Patchy percolation on a hierarchical network with small-world bonds

The bond-percolation properties of the recently introduced Hanoi networks are analyzed with the renormalization group. Unlike scale-free networks, these networks are meant to provide an analytically tractable interpolation between finite-dimensional, lattice-based models and their mean-field limits. In percolation, the hierarchical small-world bonds in the Hanoi networks impose order by uniting otherwise disconnected, local clusters. This “patchy” order results in merely a finite probability to obtain a spanning cluster for certain ranges of the bond probability, unlike the usual 0–1 transition found on ordinary lattices. The various networks studied here exhibit a range of phase behaviors, depending on the prevalence of those long-range bonds. Fixed points in general exhibit nonuniversal behavior.

Figure 1

Depiction of the three-regular network HN3 on a semi-infinite line. The entire graph becomes three-regular with a self-loop at $n=0$. Note that HN3 is planar.

Figure 2

(Color online) Depiction of the planar network HN5, comprised of HN3 (black lines) with the addition of further long-range bonds (green/shaded lines). There is no distinction made between black and shaded lines in our studies here.

Figure 3

(Color online) Depiction of the nonplanar Hanoi network HN-NP. Again, starting from a $1d$-backbone (black lines), a set of long-range bonds (blue/shaded lines) is added that break planarity but maintain the hierarchical pattern set out in Eq. (1). The RG on this network remains exact.

Figure 4

(Color online) Pattern for the recursive generation of a hierarchical lattice. In zeroth generation (top), the lattice consists of a single bond between the two “boundary” sites (squares). In successive generations each bond of that type (solid line) is replaced by a new type of bond (dashed line) in parallel with a sequence of two of the original bonds (solid lines), see middle and bottom for generation one and two.

Figure 5

(Color online) All possible combinations of reducing bonds from the $n\text{-th}$ RG step to form a new bond at step $n+1$, reversing the pattern set in Fig. 4. All graphs on the left percolate (i.e., connect end-to-end) by merit of the dashed bond alone, while on the right only the top graph provides such a connection. If $q_n$ is the probability for a solid and $p$ for a dashed bond, the weights for the graphs on the left are $p{q}_n^2$, $p{q}_n\left(1-q_n\right)$, $p{q}_n\left(1-q_n\right)$, and $p{\left(1-q_n\right)}^2$ and on the right are $\left(1-p\right)q_n^2$, $\left(1-p\right)q_n\left(1-q_n\right)$, $\left(1-p\right)q_n\left(1-q_n\right)$, and $\left(1-p\right){\left(1-q_n\right)}^2$ from top to bottom.

Figure 6

(Color online) Phase diagram for the RG flow of the simple hierarchical lattice due to Eq. (8). For any given $p$, the flow for $q_n$ evolves along the flow trajectories starting from a chosen initial value $q_0$. Except for points $\left(p<\frac{1}{2},q_0=1\right)$ on the (thick red/shaded) line of unstable fixed points, all points $\left(p,q_0<1\right)$ flow toward the (thick black) line of stable fixed points from Eq. (9) or $q^{\ast }=1$ for $p>\frac{1}{2}$. This is especially true for a homogeneous choice of bond probabilities in a network, $q_0=p$ along the (dash-dotted) diagonal line.

Figure 7

(Color online) Pattern for the recursive generation of a $2d$ hierarchical lattice, originating from the Migdal-Kadanoff bond-moving scheme in a square lattice, with the addition of a small-world bond. Each single, solid bond between the two sites (squares) of a previous generation is replaced by a small-world bond (dashed line) in parallel with two sequences of two of the original bonds (solid lines) at the next generation.

Figure 8

(Color online) Phase diagram for the RG flow of the $2d$ hierarchical lattice due to Eq. (13), using the same notation as in Fig. 6. Here, there exists an additional phase for $p<\overline{p}=\frac{5}{32}=0.15625$ due to the (thick red/shaded) line of unstable fixed points on some line $q_0\left(p\right)<1$. Points above that line flow into the now stable ordered fixed point $q^{\ast }=1$. Note that for the homogeneous choice of bond probabilities in a network, i.e., $q_0=p$ along the (dash-dotted) diagonal line, we observe a discontinuous behavior at $\overline{p}$: Rising from the partially ordered phase below, the flow onto a finite percolation probability $q^{\ast }<1$ jumps (along the thick dashed vertical line) to $q^{\ast }=1$ in the fully ordered phase above $\overline{p}$.

Figure 9

(Color online) Representation of HN3 as a branched Koch curve. The one-dimensional backbone shown in Fig. 1 is marked in black here; long-range bonds are in red/shaded. In this representation, the diameter in Eq. (2) corresponds to the baseline in the Koch curve. To prevent end-to-end connectivity, only two bonds right down the symmetry axis need to be cut, independent of size. Hence, the network is finitely ramified.

