# Bayesian inference on random simple graphs with power law degree distributions

## TL;DR

This paper introduces a Bayesian model for simple graphs with power law degree distributions using BFRY random variables, enabling automatic power law detection and scalable inference on network data.

## Contribution

It develops a novel Bayesian framework employing BFRY variables for power law graph modeling, extending to latent structures, with scalable variational inference methods.

## Key findings

- Successfully models real-world networks with power law degrees
- Automatically identifies the degree distribution parameters from data
- Scalable inference via stochastic gradient methods

## Abstract

We present a model for random simple graphs with a degree distribution that obeys a power law (i.e., is heavy-tailed). To attain this behavior, the edge probabilities in the graph are constructed from Bertoin-Fujita-Roynette-Yor (BFRY) random variables, which have been recently utilized in Bayesian statistics for the construction of power law models in several applications. Our construction readily extends to capture the structure of latent factors, similarly to stochastic blockmodels, while maintaining its power law degree distribution. The BFRY random variables are well approximated by gamma random variables in a variational Bayesian inference routine, which we apply to several network datasets for which power law degree distributions are a natural assumption. By learning the parameters of the BFRY distribution via probabilistic inference, we are able to automatically select the appropriate power law behavior from the data. In order to further scale our inference procedure, we adopt stochastic gradient ascent routines where the gradients are computed on minibatches (i.e., subsets) of the edges in the graph.