Efficient Evolutionary Models with Digraphons

TL;DR

This paper introduces efficient methods for modeling and simulating evolutionary processes in biological networks using digraphons, including a generative model, Bayesian inference, and dynamic simulation techniques.

Contribution

It presents a novel digraphon generative model, Bayesian inference approach, and efficient simulation algorithms for dynamic and evolving biological networks.

Findings

A finite basis digraphon generative model for biological networks.

A Gibbs sampling-based MAP inference method for the model.

An efficient dynamic simulation algorithm with amortized updates.

Abstract

We present two main contributions which help us in leveraging the theory of graphons for modeling evolutionary processes. We show a generative model for digraphons using a finite basis of subgraphs, which is representative of biological networks with evolution by duplication. We show a simple MAP estimate on the Bayesian non parametric model using the Dirichlet Chinese restaurant process representation, with the help of a Gibbs sampling algorithm to infer the prior. Next we show an efficient implementation to do simulations on finite basis segmentations of digraphons. This implementation is used for developing fast evolutionary simulations with the help of an efficient 2-D representation of the digraphon using dynamic segment-trees with the square-root decomposition representation. We further show how this representation is flexible enough to handle changing graph nodes and can be used to also model dynamic digraphons with the help of an amortized update representation to achieve an efficient time complexity of the update at $O(\sqrt{|V|}\log{|V|})$.