Evolution of spatially embedded branching trees with interacting nodes

TL;DR

This paper models the evolution of spatially embedded branching trees with suppressed close-node branching, revealing a transition from exponential to power-law growth and potential extinction in restricted spaces.

Contribution

It introduces a new model of cooperative branching with spatial suppression, capturing complex growth dynamics and phase transitions in embedded networks.

Findings

Initial exponential growth transitions to power-law growth after a crossover time

Spatial restrictions can lead to extinction of the branching process

Identifies a transition point analogous to a phase change in network size

Abstract

We study the evolution of branching trees embedded in Euclidean spaces with suppressed branching of spatially close nodes. This cooperative branching process accounts for the effect of overcrowding of nodes in the embedding space and mimics the evolution of life processes (the so-called "tree of life") in which a new level of complexity emerges as a short transition followed by a long period of gradual evolution or even complete extinction. We consider the models of branching trees in which each new node can produce up to two twigs within a unit distance from the node in the Euclidean space, but this branching is suppressed if the newborn node is closer than at distance $a$ from one of the previous generation nodes. This results in an explosive (exponential) growth in the initial period, and, after some crossover time $t_x \sim \ln(1/a)$ for small $a$, in a slow (power-law) growth. This special point is also a transition from "small" to "large words" in terms of network science. We show that if the space is restricted, then this evolution may end by extinction.