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Hierarchical, Regular Small-World Networks Stefan Boettcher1Bruno Gonçalves1 Hasan Guclu2 1Emory University, Dept. of Physics, Atlanta, GA 30322, USA 2Center for Nonlinear Studies, Los Alamos National Laboratory, MS-B258, Los Alamos, NM 87545, USA Abstract Two new classes of networks are introduced that resemble small-world properties. These networks are recursively constructed but retain a fixed, regular degree. They consist of a one-dimensional lattice backbone overlayed by a hierarchical sequence of long-distance links. Both types of networks, one 3-regular and the other 4-regular, lead to distinct behaviors, as revealed by renormalization group studies. The 3-regular networks are planar, have a diameter growing as $\sqrt{N}$ with the system size $N$ , and lead to super-diffusion with an exact, anomalous exponent $d_{w}=1.3057581\ldots$ , but possesses only a trivial fixed point $T_{c}=0$ for the Ising ferromagnet. In turn, the 4-regular networks are non-planar, have a diameter growing as $\sim2^{\sqrt{\log_{2}N^{2}}}$ , exhibit ``ballistic'' diffusion ($d_{w}=1$ ), and a non-trivial ferromagnetic transition, $T_{c}>0$ . It suggest that the 3-regular networks are still quite ``geometric'', while the 4-regular networks qualify as true small-world networks with mean-field properties. As an example of an application we discuss synchronization of processors on these networks. The description of the ``6-degrees-of-separation'' phenomenon in terms of small-world (SW) networks by Watts and Strogatz [Watts and Strogatz(1998)] has captured the imagination of many researchers, and was particularly timely as we suddenly found ourselves in a networked world [Boccaletti et al.(2006)Boccaletti, Latora, Moreno, Chavez, and Hwang,Barabasi(2003),Newman et al.(2001)Newman, Strogatz, and Watts]. Such a rich environment requires a diverse set of tools and models for their understanding. Statistical physics, with its notion of universality, provides powerful methods for the classification of complex systems, like the renormalization group (RG) [Goldenfeld(1992),Stanley(1999),Plischke and Bergersen(1994)]. In fact, RG was applied to the those networks to study the onset of SW behavior [Newman and Watts(1999)].   Figure 1: Display of the 3-regular network H3 (left) and 4-regular network H4 (right). H3 is planar but H4 is not.   In this Letter, we introduce and study a set of graphs which reproduce the behavior of SW networks without the usual disorder inherent in natural networks. Instead, they attain these properties in a recursive, hierarchical manner that is conducive for RG. The benefit of these features is two-fold: For one, we expect that many SW phenomena can be studied analytically on these networks, and that they will prove as useful as, say, the Migdal-Kadanoff approximation [Migdal(1976),Kadanoff(1976)] has been for physical systems in low dimensions. Furthermore, possessing a well-understood and regular set of networks with these properties is a tremendous advantage for engineering applications, as it is difficult to manufacture individual instances of random networks reliably when we can ascertain their behavior only in the ensemble average. Here, we introduce these networks by discussing elementary aspects such as geometry, diffusion, Ising model, and synchronization to highlight their properties. Each network possesses a geometric backbone, a one-dimensional line of $N=2^k$ sites, either open or closed into a ring. Every site on the lattice is connected to its nearest neighbor. To obtain a SW hierarchy, we parameterizing any integer $n$ (except for zero) uniquely in terms of two other integers $(i,j)$ , $i\geq0$ , via $$n = 2^{i}\left(2j+1\right).$$ (1) Here, $i$ denotes the level in the hierarchy whereas $j$ labels consecutive sites within each hierarchy. For instance, $i=0$ refers to all odd integers, $i=1$ to all integers once divisible by 2 (i.e., 2, 6, 10,...), and so on. In these networks, both depicted in Fig. 1, aside from the backbone, each site is also connected with (one or both) of its neighbors within the hierarchy. For example, we obtain a 3-regular network H3 by connecting first neighbors in the 1D-backbone, then 1 to 3, 5 to 7, 9 to 11, etc, for $i=0$ , next 2 to 6, 10 to 14, etc, for $i=1$ , and 4 to 12, 20 to 28, etc, for $i=2$ , and so on. The corresponding 4-regular network H4 is obtained in the same manner, but connecting to both neighbors in the hierarchy, i.e., 1 to 3, 3 to 5, 5 to 7, etc, for $i=0$ , 2 to 6, 6 to 10, etc, for $i=1$ , and so forth. For this network it is clearly preferable to extend the line to $-\infty<n<\infty$ and also connect -1 to 1, -2 to 2, etc, as well as all negative integers in the above pattern. These networks are similar in construction to models of ultra-diffusion [Huberman and Kerszberg(1985),Ogielski and Stein(1985)], yet inhibitive barriers in those models are replaced with short-cuts here. For H3, it is simple to determine geometric properties, for instance, its diameter $d$ , the longest of the shortest paths between any two sites, here the end-to-end distance. Using a sequence of networks for $k=2,4,6,\ldots$ , the diameter-path looks like a Koch curve, see Fig. 2. The length $d_{k}$ of each marked path is given by $d_{k+2}=2d_{k}+1$ for $N_{k+2}=4N_{k},$ hence $$d\sim\sqrt{N}.$$ (2) This property is reminiscent of a square-lattice of $N$ sites, whose diameter (=diagonal) is also $\sim\sqrt{N}$ . H3 is thus far from true SW behavior where $d\sim\ln N$ .   Figure 2: (Color Online) Sequence of shortest end-to-end paths (=diameter, thick lines) for H3 of size $N=2^{k}$ , $k=2,4,8$ . Whenever the system size $N$ increases by a factor of 4, the diameter $d$ increases by a factor of $\sim 2$ , leading to Eq. (2). The geometry of H4 is more subtle. We consider again the shortest path between the origin $n=0$ and the end $n=N=2^{k}$ . Due to degeneracies at each level, one has to probe many levels in the hierarchy to discern a pattern. In fact, any pattern evolves for an increasing number of levels before it gets taken over by a new one, with two patterns creating degeneracies at the crossover. We find that the paths here do not search out the longest possible jump, as in Fig. 2. Instead, the paths reach quickly to some intermediate level and follow consecutive jumps at that level before trailing off in the end. This is a key distinguishing feature between H3 and H4: Once a level is reached in H4, the entire network can be traversed at that level, while in H3 one must switch to lower levels to progress, see Fig. 1. Specifically, the first pattern holds for $k=l_{1}=1$ , the 2nd for $k=l_{2}=2$ , the 3rd for $l_{2}=2<k\leq5=l_{3}$ , and so on, with $l_{g}=l_{g-1}+g$ (initiated by $l_{1}=1$ , $l_{2}=2$ ) demarking the crossover points between patterns. Hence, $l_{g}=\frac{1}{2}g\left(g+1\right)-1,\, \left(g\geq2\right)$ . Within each pattern $g$ , we observe [Boettcher and Gonçalves()] for the end-to-end diameters $d_{k}$ : $$d_{k}^{(g)} = 2d_{k-g}^{(g-1)}+\left(2^{g-1}-1\right), \quad\left(l_{g-1}<k\leq l_{g}\right).$$ (3) At the crossover points $k=l_{g}$ (i.e., $k-g=l_{g-1}$ ) we have $d_{l_{g}}^{(g)}=2d_{l_{g-1}}^{(g-1)}+\left(2^{g-1}-1\right)$ , solved by $$d_{l_{g}}^{(g)} = \left(g-1\right)2^{g-1}+1.