A Zero-Inflated Poisson Latent Position Cluster Model

The latent position network model (LPM) is a popular approach for the statistical analysis of network data. A central aspect of this model is that it assigns nodes to random positions in a latent space, such that the probability of an interaction between each pair of individuals or nodes is determined by their distance in this latent space. A key feature of this model is that it allows one to visualize nuanced structures via the latent space representation. The LPM can be further extended to the Latent Position Cluster Model (LPCM), to accommodate the clustering of nodes by assuming that the latent positions are distributed following a finite mixture distribution. In this paper, we extend the LPCM to accommodate missing network data and apply this to non-negative discrete weighted social networks. By treating missing data as ``unusual’’ zero interactions, we propose a combination of the LPCM with the zero-inflated Poisson distribution. Statistical inference is based on a novel partially collapsed Markov chain Monte Carlo algorithm, where a Mixture-of-Finite-Mixtures (MFM) model is adopted to automatically determine the number of clusters and optimal group partitioning. Our algorithm features a truncated absorb-eject move, which is a novel adaptation of an idea commonly used in collapsed samplers, within the context of MFMs. Another aspect of our work is that we illustrate our results on 3-dimensional latent spaces, maintaining clear visualizations while achieving more flexibility than 2-dimensional models. The performance of this approach is illustrated via three carefully designed simulation studies, as well as four different publicly available real networks, where some interesting new perspectives are uncovered.

💡 Research Summary

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The paper introduces a novel statistical framework that extends the Latent Position Cluster Model (LPCM) to handle weighted, non‑negative integer networks with an excess of zero observations. By integrating a Zero‑Inflated Poisson (ZIP) distribution into the latent‑position framework, the authors simultaneously model (i) the underlying interaction intensity via a Poisson rate, (ii) the occurrence of “structural” or “unusual” zeros that arise from missing or unrecorded data, and (iii) the clustering of nodes in a latent Euclidean space.

The model assumes each directed edge yᵢⱼ follows a ZIP mixture: with probability pᵢⱼ the observation is a structural zero, otherwise it is drawn from a Poisson(λᵢⱼ). The structural‑zero indicator νᵢⱼ is introduced as a Bernoulli variable. To capture dependence of zero‑inflation on community structure, the authors replace the pair‑specific probability pᵢⱼ by a block‑matrix element p_{zᵢzⱼ}, where zᵢ denotes the latent cluster assignment of node i. This yields a stochastic‑block‑model‑like structure for the zero‑inflation component, allowing different groups to have distinct propensities for missing edges.

The Poisson rate λᵢⱼ is linked to latent positions uᵢ, uⱼ ∈ ℝᵈ (the paper focuses on d = 3) through a log‑linear distance decay:
log(λᵢⱼ) = β – ‖uᵢ – uⱼ‖.
Thus, nodes that are close in the latent space are expected to have larger interaction counts, while distant nodes have smaller rates and a higher chance of true zeros.

Node positions are modeled as draws from a finite mixture of multivariate normals: uᵢ | zᵢ = k ∼ MVN(μ_k, τ_k⁻¹ I_d). The number of mixture components K̄ is not fixed a priori; instead, a Mixture‑of‑Finite‑Mixtures (MFM) prior is placed on the clustering configuration. MFM allows the number of occupied clusters to be inferred from the data while penalizing empty clusters. However, standard Absorb‑Eject (AE) moves used in collapsed Gibbs samplers for Dirichlet‑process‑like models rely on the existence of empty clusters, which MFM forbids. To resolve this incompatibility, the authors devise a Truncated Absorb‑Eject (TAE) move. TAE operates within the constrained space of non‑empty clusters, proposing to either create a new cluster (by splitting an existing one) or merge two existing clusters, while respecting the MFM prior. This move dramatically improves mixing over the number‑of‑clusters dimension.

Inference proceeds via a partially collapsed Gibbs sampler combined with Metropolis‑within‑Gibbs updates. The algorithm cycles through:

1. Data augmentation – sampling νᵢⱼ (structural‑zero indicators) and the latent true counts xᵢⱼ for edges flagged as structural zeros.
2. Poisson parameters – updating λᵢⱼ and the intercept β using conjugate or Metropolis steps based on the augmented counts.
3. Latent positions and cluster parameters – drawing uᵢ, μ_k, τ_k from their full conditional distributions, exploiting the normal‑inverse‑gamma conjugacy.
4. Cluster assignments – updating zᵢ via a multinomial draw conditioned on current μ, τ, and the block‑matrix P.
5. Zero‑inflation block matrix – updating each p_{gh} with a Beta‑Bernoulli conjugate update.
6. TAE move – proposing a change in the number of occupied clusters and accepting/rejecting it with a Metropolis‑Hastings ratio that incorporates the MFM prior.

The authors evaluate the method through three carefully designed simulation studies. The first examines performance under extreme sparsity (80%+ zeros); the second varies the block‑specific zero‑inflation probabilities to test the model’s ability to recover heterogeneous missing‑edge patterns; the third demonstrates the benefits of a three‑dimensional latent space for visual interpretation. Across all scenarios, the ZIP‑LPCM accurately distinguishes structural zeros from Poisson zeros, recovers the true number of clusters, and yields latent positions that reflect the simulated community geometry.

Four real‑world networks are then analyzed: (a) a criminal‑suspect summit co‑attendance network, (b) an email exchange network, (c) a phone‑call network, and (d) a collaboration network. All exhibit non‑negative integer edge weights and a high proportion of zeros. The model uncovers interpretable block structures in the zero‑inflation matrix—e.g., certain groups in the summit network have markedly lower probabilities of unusual zeros, suggesting deliberate concealment of meetings. In the email network, core departments form a tight cluster with almost no structural zeros, while peripheral units show elevated missing‑edge rates, highlighting organizational boundaries. The phone‑call data reveal a clear inverse relationship between latent distance and call frequency, and the collaboration network displays discipline‑specific clusters with distinct patterns of unrecorded collaborations. The three‑dimensional visualizations provide intuitive plots where inter‑cluster distances correspond to interaction intensity, facilitating exploratory analysis for practitioners.

In summary, the paper makes three substantive contributions:

1. Methodological integration – a unified ZIP‑LPCM that simultaneously models interaction intensity, zero‑inflation, and community structure for weighted networks.
2. Bayesian computation – an innovative partially collapsed MCMC scheme with the Truncated Absorb‑Eject move, enabling efficient inference under an MFM prior without empty clusters.
3. Practical demonstration – extensive simulation validation and application to diverse real datasets, showcasing the model’s ability to reveal hidden missing‑edge mechanisms and to produce clear latent‑space visualizations.

The authors acknowledge limitations, such as the reliance on a distance‑based log‑linear link (which imposes symmetry) and the use of a single block matrix for zero‑inflation rather than a more flexible latent‑space‑dependent formulation. Future work could explore hierarchical models for νᵢⱼ, non‑symmetric link functions, incorporation of covariates, and scalability to massive online social networks. Nonetheless, the proposed framework represents a significant advance for statistical network analysis where zero‑inflated weighted data and unknown community structure coexist.