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Simplicial complexes are gaining increasing scientific attention as they aregeneralized network structures that can represent the many-body interactionsexisting in complex systems raging from the brain to high-order socialnetworks. Simplicial complexes are formed by simplicies, such as nodes, links,triangles and so on. Cell complexes further extend these generalized networkstructures as they are formed by regular polytopes such as squares, pentagonsetc. Pseudo-fractal simplicial and cell complexes are a major example ofgeneralized network structures and they can be obtained by gluing$2$-dimensional $m$-polygons ($m=2$ triangles, $m=4$ squares, $m=5$ pentagons,etc.) along their links according to a simple iterative rule. Here weinvestigate the interplay between the topology of pseudo-fractal simplicial andcell complexes and their dynamics by characterizing the critical properties oflink percolation defined on these structures. By using the renormalizationgroup we show that the pseudo-fractal simplicial and cell complexes have acontinuous percolation threshold at $p_c=0$. When the pseudo-fractal structureis formed by polygons of the same size $m$, the transition is characterized byan exponential suppression of the order parameter $P_{\infty}$ that depends onthe number of sides $m$ of the polygons forming the pseudo-fractal cellcomplex, i.e., $P_{\infty}\propto p\exp(-\alpha/p^{m-2})$. Here these resultsare also generalized to random pseudo-fractal cell-complexes formed by polygonsof different number of sides $m$.

Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes