Complex networks from space-filling bearings

Two-dimensional space-filling bearings are dense packings of disks that can rotate without slip. We consider the entire first family of bearings for loops of four disks and propose a hierarchical construction of their contact network. We provide analytic expressions for the clustering coefficient and degree distribution, revealing bipartite scale-free behavior with a tunable degree exponent depending on the bearing parameters. We also analyze their average shortest path and percolation properties.

Figure 1

Section of the $m,n=0$ bearing and the first generation of its contact network. All loops of touching disks consist of an even number of disks and the network is bipartite; i.e., one can label the nodes as $a$ and $b$ such that two touching nodes are always of a different type.

Figure 2

Hierarchical rule to construct the contact network for $m=n=0$. (a) One starts with a loop of four nodes (two $a$ and two $b$ nodes) in a loop arrangement. (b) A new $a$ node is added to the center of the loop and connected to the $b$ nodes on the top and bottom of the loop, dividing it into two loops. (c) Two new $b$ nodes are also added, one inside of each loop and connected to the two closest $a$ nodes. The initial loop is now divided into four equal loops. (d) The new generation is constructed in the same way inside each loop.

Figure 3

Size dependence of the average shortest path $l$, for the contact network of bearings of $m=n=0$, where $N$ is the number of nodes. As expected for small-world networks, the shortest path scales logarithmically with the number of nodes, over more than three orders of magnitude. By regression, we estimate $l=0.61lnN+0.65$.

Figure 4

Clustering coefficient $C_4$ for the contact network of bearings of $m=n=0$ as a function of the number of nodes, $N$. The values of $C_4$ were obtained by numerically evaluating the sum (19). The value of the clustering coefficient converges towards $\approx 0.8625$ in the thermodynamic limit.

Figure 5

Dependence of the percolation threshold $p_c$ on the network size $N$. We estimate the threshold $p_c$ as the value of $p$ at which the probability of spanning $P\left(p_c\right)$ is $1/2$. We applied the bisection method and simulated different values of $p^{*}$ until $\left|P\left(p^{*}\right)-1/2\right|<{10}^{-3}$. The black line corresponds to the least-squares fit to the data, $log{p}_c\sim -\frac{1}{2\nu }logN$, with exponent $\nu =7\pm 1$ and coefficient of determination $0.992$.

Figure 6

Examples of space-filling bearings of the first family, for different values of $m$ and $n$.

Figure 7

Left: A sketch of the initial network setup (blue, sites drawn as squares) and the first generation in the general case (red, sites as dots). Right: The first two generations and part of the second (green) for $m=1$ and $n=2$.

Figure 8

Left: Degree exponent $\gamma$ as a function of $m$ and $n$. Note that $1<\gamma \le 2$ and $\gamma =2$ for $m=n$. Right: Degree exponent $\gamma$ as a function of $n$ for $m=100$.

Figure 9

Left: Average shortest path $l$ as a function of the number of nodes, $N$, for $n=m$. Lines were shifted vertically by adding $n$ for better visualization. Right: Average shortest path $l$ as a function of the number of sites, $N$, for $m=0$. Lines were also shifted vertically. The lines are just guides to the eye.