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

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 nonamenable 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 interlayer 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.

Figure 1

We show the first iteration $n=1$ of the branching process in which the initial single link (shown as a thick red line) branches out to $k, m$-polygons with $k$ drawn from the distribution $r_k$. Here the first iteration is shown for different values of $m=3,4$ (corresponding to the attachment of triangles and rectangles, respectively) and $k=3,4$.

Figure 2

The result of the first $n=2$ iterations are shown for: (a) the flower network with $m=3$ and $r_k={\delta }_{k,2}$, and (b) for a random branching cell complex with $m=3$ with nonzero probabilities $r_1,r_2,r_3$. In both cases nodes 1,2 are the nodes present at the iteration $n=0$, nodes 3,4 are the nodes added at the iteration $n=1$, and all the other nodes are added at iteration $n=2$.

Figure 3

The percolation probability $T$ and $1-\psi$ versus the occupation probability of the links $p$, where $\psi$ is the fractal exponent. The considered branching simplicial complex ($m=3$) has three percolation thresholds: the lower percolation threshold at $p^{★}=0$, the upper percolation threshold at $p_c=0.1935\left(5\right),$ and one intermediate percolation threshold at $p_c^{★}=0.0395\left(6\right)$. At the intermediate percolation threshold both $T$ and $\psi$ have a discontinuity but remain smaller than one. Here the branching simplicial complex has $r_k=0.62{\delta }_{k,1}+0.07{\delta }_{k,2}+0.31{\delta }_{k,20}.$

Figure 4

The barycentric plot characterizing the phase diagram of link percolation for the branching simplicial complex with $m=3$ and $r_k=r_1{\delta }_{k,1}+r_{k,2}{\delta }_{k,2}+r_{\widehat{k}}{\delta }_{k,\widehat{k}}$, where $\widehat{k}=3,5,15,$ or is diverging. The parameter space $(r_1,r_2,r_{\widehat{k}})$ is partitioned into phases ${\mathrm{\Omega }}_{ij}$ at which the percolation probability $T$ has $i$ continuous and $j$ discontinuous transitions. When $k\to \infty$ in the rightmost barycentric plot, the first two critical points merge and the domains ${\mathrm{\Omega }}_{21},{\mathrm{\Omega }}_{11},{\mathrm{\Omega }}_{12}$ switch correspondingly to ${\mathrm{\Omega }}_{11},{\mathrm{\Omega }}_{01},{\mathrm{\Omega }}_{02}$.

Figure 5

The barycentric plot characterizing the phase diagram of link percolation for the branching simplicial complex with $m=3$ and $r_k=r_1{\delta }_{k,1}+r_2{\delta }_{k,2}+r_{20}{\delta }_{k,20}$. The percolation probability $T$ versus $p$ is shown at points A, B, C, and D that belong to phases ${\mathrm{\Omega }}_{20},{\mathrm{\Omega }}_{21},{\mathrm{\Omega }}_{21}$, and ${\mathrm{\Omega }}_{12}$, respectively, at the shared accumulation points $E\in {\mathrm{\Omega }}_{11}\cap {\mathrm{\Omega }}_{21}\cap {\mathrm{\Omega }}_{12}$ and $F\in {\mathrm{\Omega }}_{20}\cap {\mathrm{\Omega }}_{11}\cap {\mathrm{\Omega }}_{21}\cap {\mathrm{\Omega }}_{12}$. The sample points are given by the following barycentric coordinates ($r_1,r_2,r_{20}$): $A=(0.8,0.1,0.1), B=(0.33,0.33,0.33), C=(0.62,0.07,0.31),D=(0.55,0.22,0.23),E=(0.52,0.13,0.35)$, and $F=(0.65,0.16,0.19)$. The dashed lines indicate the unstable branches and the vertical lines indicate the predicted positions of the discontinuous phase transitions.

Figure 6

The barycentric plot characterizing the phase diagram of link percolation for the branching simplicial complex with $m=3$ and $r_k=r_1{\delta }_{k,1}+r_2{\delta }_{k,2}+r_{\infty }{\delta }_{k,\widehat{k}}$ with $\widehat{k}\to \infty$. The percolation probability $T$ versus $p$ is shown at points A, B, and C that belong to phases ${\mathrm{\Omega }}_{11},{\mathrm{\Omega }}_{02}$, and ${\mathrm{\Omega }}_{01}$, respectively, and at their shared accumulation point $D\in {\mathrm{\Omega }}_{11}\cap {\mathrm{\Omega }}_{01}\cap {\mathrm{\Omega }}_{02}$. The sample points are given by their barycentric coordinates ($r_1,r_2,r_{\infty }$): $A=(0.70,0.08,0.22), B=(0.45,0.45,0.1), C=(0.25,0.25,0.50)$, and $D=(0.50,0.13,0.37)$. The dashed lines indicate the unstable branches and the vertical lines indicate the predicted positions of the discontinuous phase transitions.

Figure 7

The percolation probability $T$ and the fractional exponent $\psi$ are plotted as a function of occupation probability $p$ of the links. Here we consider a branching simplicial complex $(m=3)$ with $r_k$ distribution $r_k=r_1{\delta }_{k,1}+r_2{\delta }_{k,2}+r_3{\delta }_{k,3}$ with $(r_1,r_2,r_3)$ given by (0.9,0.05,0.05) for the solid blue line [continuous critical point of Eq. (10)], (0.8,0.2,0) for the orange dashed line [tricritical point of Eq. (10)], (0.727273,0.181818,0.0909091) for the green dot-dashed line $[s$ order critical point of Eq. (10) with $s=4]$, and (0.15,0.54,0.31) for the red dotted line [discontinuous critical point of Eq. (10)].

Figure 8

The percolation order parameter $P_{\infty }$ as a function of occupation probability $p$ of the links. Here we consider a branching simplicial complex $(m=3)$ with $r_k$ distribution $r_k=r_1{\delta }_{k,1}+r_2{\delta }_{k,2}+r_3{\delta }_{k,3}$ with $(r_1,r_2,r_3)$ given by (0.9,0.05,0.05) for the solid blue line (discontinuous percolation transition), (0.8,0.2,0) for the orange dashed line (second-order transition), (0.727273,0.181818, 0.0909091) for the green dot-dashed line (anomalous transition following Eq. (61) with $\sigma =3$) and (0.15,0.54,0.31) for the red dotted line (BKT transition). The figure has been obtained by iterating Eq. (42) for $n=1000$ iterations.

Figure 9

The scaling of the exponent ${\psi }_n$ as a function of $n$ is plotted at the critical point $p=p_c$ corresponding to the BKT transition and slightly above the critical point for $p=p_c+\epsilon$ with $\epsilon ={10}^{-4}$. The $r_k$ distribution is given by $r_k=0.1{\delta }_{k,1}+0.5{\delta }_{k,2}+0.3{\delta }_{k,3}+0.1{\delta }_{k,4}$ and $p_c=0.1261\left(7\right)$.

Figure 10

The critical scaling of the order parameter $P_{\infty }$ versus $p-p_c$ is shown for branching simplicial complexes ($m=3$) with distribution $r_k=r_1{\delta }_{k,1}+r_2{\delta }_{k,2}+r_3{\delta }_{k,3}+r_4{\delta }_{k,4}+r_5{\delta }_{k,5}$ and parameter values corresponding to the critical points of order $s$ of Eq. (10). Here the data are obtained by iterating Eq. (42) $n=2\times {10}^4$ times. The thin dotted line is the theoretically predicted scaling given by Eq. (61) with $\sigma =1/3$.