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

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.

💡 Research Summary

The paper presents a rigorous analytical treatment of the Unanimity Rule (UR) on three canonical network topologies—exponential, uniform (regular), and scale‑free—and demonstrates how the resulting equations can be used to predict biodiversity collapse in ecological food webs. The Unanimity Rule is a binary state dynamics in which a node changes its state only when all of its neighbors are already in the opposite state. This rule captures the idea that a species (node) can survive only if every directly interacting species (its neighbors) is present; the loss of a single neighbor can trigger a cascade of extinctions when the rule is applied globally.

Methodology
The authors first formulate the UR dynamics in terms of the fraction of active nodes, ρ(t), and derive a closed‑form recursion for the asymptotic value ρ∞ = lim t→∞ ρ(t). The derivation relies on generating‑function techniques and a mean‑field approximation that becomes exact for networks with narrow degree distributions. For each degree distribution they obtain a specific functional relationship between the initial active fraction ρ₀ and the final active fraction ρ∞:

1. Exponential networks (P(k) ∝ e^{-λk}): The degree distribution decays rapidly, allowing the independence assumption for neighbor states to hold. The authors derive an exact equation
   ρ∞ = 1 − exp