Percolation on branching simplicial and cell complexes and its relation to interdependent percolation

TL;DR

This paper explores how percolation behaves on complex branching simplicial and cell complexes, revealing multiple phase transitions and linking these phenomena to interdependent percolation in multiplex networks, with implications for network geometry effects.

Contribution

It establishes a novel relation between percolation in branching complexes and interdependent percolation, uncovering multiple phase transitions and universality classes including BKT.

Findings

Branching cell complexes exhibit more than two percolation phase transitions.

Percolation probability and fractal exponent show discontinuities at these transitions.

Upper percolation transition can belong to various universality classes, including BKT.

Abstract

Network geometry has strong effects on network dynamics. In particular, the underlying hyperbolic geometry of discrete manifolds has recently been shown to affect their critical percolation properties. Here we investigate the properties of link percolation in non-amenable two-dimensional branching simplicial and cell complexes, i.e., simplicial and cell complexes in which the boundary scales like the volume. We establish the relation between the equations determining the percolation probability in random branching cell complexes and the equation for interdependent percolation in multiplex networks with inter-layer degree correlation equal to one. By using this relation we show that branching cell complexes can display more than two percolation phase transitions: the upper percolation transition, the lower percolation transition, and one or more intermediate phase transitions. At these additional transitions the percolation probability and the fractal exponent both feature a discontinuity. Furthermore, by using the renormalization group theory we show that the upper percolation transition can belong to various universality classes including the Berezinskii-Kosterlitz-Thouless (BKT) transition, the discontinuous percolation transition, and continuous transitions with anomalous singular behavior that generalize the BKT transition.