Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.22148

Ordering Stochastic Block Models via prior transitivity

Lapo Santi·Nial Friel·Pierpaolo De Blasi
View original source ↗Read source document ↗Export metadata ↓

Abstract

In directed networks, nodes may form groups with similar interaction patterns, while these groups may themselves follow an ordered structure. Existing methods typically treat these features separately, either clustering nodes without enforcing a coherent block order, or ranking individual nodes without allowing for structurally equivalent groups. We introduce the Transitive Stochastic Block Model (TSBM), a Bayesian model for directed weighted networks that uses transitivity-inducing priors to infer ordered blocks. The model separates the total volume of interaction between two nodes from the direction of interaction conditional on interaction occurring, so that hierarchy is imposed on directional imbalance rather than interaction frequency. We consider two order-restricted specifications: a flexible weak-stochastic-transitivity version, which excludes cyclic dominance patterns while allowing heterogeneous block-pair strengths, and a Toeplitz strong-stochastic-transitivity version, in which directional advantage increases with rank separation. Posterior inference is performed through a Gibbs sampler using Pólya-Gamma data augmentation. Since ordered block labels are not exchangeable, we introduce an age-ordered partition prior to infer the number of blocks jointly with node allocation. Simulation studies show that order-constrained priors improve prediction and partition recovery, especially in sparse networks. Across six empirical directed networks, the TSBM improves predictive performance in four cases and yields partitions with clearer ordered structure. The results also identify cases, such as nearly deterministic dominance networks or non-transitive citation networks, where imposing ordered blocks can harm prediction. The TSBM therefore provides a probabilistic framework for estimating ordered groups and assessing when a transitive block structure is supported by the data.

Explore related subjects

stat.MEstat.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Lapo Santi, Nial Friel, Pierpaolo De Blasi. 2026-06-20. Ordering Stochastic Block Models via prior transitivity. https://arxiv.org/abs/2606.22148

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Likelihood Based Inference in Fully and Partially Observed Exponential Family Graphical Models with Intractable Normalizing Constants

Probabilistic graphical models that encode an underlying Markov random field are fundamental building blocks of generative modeling to learn latent representations in modern multivariate data sets with complex dependency structures. Among these, the exponential family graphical models are especially popular, given their fairly well-understood statistical properties and computational scalability to high-dimensional data based on pseudo-likelihood methods. These models have been successfully applied in many fields, such as the Ising model in statistical physics and count graphical models in genomics. Another strand of models allows some nodes to be latent, so as to allow the marginal distribution of the observable nodes to depart from exponential family to capture more complex dependence. These approaches form the basis of generative models in artificial intelligence, such as the Boltzmann machines and their restricted versions. A fundamental barrier to likelihood-based (i.e., both maximum likelihood and fully Bayesian) inference in both fully and partially observed cases is the intractability of the likelihood. The usual workaround is via adopting pseudo likelihood-based approaches, following the pioneering work of Besag(1974). The goal of this paper is to demonstrate that full likelihood-based analysis of these models is feasible in a computationally efficient manner under a logarithmically sparse setting. The chief innovation lies in utilizing the tractable independence model underlying an intractable graphical model, to estimate the normalizing constant, as well as its gradient. Extensive numerical results, supporting theory and comparisons with pseudo likelihood-based approaches demonstrate the applicability of the proposed method.

stat.ME↗
arXiv2026

Interpretable Deep Neural Network for Modeling Functional Surrogates

Developing surrogates for computer models has become increasingly important for addressing complex problems in science and engineering. This article introduces an artificial intelligent (AI) surrogate, referred to as the DeepSurrogate, for analyzing functional outputs with vector-valued inputs. The relationship between the functional output and vector-valued input is modeled as an infinite sequence of unknown functions, each representing the relationship at a specific location within the functional domain. These spatially indexed functions are expressed through a combination of basis functions and their corresponding coefficient functions, both of which are modeled using deep neural networks (DNN). The proposed framework accounts for spatial dependencies across locations, while capturing the relationship between the functional output and scalar predictors. It also integrates a Monte Carlo (MC) dropout strategy to quantify prediction uncertainty, enhancing explainability in the deep neural network architecture. The proposed method enables efficient inference on datasets with approximately 50,000 spatial locations and 20 simulations, achieving results in under 10 minutes using standard hardware. The approach is validated on extensive synthetic datasets and a large-scale simulation from the Sea Lake and Overland Surge from Hurricanes (SLOSH) simulator. An open-source Python package implementing the method is made available.

stat.ME↗
arXiv2026

Deep Generative Modeling with Spatial and Network Images: An Explainable AI (XAI) Approach

This article addresses the challenge of modeling the amplitude of spatially indexed low frequency fluctuations (ALFF) in resting state functional MRI as a function of cortical structural features and a multi-task coactivation network in the Adolescent Brain Cognitive Development (ABCD) Study. It proposes a generative model that integrates effects of spatially-varying inputs and a network-valued input using deep neural networks to capture complex non-linear and spatial associations with the output. The method models spatial smoothness, accounts for subject heterogeneity and complex associations between network and spatial images at different scales, enables accurate inference of each images effect on the output image, and allows prediction with uncertainty quantification via Monte Carlo dropout, contributing to one of the first Explainable AI (XAI) frameworks for heterogeneous imaging data. The model is highly scalable to high-resolution data without the heavy pre-processing or summarization often required by Bayesian methods. Empirical results demonstrate its strong performance compared to existing statistical and deep learning methods. We applied the XAI model to the ABCD data which revealed associations between cortical features and ALFF throughout the entire brain. Our model performed comparably to existing methods in predictive accuracy but provided superior uncertainty quantification and faster computation, demonstrating its effectiveness for large-scale neuroimaging analysis. Open-source software in Python for XAI is available.

stat.ME↗