Failure tolerance of load-bearing hierarchical networks

We investigate the statistics and dynamics of failure in a two-dimensional load-bearing network with branching hierarchical structure, and its variants. The variants strengthen the original lattice by using connectivity strategies which add new sites to the maximal cluster in top-to-bottom or bottom-to-top versions. We study the load-bearing capacity and the failure tolerance of all versions, as well as that of the strongest realization of the original lattice, the V lattice. The average number of failures as a function of the test load shows power-law behavior with power $5/2$ for the V lattice, but sigmoidal behavior for all other versions. Thus the V lattice turns out to be the critical case of the load-bearing lattices. The distribution of failures is Gaussian for the original lattice, the V lattice, and the bottom-to-top strategy, but is non-Gaussian for the top-to-bottom one. The bottom-to-top strategy leads to stable and strong lattices, and can resist failure even when tested with weights which greatly exceed the capacity of its backbone. We also examine the behavior of asymmetric lattices and discover that the mean failure rates are minimized if the probability of connection $p$ is symmetric with respect to both neighbors. Our results can be of relevance in the context of realistic networks.

Figure 1

A network of $M=8$ layers with 8 sites per layer with connection probability $p=1/2$. The beaded line is the trunk of maximal cluster. The weight-bearing capacity of the trunk is $W_T=99$.

Figure 2

(a) A strategy-I lattice network with $p=0.5$ in which $l_1,l_2,l_3$ reconnections are made to increase the capacity of sites on trunk, and (b) a strategy-II network with $p=0.5$ where $l_1,l_2,l_3,l_4,l_5,l_6,l_7$ reconnections are made to the sites on the trunk.

Figure 3

The V-lattice network of $D=8$ layers with 8 sites per layer, which is the critical case of our original network. The beaded line is the trunk of maximal cluster with weight-bearing capacity $W_T=120$.

Figure 4

The distributions of capacities displaying power-law behavior are shown for the lattice size $L=3000$ ($p=0.5$) at depths (a) $D=12$ and (b) $D=2500$ with standard deviation $\sigma =0.2306$ and (c) $D=2900$ with $\sigma =0.2318$ for the original lattice, (d) $D=12$, (e) $D=2500$ with $\sigma =0.2306$ and (f) $D=2900$ with $\sigma =0.2283$ for strategy-I lattice, (g) $D=12$ and (h) $D=2900$ with $\sigma =0.2423$ for strategy-II lattice. The capacity distribution for the V lattice at (i) $D=2900$ is not a power law. The value of power-law exponent $\alpha$ is found to be $4/3$ for the original lattice and the $D<D_s$ case of strategy I, and $1.24$ in the strategy-II lattice and the $D>D_s$ case of strategy I, where $D_s=2715$.

Figure 5

The percentage of the number of the reconnections made in strategy I, averaged over $50$ network realizations, is plotted against lattice side $L$. The percentage of reconnections falls with lattice side and starts saturating.

Figure 6

A plot of the number of failures averaged over 1000 realizations for (a) original lattice with $p=0.5$, (b) strategy-I lattice with $p=0.5$, and (c) strategy-II lattice with $p=0.5$ networks against the increment in test weight $w_t$ for $1000L$. The graphs show sigmoidal behavior for both the original lattice and the strategy-I lattice, whereas strategy II shows cubic behavior $(p_1{r}^3+p_2{r}^2+p_{3}r+p_4)$ with $p_1=4.76\times {10}^{-24}$, $p_2=-2.87\times {10}^{-15}$, $p_3=9.74\times {10}^{-07}$, $p_4=641.8,$ and ${\chi }^2=15.42$ before saturation is achieved. In the corresponding insets of the original and strategy-I lattice networks, the presaturation parts of mean failures on log-log scale are seen to show a power law over a very short range with exponents of $1.95$ and $1.77,$ respectively.

Figure 7

The failure rate for the V lattice (a) saturates to maximum failure with a steep rise and its presaturation part on a log-log scale. (b) The failure rate scaled by lattice side $L$ is seen to follow a good power law with an exponent of $5/2$ and ${\chi }^2=0.1925$. In (c) again the failure rate of the V lattice for $L=1000$ is plotted against $w_t$, with failure rates for lattices perturbed from the V lattice by introducing connections with $p=0.15$, $p=0.35$, and $p=0.50$.