$$ (4) With $k=l_{g}$ and $g\sim\sqrt{2l_{g}}\sim\sqrt{2k}\sim\sqrt{\log_{2}N^{2}}$ , it is $$d_{k} \sim\frac{1}{2} \sqrt{\log_{2}N^{2}}\,\,2^{\sqrt{\log_{2}N^{2}}} \qquad\left(N\to\infty\right)$$ (5) for the diameter of H4. Expecting the diameter of a small world to scale as $d\sim\log N$ , we rewrite Eq. (5): $$d_{k} \sim \left(\log_{2}N\right)^{\alpha}\quad{\rm with} \quad\alpha\sim\frac{\sqrt{\log_{2}N^{2}}}{\log_{2}\log_{2}N^{2}}+\frac{1}{2}.$$ (6) Technically, $\alpha$ diverges with $N$ and the diameter grows faster than any power of $\log_{2}N$ [but less than any power of $N$ , unlike Eq. (2)]. In reality, though, $\alpha$ varies only very slowly with $N$ , ranging merely from $\alpha\approx1.44$ to $\approx1.84$ over nine orders of magnitude, $N=10-10^{10}$ . As a demonstration of the rich dynamics facilitated by these networks, we have studied random walks on H3 and H4. Starting all walks at $n=0$ , and we have focused in our simulations only on the mean distance $$\left\langle \left\vert n\right\vert\right\rangle \sim t^{1/d_{w}}.$$ (7) All walks are controlled by the probability $p$ of a walker to step off the lattice into the direction of a long-range jump. In particular, the walker always jumps either to the left or right neighbor with probability $\left(1-p\right)/2$ , but makes a long-range jump with probability $p$ on H3, or $p/2$ to either the left or right on H4. In both cases, a simple 1D nearest-neighbor walk results for $p=0$ , where $d_{w}=2$ for ordinary diffusion. For any probability $p>0$ , the long-range links will dominate the asymptotic behavior, and leading scaling behavior is independent of $p$ .   Figure: Rescaled plot of the mean distance $\left\langle \left\vert n\right\vert\right\rangle$ in H4 for walks up to $t=10^{6}$ . We demonstrate that $d_{w}=1$ but with log-corrections by rewriting Eq. (7) as $\left\langle \left\vert n\right\vert\right\rangle /t\sim V\left[\ln t\right]^{\beta}$ . Then we obtain $\ln\left(\left\langle \left\vert n\right\vert\right\rangle /t\right)/\ln\left[\ln t\right]\sim\beta+\ln V/\ln\left[\ln t\right]$ and linearly extrapolate (dashed lines) $1/\ln\left[\ln t\right]\to0$ , estimating $\beta \approx -0.18$ at the intercept, independent of $p$ . An effective ``velocity'' $V$ could be extracted from the slope. For any value besides $d_{w}=1$ , these extrapolations would not converge.   Adapting the RG for random walks in Refs. [Kahng and Redner(1989),Redner(2001)], we find exact results for H3, for instance, an anomalous exponent in Eq. (7) of $d_{w}=2-\ln\left(\phi\right)/\ln\left(2\right)=1.3057581\ldots$ with $\phi=\left(1+\sqrt{5}\right)/2$ [Boettcher and Gonçalves()]. These calculations are lengthy but parallel those depicted below in Fig. 4 and will be presented elsewhere. This is a truly remarkable exponent because it is a rare example of a nearest-neighbor fractal walk with super-diffusive ($1<d_{w}<2$ ) behavior [Havlin and Ben-Avraham(1987)], aside from well-tuned Levy flights in which the probability to jump between any two sites decays with a power of their separation [Shlesinger et al.(1993)Shlesinger, Zaslavsky, and Klafter]. We have not been able to extend this RG calculation to obtain analytic results for H4 yet, although the high degree of symmetry inherent in these networks (and the simple result obtained) suggest the possibility. Simulations for H4, evolving some $2\times10^{7}$ walks for $10^{6}$ time steps each, yield a value of $d_{w}=1$ with high confidence, see Fig. 3. Hence, a walk on H4 proceeds effectively ballistic, but hardly with linear motion: widely fluctuating jumps conspire just so that a single walker extends outward with an on-average constant velocity in both directions. Clearly, it is easier to