Topological Percolation on Hyperbolic Simplicial Complexes

Abstract

Simplicial complexes are increasingly used to understand the topology of complex systems as different as brain networks and social interactions. It is therefore of special interest to extend the study of percolation to simplicial complexes. Here we propose a topological theory of percolation for discrete hyperbolic simplicial complexes. Specifically we consider hyperbolic manifolds in dimension $d=2$ and $d=3$ formed by simplicial complexes, and we investigate their percolation properties in the presence of topological damage, i.e., when nodes, links, triangles or tetrahedra are randomly removed. We show that in $d=2$ simplicial complexes there are four topological percolation problems and in $d=3$, there are six. We demonstrate the presence of two percolation phase transitions characteristic of hyperbolic spaces for the different variants of topological percolation. While most of the known results on percolation in hyperbolic manifolds are in $d=2$, here we uncover the rich critical behavior of $d=3$ hyperbolic manifolds, and show that triangle percolation displays a Berezinskii-Kosterlitz-Thouless (BKT) transition. Finally we provide evidence that topological percolation can display a critical behavior that is unexpected if only node and link percolation are considered.

(17 pages, 9 figures)

References in corpus (18)

• Uncovering the overlapping community structure of complex networks in nature and society
• Critical phenomena in complex networks
• Hyperbolic Geometry of Complex Networks
• Clique percolation in random networks
• Sustaining the Internet with Hyperbolic Mapping
• Random hypergraphs and their applications
• Simplicial complexes and complex systems
• Recent advances and open challenges in percolation
• Emergent Complex Network Geometry
• Recent advances of percolation theory in complex networks
• Topological implications of negative curvature for biological and social networks
• BKT-like transition in the Potts model on an inhomogeneous annealed network
• Small-World Bonds and Patchy Percolation on the Hanoi Network
• Network Geometry and Complexity
• Monte-Carlo simulation study of the two-stage percolation transition in enhanced binary trees
• Generating-function approach for bond percolations in hierarchical networks
• Percolation Thresholds in Hyperbolic Lattices
• From explosive to infinite-order transitions on a hyperbolic network

Cited by in corpus (32)

• Networks beyond pairwise interactions: structure and dynamics
• Simplicial models of social contagion
• Percolation on complex networks: Theory and application
• Explosive higher-order Kuramoto dynamics on simplicial complexes
• Higher-order percolation processes on multiplex hypergraphs
• Beyond the clustering coefficient: A topological analysis of node neighbourhoods in complex networks
• Simplicial complexes: higher-order spectral dimension and dynamics
• Synchronization in Network Geometries with Finite Spectral Dimension
• The dynamic nature of percolation on networks with triadic interactions
• Global topological synchronization on simplicial and cell complexes
• The spectral dimension of simplicial complexes: a renormalization group theory
• D-dimensional oscillators in simplicial structures: odd and even dimensions display different synchronization scenarios
• Homological Percolation and the Euler Characteristic
• Higher-Order Components Dictate Higher-Order Contagion Dynamics in Hypergraphs
• Percolation on branching simplicial and cell complexes and its relation to interdependent percolation
• The higher-order spectrum of simplicial complexes: a renormalization group approach
• The nature of hypergraph $k$-core percolation problems
• Simplex2Vec embeddings for community detection in simplicial complexes
• Renormalization group for link percolation on planar hyperbolic manifolds
• Unveiling the Importance of Nonshortest Paths in Quantum Networks
• Direct Visualization of Spatial inhomogeneity of Spin Stripes Order in La1.72Sr0.28NiO4
• Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes
• Network Renormalization
• Quantum statistics in Network Geometry with Fractional Flavor
• Higher-order triadic percolation on random hypergraphs
• Combinatorial Properties for a Class of Simplicial Complexes Extended from Pseudo-fractal Scale-free Web
• Critical time-dependent branching process modelling epidemic spreading with containment measures
• Nearest-neighbour directed random hyperbolic graphs
• Edge corona product as an approach to modeling complex simplical networks
• Deterministic simplicial complexes
• Construction of simplicial complexes with prescribed degree-size sequences
• Extended-range percolation in five dimensions