Universality in survivor distributions: Characterising the winners of competitive dynamics

TL;DR

This paper explores the universal patterns in survivor distributions within a spatially extended competitive model, revealing how survival probabilities depend on network properties and a key concept called dynamical fugacity.

Contribution

It introduces an analytical inhomogeneous mean-field approach to predict survival probabilities and demonstrates universality in survivor distributions across different network geometries.

Findings

Survivor distributions grow exponentially with system size.

In the large-mass limit, survival probability depends only on node mass and degree.

The concept of dynamical fugacity explains universality in survivor patterns.

Abstract

We investigate the survivor distributions of a spatially extended model of competitive dynamics in different geometries. The model consists of a deterministic dynamical system of individual agents at specified nodes, which might or might not survive the predatory dynamics: all stochasticity is brought in by the initial state. Every such initial state leads to a unique and extended pattern of survivors and non-survivors, which is known as an attractor of the dynamics. We show that the number of such attractors grows exponentially with system size, so that their exact characterisation is limited to only very small systems. Given this, we construct an analytical approach based on inhomogeneous mean-field theory to calculate survival probabilities for arbitrary networks. This powerful (albeit approximate) approach shows how universality arises in survivor distributions via a key concept -- the {\it dynamical fugacity}. Remarkably, in the large-mass limit, the survival probability of a node becomes independent of network geometry, and assumes a simple form which depends only on its mass and degree.