traverse H4 than H3 because of the above-stated fact that on H4 a walker can progress repeatedly within a hierarchical level. We have also studied Ising spin models on H3 and H4, with RG and with Monte Carlo simulations. First, we consider the RG for the Ising model on H3. In this case, all steps can be done exactly but the result turns out to be trivial (for uniform bonds) in the sense that there are no finite-temperature fixed points of the RG flow. Yet, the calculation is instructive, highlighting the large number of statistical models that can be accessed through the hierarchical nature of the process, and it is almost identical in outcome to the treatment below for H4. That small difference is just enough to provide H4 with a non-trivial $T_{c}>0$ , which we confirm numerically. The RG consists of recursively tracing out odd-relabeled spins $x_{n\pm 1}$ , see Fig. 4. The $x_{n\pm 1}$ are connected to their even-labeled nearest neighbors on the lattice backbone by a coupling $K_{0}$ . At any level, each $x_{n\pm 1}$ is also connected to one other such spin $x_{n\mp1}$ across an even-labeled spin $x_{n}$ with $n=2\left(2j+1\right)$ in Eq. (1) that is exactly once divisible by 2. Let us call that coupling $K_{1}$ , all other couplings are $K_{i>1}$ . During the RG process, a new coupling $L$ (green, dashed line in Fig. 4) between next-nearest even-labeled neighbors emerges. Putting all higher level terms into $\mathcal{R}$ , we can section the Hamiltonian $$-\beta\mathcal{H}\! = \sum_{\left\{ n=2(2j+1)\right\} } \left(-\beta\mathcal{H}_{n}\right)+\mathcal{R}\left(K_{2},K_{3},\ldots\right),$$ (8) where each sectional Hamiltonian $-\beta\mathcal{H}_{n}$ is given by $$\sum_{m=n-2}^{n+1}K_{0}x_{m}x_{m+1} +K_{1}x_{n-1}x_{n+1}+L\left(x_{n-2}x_{n}+x_{n}x_{n+2}\right)$$ with $\left(K_{0},K_{1},L\right)$ as unrenormalized couplings and we neglected an overall energy scale $I$ . After tracing out the odd-labeled spins in each $\exp\left[-\beta\mathcal{H}_{n}\right]$ , we identify the renormalized couplings (neglecting $I'$ ): $$K'_{0} =\!\! L\!+\!\frac{1}{2}\ln\cosh\left(2K_{0}\right)\!+\!\frac{1}{4}\ln\left[1 +\!\tanh\left(K_{1}\right)\tanh^{2}\left(2K_{0}\right)\right],$$ $$L' =\!\! \frac{1}{4}\ln\left[1 +\!\tanh\left(K_{1}\right)\tanh^{2}\left(2K_{0}\right)\right],$$ (9) and $K'_{i}=K_{i+1}$ f. a. $i\geq1$ . The high-$T$ solution $K_{0}^{*}=L^{*}=0$ is a trivial fixed point of Eq. (9). Excluding that and eliminating $L^{*}$ yields $1=\tanh\left(K_{1}\right)\tanh\left(2K_{0}^{*}\right)$ , which has only the $T_{c}=0$ solution $K_{0}^{*}=\infty$ (where also $K_{1}=J_{1}/T\to\infty$ ). Note, however, that the RG recursions (9) have a remarkable property due to the hierarchical structure of the network: The next-level coupling $K_{1}$ appears as a free parameter and acts as ``source term'' that could be chosen to represent physically interesting situations, e. g. disorder or distance-dependence. For instance, with $K_{i}$ as an increase function of distance $r_{i}=2^{i+1}$ , a non-trivial fixed point could be created.   Figure 4: Depiction of (exact) RG step for the Ising model on H3. Tracing out odd-labeled variables $x_{n\pm 1}$ for all $n=2(2j+1)$ , in the left plot leads to the renormalized couplings $(L',K'_{0},K'_{1})$ on the right in terms of the old couplings $(L,K_{0},K_{1})$ . Unlabeled bonds correspond to $K_{i\geq 2}$ . H3 does not contain couplings of type $(L,L')$ , but they become relevant during the RG process. Random walks on H3 lead to a topologically equivalent, but more involved RG step [Boettcher and Gonçalves()].   