Renormalization group for link percolation on planar hyperbolic manifolds

Network geometry is currently a topic of growing scientific interest, as it opens the possibility to explore and interpret the interplay between structure and dynamics of complex networks using geometrical arguments. However, the field is still in its infancy. In this work we investigate the role of network geometry in determining the nature of the percolation transition in planar hyperbolic manifolds. Boettcher et al. [Nat. Comm. 3, 787 (2012)] have shown that a special type of two-dimensional hyperbolic manifolds, the Farey graphs, display a discontinuous transition for ordinary link percolation. Here we investigate using the renormalization group the critical properties of link percolation on a wider class of two-dimensional hyperbolic deterministic and random manifolds constituting the skeletons of two-dimensional cell complexes. These hyperbolic manifolds are built iteratively by subsequently gluing $m$-polygons to single edges. We show that when the size $m$ of the polygons is drawn from a distribution $q_m$ with asymptotic power-law scaling $q_m\simeq C{m}^{-\gamma }$ for $m\gg 1$, different universality classes can be observed depending on the value of the power-law exponent $\gamma$. Interestingly, the percolation transition is hybrid for $\gamma \in (3,4)$ and becomes continuous for $\gamma \in (2,3]$.

Figure 1

The first iterations $n=0,n=1$, and $n=2$ for the constructions of the considered hyperbolic manifolds are shown in the case in which the hyperbolic manifold is deterministic and formed only by triangles ($m=3$) or squares $(m=4)$ and in the case in which the hyperbolic manifold is random with $q_m=0.5{\delta }_{m,3}+0.5{\delta }_{m,4}.$

Figure 2

The percolation probability $T$ is plotted versus $p$ for deterministic hyperbolic manifolds with $m=3,4,5$ (a) and random hyperbolic manifolds with scale-free distribution $q_m=C{m}^{-\gamma }$ with $m\ge 3$ and $\gamma =4.5,3.5,2.5$ (b).

Figure 3

The results of the first $n$ iterations generating the regular hyperbolic manifolds with $m=3$ (a) and $m=4$ (b) are shown in blue (thick line), together with the results of the first $n$ iteration generating a random hyperbolic manifold with $q_m=1/2\delta (m,3)+1/2\delta (m,4)$ (c). The initial link is indicated in black, and the dual network (the tree) is indicated in red (thin line). The number of iterations $n$ is $n=5$ for panel (a) and $n=4$ for panels (b) and (c).

Figure 4

Diagrammatic representation of generating functions ${\widehat{T}}_n\left(x\right)$ (a) and ${\widehat{S}}_n(x,y)$ (b). Filled areas indicate clusters that either connect $\left[{\widehat{T}}_n\left(x\right)\right]$ or do not connect $\left[{\widehat{S}}_n(x,y)\right]$ the end nodes.

Figure 5

The diagrams contributing to ${\widehat{T}}_{n+1}\left(x\right)$ for the deterministic manifold with $m=4$ are shown. The contributions from configurations (a), (b), and (d)–(h) are $p{x}^2{\widehat{T}}_n^3\left(x\right)$ (a), $(1-p)x^2{\widehat{T}}_n^3\left(x\right)$ (b), $p{x}^2{\widehat{T}}_n^2\left(x\right){\widehat{S}}_n(x,x)$ (d–f), $px{\widehat{T}}_n\left(x\right){\widehat{S}}_n^2(x,1)$ (g, h). The comprehensive contribution of configurations (c) and (i) is ${\widehat{S}}_n^2(x,1)$.

Figure 6

The diagrams contributing to ${\widehat{S}}_{n+1}\left(x\right)$ for the deterministic manifold with $m=4$ are shown. The comprehensive contribution from configurations (a) and (g) is $(1-p){\widehat{S}}_n(x,1){\widehat{S}}_n(y,1)$. The contributions from the other configurations are $(1-p)y^2{\widehat{T}}_n^2\left(y\right){\widehat{S}}_n(x,y)$ (b), $(1-p)xy{\widehat{T}}_n\left(x\right){\widehat{T}}_n\left(y\right){\widehat{S}}_n(x,y)$ (c), $(1-p)x^2{\widehat{T}}_n^2\left(x\right){\widehat{S}}_n(x,y)$ (d), $(1-p)y{\widehat{T}}_n\left(y\right){\widehat{S}}_n(x,1){\widehat{S}}_n(y,1)$ (e), $(1-p)x{\widehat{T}}_n\left(x\right){\widehat{S}}_n(x,1){\widehat{S}}_n(y,1)$ (f).

Figure 7

The exponent ${\psi }_n$ is shown as a function of $p$ for the deterministic hyperbolic manifold with $m=3,4,5$. All curves are obtained for $n=1000$.

Figure 8

The discontinuity $P_{\infty }\left(p_c\right)$ of the order parameter $P_{inf}$ at the critical point $p=p_c$ is numerically evaluated for deterministic hyperbolic manifolds with $3\le m\le 20$ evolved up to $n=1000$ iterations. These numerical results agree well with the analytical expression in Eq. (95) up to a multiplicative constant $C=1.021...$.

Figure 9

The critical scaling of $P_{\infty }\left(p\right)$ for $0<\mathrm{\Delta }p=p-p_c\ll 1$ is numerically investigated for deterministic hyperbolic manifolds with $m=3$ (a) and $m=4$ (b). By investigating the finite-size effects by considering a number of iterations $n=500,1000,2000$ we validate our analytical expression Eq. (92). The constant indicates $c=4ln9$ for both panels (a) and (b).