Rapidly Varying Completely Random Measures for Modeling Extremely Sparse Networks

TL;DR

This work introduces rapidly varying completely random measures (CRMs) by mixing the Lévy intensity over the stability index α ∈ (0,1], yielding a new class (mGG) that includes stable and generalized gamma processes as limits. The canonical mixed Stable (mSt) and the full five-parameter mixed Generalized Gamma (mGG) constructions provide tractable Laplace exponents and a size-biased representation, enabling efficient simulation and posterior inference. Applying the mGG framework to Caron–Fox sparse graphs yields models where the number of edges grows near-linearly with the number of nodes, aligning with empirical observations of real-world networks, and is accompanied by scalable MCMC algorithms and demonstrated on synthetic and large real networks. The results offer a flexible, mathematically tractable approach to modeling extreme sparsity in networks and potentially other domains where near-linear growth of latent structures is expected.

Abstract

Completely random measures (CRMs) are fundamental to Bayesian nonparametric models, with applications in clustering, feature allocation, and network analysis. A key quantity of interest is the Laplace exponent, whose asymptotic behavior determines how the random structures scale. When the Laplace exponent grows nearly linearly - known as rapid variation - the induced models exhibit approximately linear growth in the number of clusters, features, or edges with sample size or network nodes. This regime is especially relevant for modeling sparse networks, yet existing CRM constructions lack tractability under rapid variation. We address this by introducing a new class of CRMs with index of variation $α\in(0,1]$, defined as mixtures of stable or generalized gamma processes. These models offer interpretable parameters, include well-known CRMs as limiting cases, and retain analytical tractability through a tractable Laplace exponent and simple size-biased representation. We analyze the asymptotic properties of this CRM class and apply it to the Caron-Fox framework for sparse graphs. The resulting models produce networks with near-linear edge growth, aligning with empirical evidence from large-scale networks. Additionally, we present efficient algorithms for simulation and posterior inference, demonstrating practical advantages through experiments on real-world sparse network datasets.

Figures (20)

• Figure 1: Point process representation of a random graph. Each node $i$ is embedded in $\mathbb{R}_{+}$ at some location $\theta_{i}$ and is associated with a sociability parameter $W_{i}$. An edge between nodes $\theta_{i}$ and $\theta_{j}$ is represented by a point at locations $(\theta_{i},\theta_{j})$ and $(\theta_{j},\theta_{i})$ in $\mathbb{R}_{+}^{2}$ (Figure 1, Caron2017). The corresponding graph is plotted in (b).
• Figure 2: Sampled mGG graphs (a) with parameters $\alpha=1$, $\tau=0$, $\beta=1$, $c=1$, and $\eta=100$, 2122 nodes, and 2583 edges; (b) with parameters $\alpha=1$, $\tau=0$, $\beta=1.5$, $c=5$, and $\eta=20$, 1249 nodes, and 1366 edges. The node size is proportional to its degree. The graphs were generated using our size-biased method and displayed using NetworkX.
• Figure 3: Examination of the mGG graph properties ($\bullet$) with parameters $\alpha=1$, $\tau=0$, $c=1$, and $\beta=1$ for various values of $\eta$ ranging from 50 to 6000, resulting in graphs of different sizes. Comparison with the Generalized Gamma CRM ($\blacksquare$) with parameters $\tau=1$ and $\sigma=0.5$, and with the Barabási–Albert model ($\blacktriangle$). For every configuration we simulate 20 graph samples and plot the mean of the quantity of interest.
• Figure 4: MCMC traceplot of parameters (a) $\beta$, (b), c, (c), $\eta$ and (d) the total sum of the weights $w_i$ for a mGG graph with model parameters $\alpha=1, \tau=0, \beta=1, c=2$ and $\eta=130$. Three chains are displayed and the true value is in red dashed line.
• Figure 5: MCMC traceplot of four weights for a graph generated from a mGG graph model with parameters $\alpha=1$, $\tau=0$, $\beta=1$, $c=2$ and $\eta=130$. Three chains are displayed and the true value is in red dashed line. The degree of the corresponding node is (a) 536, (b) 496, (c) 49 and (d) 2.

Theorems & Definitions (24)

• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 3.4
• Proposition 3.5
• Proposition 4.1: Asymptotic number of nodes and edges
• Corollary 4.2: Extreme sparsity
• Proposition 4.3: Asymptotic degree distribution
• Lemma S1
• Lemma S2