Renormalization group theory of percolation on pseudofractal simplicial and cell complexes

Simplicial complexes are gaining increasing scientific attention as they are generalized network structures that can represent the many-body interactions existing in complex systems ranging from the brain to high-order social networks. Simplicial complexes are formed by simplicies, such as nodes, links, triangles, and so on. Cell complexes further extend these generalized network structures as they are formed by regular polytopes, such as squares, pentagons, etc. Pseudofractal simplicial and cell complexes are a major example of generalized network structures and they can be obtained by gluing two-dimensional m-polygons (m=3 triangles, m=4 squares, m=5 pentagons, etc.) along their links according to a simple iterative rule. Here we investigate the interplay between the topology of pseudofractal simplicial and cell complexes and their dynamics by characterizing the critical properties of link percolation defined on these structures. By using the renormalization group we show that the pseudofractal simplicial and cell complexes have a continuous percolation threshold at p_{c}=0. When the pseudofractal structure is formed by polygons of the same size m, the transition is characterized by an exponential suppression of the order parameter P_{∞} that depends on the number of sides m of the polygons forming the pseudofractal cell complex, i.e., P_{∞}∝pexp(-α/p^{m-2}). Here these results are also generalized to random pseudofractal cell complexes formed by polygons of different number of sides m.