Figure 10

(Color online) Depiction of the (exact) renormalization-group step for percolation on HN3. The step consists of eliminating every second site (with label $b$) in the left diagram and expressing the renormalized connection probabilities $\left(R_{n+1},S_{n+1},T_{n+1},U_{n+1},N_{n+1}\right)$ on the right diagram in terms of the old values $\left(R_n,S_n,T_n,U_n,N_n\right)$. Here, $R_n$ refers to the probability that three consecutive points $abc$ are connected, $S_n$ refers to $ab$ being connected but not $c$, $T_n$ to $bc$ being connected but not $a$, $U_n$ to $ac$ being connected but not $b$, whereas $N_n$ refers to the probability that neither $a$, $b$, or $c$ are mutually connected. Upper case letters $A,B,C$ refer to the renormalized sites at $n+1$. Note that the original network in Fig. 1 does not contain bonds of type $U_n$ (long-dashed line), but that they become relevant during the RG process.

Figure 11

(Color online) Example of graphlets contributing to the RG step in HN3. Eliminating every second site in the network (dots labeled $b$), the connectedness of the renormalized sites $\left(A,B,C\right)$ have to be assessed. There are seven bonds in the elementary graphlet: two each for pair connections $ab$, $bc$, and $ac$, and one previously unrenormalized bond with raw probability $p$ stretching between the two sites $b$. Bonds of higher level in the hierarchy (vertical lines) remain unaffected until a later RG step. Thus, $2^7=128$ graphlets with combinations of these bond must be considered. Only four examples are displayed here. In the first, all bonds are present with a direct bond between $abc$ on both sides of $B$; each such triangle has a probability $R_n/4$, and the bond $p$ is present, making the weight $p{R}_n^2/16$. Because, after renormalization, the remaining sites $ABC$ are also directly linked, this graphlet contributes to $R_{n+1}$. In the second example, only $ac$ but not $b$ are linked on the left, contributing a weight of $U_n$, but there is a path connecting $abc$ on the right with weight $R_n/4$, and there is no bond $p$, making the total weight $\left(1-p\right)U_n{R}_n$. Again, there is a path between $ABC$, so it contributes to $R_{n+1}$. In the third example, the $S_n$ bond between $ab$ on the left and the $T_n$ bond on the right together with the $p$-bond ensure a connection between $AB$, thus making a contribution to $S_{n+1}$. The fourth example is the only graphlet that contributes to $U_{n+1}$ with a direct link between $AC$ but not $B$.

Figure 12

(Color online) Depiction of the (exact) RG step for percolation on HN5. In contrast to the corresponding step for HN3 in Fig. 10, the long-dashed bonds are germane to the network, not emergent properties. Thus, the long-dashed bond on top (left) has still probability $p$; the long-dashed bonds of the previous hierarchical level below have a probability of $p$ plus the emergent contribution from the previous step. The graphlet after the RG step (right) is identical to that in Fig. 10.

Figure 13

(Color online) Phase diagram for the RG flow of HN5 due to Eq. (31). The all-connected operator $R$ is used as an indicator of percolative order in the system. For any given $p$, $R_n$ evolves along the shown trajectories starting from an initial value $R_0$. Except for points $\left(p<\overline{p}=2-\varphi ,R_0=1\right)$ on the (thick red/shaded) line of unstable fixed points, all points $\left(p,R_0<1\right)$ flow toward the (thick black) line of stable fixed points obtained from Eq. (31) or to $R^{\ast }=1$ for $p>\overline{p}$. For a homogeneous choice of bond probabilities in the network given in Eqs. (32), i.e., $R_0=p^2\left(3-2p\right)$ on the dash-dotted line, the stable fixed point is always approached from below.

Figure 14

(Color online) Depiction of the (exact) RG step for percolation on HN-NP. In contrast to HN3, the long-range bond connecting $bb$ in Fig. 10 is replaced by a crossing pair of such bonds connecting $ab$. The $U$ bond connecting $ac$ is an emergent operator, like in HN3. The graphlet after the RG step (right) is identical to that for HN3 (5) in Fig. 10 (12).

Figure 15

(Color online) Phase diagram for the RG flow of HN-NP due to Eq. (37). Here, the all-connected operator $R$ is used as in indicator of percolative order in the system. For any given $p$, $R_n$ evolves along the shown trajectories starting from an initial value $R_0$. The homogeneous choice of bond probabilities in the network given in Eq. (21), i.e., $R_0=p^2$ along the (dash-dotted) diagonal line, crosses the line of unstable fixed points just below the branch-point singularity.