Emergence of strongly connected giant components in continuum disk-spin percolation

TL;DR

This paper introduces a continuum percolation model for overlapping disks with spins, inspired by neuronal networks, studying how giant strongly connected components emerge and how critical exponents vary with temperature.

Contribution

It presents a novel continuum model incorporating spin and orientation effects in disk percolation, inspired by biological neuronal networks, and analyzes phase transitions and critical behavior.

Findings

Emergence of a giant strongly connected component in the model.

Critical exponents depend on temperature.

Phase diagram characterization of the percolation transition.

Abstract

We propose a continuum model of percolation in two dimensions for overlapping disks with spin. In this model the existence of bonds is determined by the distance between the centers of the disks, and by the scalar product of the (randomly) directed spin with the direction of the vector connecting the centers of neighboring disks. The direction of a single spin is controlled by a "temperature", representing the amount of polarization of the spins in the direction of an external field. Our model is inspired by biological neuronal networks and aims to characterize their topological properties when axonal guidance plays a major role. We numerically study the phase diagram of the model observing the emergence of a giant strongly connected component, representing the portion of neurons that are causally connected. We provide strong evidence that the critical exponents depend on the temperature.