In contrast, H4 provides a non-trivial solution for the Ising model even for uniform bonds, as expected for a mean-field system. Again, an exact result for H4 is elusive, although in light of the inherent symmetries such a solution may be possible. Instead, here we proceed to a Niemeijer-van Leeuwen cumulant expansion [Plischke and Bergersen(1994)] and compare with our numerical simulations. The Hamiltonian $-\beta\mathcal{H}$ indeed has an elegant hierarchical form separating the lattice backbone and long-range couplings: $$\sum_{n=1}^{2^{k}}K_{0}x_{n-1}x_{n}+\sum_{i=1}^{k} \sum_{j=1}^{2^{k-i}}K_{i}x_{2^{i-1}\left(2j-1\right)}x_{2^{i-1}\left(2j+1\right)}.$$ (10) For the RG, we set $-\beta\mathcal{H}=-\beta\mathcal{H}_{0}-\beta\mathcal{V}+\mathcal{R}$ with $$-\beta\mathcal{H}_{0} = \sum_{j=1}^{2^{k-1}}K_{0}x_{2j-1}\left(x_{2j-2}+x_{2j}\right) +\sum_{j=1}^{2^{k-1}}Lx_{2j-2}x_{2j},$$ $$-\beta\mathcal{V} = \sum_{j=1}^{2^{k-1}}K_{1}x_{2j-1}x_{2j+1},$$ (11) added new couplings $L$ that emerge during RG, as in Fig. 4. Tracing out odd spins and relabeling all remaining even spin variables $x_{n}\to x'_{n/2}$ , the cumulant expansion applied to Eq. (11) yields a new Hamiltonian $-\beta\mathcal{H}'$ , formally identical to Eq. (10), with the rescaled couplings $$K'_{0} = L+\frac{1}{2}\ln\cosh\left(2K_{0}\right) +\frac{K_{1}}{2}\tanh^{2}\left(2K_{0}\right),$$ (12) $$K'_{1} = K_{2}+\frac{K_{1}}{4}\tanh^{2}\left(2K_{0}\right),\quad L'=\frac{K_{1}}{4}\tanh^{2}\left(2K_{0}\right).$$ and $K'_{i}=K_{i+1}$ for $(i\geq2)$ . These are the same relation one would obtain for the 1D-Ising model with nnn couplings, if $K_{1}\equiv L$ and $K_{i}=0$ for $i\geq2$ . In that case, one would find - correctly - that there are no non-trivial fixed points. The $K_{2}-$ term, which appears essentially as an arbitrary source again at every RG step, if chosen appropriately, provides the sole ingredient for a non-trivial outcome. But unlike for H3, here already uniform ($i$ -independent) $K_{i}$ obtain $T_{c}>0$ . Holding the source terms fixed, $K_{i\geq2}\equiv1$ , we find a single nontrivial fixed point at $K_{0}^{*}\approx0.2781$ , $K_{1}^{*}\approx1.0681$ , $L^{*}\approx0.0681$ . An analysis of the RG flow [Plischke and Bergersen(1994)] in Eqs. (13), starting with identical $K_{i}\equiv\beta J$ f. a. $i$ , yields $T_{c}\approx2.2545J$ . Simulations on H4 with uniform bonds for increasing system sizes $N=2^{k}$ accumulate to $T_{c}=2.1(1)J$ . Finally, we demonstrate the potential usefulness of having a regular (i.e., non-random) network at hand with fixed, predictable properties. Synchronization is a fundamental problem in natural and artificially coupled multicomponent systems [Strogatz(2001)]. Since the introduction of SW networks [Watts and Strogatz(1998)], it has been established that such networks can facilitate autonomous synchronization [Barahona and Pecora(2002)]. A particular synchronization problem emerges in the context of parallel discrete-event simulations (PDES) [Korniss et al.(2003)Korniss, Novotny, Guclu, Toroczkai, and Rikvold]. Nodes must frequently ``synchronize'' with their neighbors (on a given network) to ensure causality in the underlying simulated dynamics. The local synchronizations, however, can introduce correlations in the resulting synchronization landscape, leading to strongly nonuniform progress at the individual processing nodes. The above is a prototypical example for synchronization in many systems such as causally constrained queuing networks, supply-chain networks based on electronic transactions [Nagurney et al.