[Paper Review] On decomposable random graphs

Mohamad Elmasri|arXiv (Cornell University)|Oct 9, 2017

Complex Network Analysis Techniques10 references3 citations

TL;DR

This paper introduces a novel framework for modeling decomposable random graphs using tree-dependent bipartite graphs, which simplifies Markov chain Monte Carlo updates by enforcing conditional independence of nodes given a latent tree. The approach enables efficient sampling and exact likelihood factorization, while preserving key properties like junction tree structure and allowing explicit modeling of sub-clustering dynamics.

ABSTRACT

Decomposable graphs are known for their tedious and complicated Markov update steps. Instead of modelling them directly, this work introduces a class of tree-dependent bipartite graphs that span the projective space of decomposable graphs. This is achieved through dimensionality expansion that causes the graph nodes to be conditionally independent given a latent tree. The Markov update steps are thus remarkably simplified. Structural modelling with tree-dependent bipartite graphs has additional benefits. For example, certain properties that are hardly attainable in the decomposable form are now easily accessible. Moreover, tree-dependent bipartite graphs can extract and model extra information related to sub-clustering dynamics, while currently known models for decomposable graphs do not. Properties of decomposable graphs are also transferable to the expanded dimension, such as the attractive likelihood factorization property. As a result of using the bipartite representation, tools developed for random graphs can be used. Hence, a framework for random tree-dependent bipartite graphs, thereupon for random decomposable graphs, is proposed.

Motivation & Objective

• To address the computational complexity of Markov chain updates in decomposable graphs, which are hindered by intricate conditional dependencies.
• To simplify the structural modeling of decomposable graphs by embedding them in a higher-dimensional bipartite space with a latent tree structure.
• To enable fast, parallelizable sampling by decoupling node updates through conditional independence given the tree.
• To provide explicit access to maximal cliques and junction trees, which are otherwise difficult to compute in high-dimensional settings.
• To extend tools from random graph theory—particularly point process representations—to decomposable graph models.

Proposed method

• Model decomposable graphs via a tree-dependent bipartite representation where nodes are conditionally independent given a latent tree structure.
• Use dimensionality expansion to represent the adjacency structure as a binary matrix Z* between nodes and latent communities, with edge indicators s_ki governed by exponential and truncated exponential distributions.
• Define the joint conditional density of (Z*, S) given the tree T_G as a product of exponential terms involving affinity parameters ϑ’_k and ϑ_i, with indicator functions for edge presence.
• Implement iterative Gibbs sampling by conditionally updating Z* given T, then T given Z*, using conjugate priors (e.g., Gamma) for affinity parameters.
• Leverage the Kallenberg representation theorem to model the graph as a continuous point process, treating the affinity function W as fixed with known distributions.
• Use truncated exponential distributions for edge indicators s_ki, with different parameterizations depending on whether z*_{ki} = 0 or 1.

Experimental results

Research questions

• RQ1 Can Markov chain updates for decomposable graphs be simplified by embedding them in a higher-dimensional bipartite space?
• RQ2 How can conditional independence among nodes be induced via a latent tree to decouple update steps?
• RQ3 What is the impact of this representation on likelihood factorization and model inference efficiency?
• RQ4 Can sub-clustering dynamics and maximal clique structures be explicitly modeled in this framework?
• RQ5 To what extent can existing random graph tools, such as point process representations, be adapted to decomposable graph models?

Key findings

• The proposed tree-dependent bipartite model enables simultaneous, parallel updates of graph nodes due to conditional independence given the latent tree, drastically reducing computational complexity.
• The joint conditional density of (Z*, S) given the tree T_G is explicitly factorized as a product of exponential terms involving affinity parameters and edge indicators.
• Posterior distributions for affinity parameters ϑ’_k and ϑ_i are conjugate and tractable, with full conditionals proportional to Gamma-like forms.
• Edge indicators s_ki are updated via a truncated exponential distribution, with different parameterizations based on the presence or absence of an edge in Z*.
• The framework allows exact likelihood factorization similar to the original decomposable graph property, preserving the key statistical advantage of factorization over cliques and separators.
• The model provides direct access to the set of maximal cliques and junction tree structure, which are otherwise hard to compute or sample in high-dimensional decomposable graphs.