Polylogarithm models for heavy-tailed counts: approximate Bayesian inference and application to ecological networks

Abstract Ecological surveys often generate count data to assess biodiversity and ecosystem dynamics, capturing species abundances or behavioural interactions such as pollination and resource competition. In this study, we analyse flower–invertebrate interactions in the Brazilian Atlantic Forest using a dataset compiled from multiple regional surveys. We examine 5,150 recorded interactions spanning four plant orders and 28 invertebrate orders, including effective pollination, general floral visits, and contact-based reproductive behaviours. Our main objectives are to model the distribution of interaction frequencies within the resulting network and to cluster flower–invertebrate interactions. These tasks are complicated by the heavy-tailed nature of the data. To address this, we develop a toolkit for sampling and fitting the Polylogarithm distribution, a discrete model capable of capturing heavy-tailed behaviour. We show that the Polylogarithm distribution unifies several well-known discrete heavy-tailed distributions, but its probability function is intractable. We overcome this limitation by designing a rejection sampler to generate exact draws, which forms the basis of a Bayesian inferential framework using Approximate Bayesian Computation. Finally, we extend the model to a finite mixture formulation to identify ecological interaction patterns. This clustering approach reveals three main plant groups and improves the fit to the empirical degree sequence.