(2005)Nagurney, Cruz, Dong, and Zhang], etc. Consider an arbitrary network in which the nodes interact through the links. The nodes are assumed to be task processing units, such as computers or manufacturing devices. Each node has completed an amount of task $h_{i}$ and these together at all nodes constitute the task-completion (synchronization) landscape $\{ h_{i}(t)\}_{i=1}^{N}$ . Here $t$ is the discrete number of parallel steps executed by all nodes, which is proportional to the real time, and $N$ is the number of nodes. In this particular model the nodes whose local field variables are incremented by an exponentially distributed random amount at a given step are those whose completed task amount is not greater than the tasks at their neighbors. Thus, denoting the neighborhood of the node $i$ by $S_{i}$ , if $h_{i}(t)\leq\min_{j\in S_{i}}\{ h_{j}(t)\}$ , the node $i$ completes some additional exponentially distributed random amount of task; otherwise, it idles. In its simplest form the evolution equation for the amount of task completed at the node $i$ can be written as $$h_{i}(t+1)=h_{i}(t)+\eta_{i}(t)\prod_{j\in S_{i}}\Theta \left(h_{j}(t)-h_{i}(t)\right),$$ (13) where $h_{i}(t)$ is the local field variable (amount of task completed) at node $i$ at time $t$ ; $\eta_{i}(t)$ are iid random variables with unit mean, delta correlated in space and time (the new task amount); and $\Theta(...)$ is the Heaviside step function. Despite its simplicity, this rule preserves unaltered the asynchronous dynamics of the underlying system. The average steady-state spread or width of the synchronization landscape (average degree of de-synchronization) can be written as [Korniss et al.(2003)Korniss, Novotny, Guclu, Toroczkai, and Rikvold] $$\left<w^{2}\right>\equiv\left<\frac{1}{N}\sum_{i=1}^{N}(h_{i}-\bar{h})^{2}\right>.$$ (14) In low dimensional regular lattices the synchronization landscape Eq. 13 belongs to the Kardar-Parisi-Zhang [Kardar et al.(1986)Kardar, Parisi, and Zhang] universality class, a rough desynchronized state dominated by large-amplitude long-wavelength fluctuations, where $\left <w^{2}\right >$ diverges with $N$ . This divergence hinders the synchronization in low-dimensional regular task-completion networks. Ref. [Korniss et al.(2003)Korniss, Novotny, Guclu, Toroczkai, and Rikvold] showed that $\left <w^{2}\right >$ becomes finite on a SW model similar to Watts-Strogatz [Watts and Strogatz(1998)] construction but with fixed nearest-neighbor links and one additional random link per node, also known as Newman-Watts model [Newman and Watts(1999)]. In Fig. 5, we show the width as a function of system size for 1D loop, Newman-Watts SW and our hierarchical regular SW networks. As it can be seen from the figure the width for H4 behaves very similar to SW, with at most a logarithmic divergence in $N$ , while it diverges with a power of $N$ for H3, but weaker than for a pure 1D system. In contrast to the random SW network, H4 provides very similar properties with the benefit of a regular and reproducible structure that is easy to manufacture, and that is potentially analytically tractable.   Figure 5: Plot of the width $\left <w^{2}\right >$ in Eq. (14) as a function of $N$ from $2^7$ to $2^{17}$ . The pure 1D-loop without long-range connections diverges most strongly, linear in $N$ (top graph), while the random long-range connections (one per node) in a Watts-Strogatz network SW keep its width finite (bottom graph). Between those are the results for the same loop with H3 (upper graph) and H4 (lower graph) connections. The width diverges with a weak power of $N$ for H3, but merely logarithmically for H4. In conclusion, we introduced a new set of hierarchical networks with regular, small world properties and demonstrated their usefulness, for theory and engineering applications, with all but a few examples. 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