonidentifiability of latent positions
• Remark 3.1
• Theorem 3.1
• Proposition 3.2
• proof : Proof of Theorem \ref{['theorem:consistency']}
• Theorem 3.3