Solution of the Unanimity Rule on exponential, uniform and scalefree networks: A simple model for biodiversity collapse in foodwebs

TL;DR

This paper analytically solves the Unanimity Rule on various network types to understand biodiversity collapse in foodwebs, providing equations that predict species loss leading to system collapse.

Contribution

It offers exact solutions for exponential and uniform networks and a good approximation for scalefree networks, linking network theory to biodiversity loss.

Findings

Exact solutions for exponential and uniform networks.

Approximate solution for scalefree networks matching simulations.

Model can estimate critical species removal for collapse.

Abstract

We solve the Unanimity Rule on networks with exponential, uniform and scalefree degree distributions. In particular we arrive at equations relating the asymptotic number of nodes in one of two states to the initial fraction of nodes in this state. The solutions for exponential and uniform networks are exact, the approximation for the scalefree case is in perfect agreement with simulation results. We use these solutions to provide a theoretical understanding for experimental data on biodiversity loss in foodwebs, which is available for the three network types discussed. The model allows in principle to estimate the critical value of species that have to be removed from the system to induce its complete collapse.