Figure 8

The plots of trunk capacities averaged over $200$ realizations against increment in lattice side $L$ can be fitted by a power law for (a) original lattice ($p=0.5$) with exponent $=2.5$, for (b) strategy I ($p=0.5$) with exponent $=2.56$, for (c) strategy II ($p=0.5$) with exponent $=3.07$, and for (d) V lattices with exponent $=2.97$.

Figure 9

Plots of mean failures averaged over $200$ realizations, corresponding to trunk weights against increment in lattice side $L$ for (a) original lattice ($p=0.5$) and (b) strategy-I lattice ($p=0.5$), which follow a power law with exponents of $1.24$ and $0.91,$ respectively. Mean failures display linearity for (c) strategy II ($p=0.5$) with slope $=$$0.93$ and (d) V lattice with slope $=$$0.86$.

Figure 10

The distribution $P(r)$ of the mean number of failures $r$ with the trunk weight taken as test weight corresponding to $1000$ realizations for $1000L$ for (a) the original lattice ($p=0.5$), (b) the strategy-I lattice ($p=0.5$), and (d) the V-lattice fitted to Gaussian, whereas (c) strategy II ($p=0.5$) does not conform to any standard distribution form.

Figure 11

The distribution $P(W_T)$ of the capacities $W_T$ corresponding to $1000$ realizations for $1000L$ for (a) the original lattice ($p=0.5$) and (b) the strategy-I lattice ($p=0.5$) fitted to Gaussian, whereas (c) strategy II ($p=0.5$) does not conform to any standard distribution form.

Figure 12

The ratio of trunk weight and saturation weight for (a) original and strategy-I lattices ($1000L$) plotted against connection probability $p$. This ratio in strategy II remains approximately constant at the value $0.91$.

Figure 13

Plots of trunk capacities $W_T$, averaged over 200 realizations, versus all allowed values of connection probability $p$ for the (a) original, (b) strategy-I, and (c) strategy-II lattice networks for $1000L$.

Figure 14

Plots of mean failures $r$ averaged over 200 realizations, with the trunk weights $W_T$ of the original lattice taken as test weight versus all allowed values of connection probability $p$ for the (a) original, (b) strategy-I, and (c) strategy-II lattice networks for $1000L$. The plot for strategy-I fits to sixth-degree Fourier series function $f(r)=a_0+{\sum }_{i=1}^6[a_i\cos (\mathit{irw})+b_i\sin (\mathit{irw})]$ with $w=6.49$, $a_0=103.95$, $a_1=53.32$, $b_1=5.92$, $a_2=-30.8$, $b_2=-6.52$, $a_3=8.19$, $b_3=2.26$, $a_4=5.62$, $b_4=2.48$, $a_5=-2.0$, $b_5=-1.0$, $a_6=1.84$, $b_6=1.49$, and ${\chi }^2=7.571$.

Figure 15

Plots of (a) difference in number of left and right connections $l-r$ in original lattice, and differences in number of left and right reconnections $|l_c-r_c|$ in (b) strategy I and (c) strategy II, averaged over $100$ realizations for lattice of side $L=1000$. For strategy I (b), the segments of the $|l_c-r_c|$ versus $p$ curve ranging from $p=0.05$ to 0.35 and from $p=0.65$ to 0.95 are fitted to a fifth-degree polynomial function $p(l_c-r_c)={\sum }_{i=1}^6{p}_i(l_c-r_c){}^{6-i}$, with $p_1=51.48$, $p_2=2112$, $p_3=-4365$, $p_4=3208$, $p_5=-1006$, $p_6=244.8$, and ${\chi }^2=37.16$. However, for strategy II (c), the segments of the $|l_c-r_c|$ versus $p$ curve from $p=0.05$ to 0.35 and from $p=0.65$ to 0.95 remain flat at $|l_c-r_c|=999$.

Figure 16

$|Z_r(\sigma )-1|$ plotted against $\sigma$ displays power-law behavior for original lattice and $D<D_s$ case of strategy I (blue squares) with cumulative exponent $=$$0.33$, ${\chi }^2=0.22508$; and for strategy-II and $D>D_s$ case of strategy-I lattices (red circles) with cumulative exponent $=$$0.3287$, ${\chi }^2=0.00041$, for $100L$. The $d=12$ with standard deviation $=$$12.677$ averaged over $50$ realizations has been incorporated into strategy-II numerical analysis.