Usage leading to an abrupt collapse of connectivity

Network infrastructures are essential for the distribution of resources such as electricity and water. Typical strategies to assess their resilience focus on the impact of a sequence of random or targeted failures of network nodes or links. Here we consider a more realistic scenario, where elements fail based on their usage. We propose a dynamic model of transport based on the Bak–Tang–Wiesenfeld sandpile model where links fail after they have transported more than an amount $\mu$ (threshold) of the resource and we investigate it on the square lattice. As we deal with a new model, we provide insight on its fundamental behavior and dependence on parameters. We observe that, for low values of the threshold due to a positive feedback of link failure, an avalanche develops that leads to an abrupt collapse of the lattice. By contrast, for high thresholds the lattice breaks down in an uncorrelated fashion. We determine the critical threshold ${\mu }^{*}$ separating these two regimes and show how it depends on the toppling threshold of the nodes and the mass increment added stepwise to the system. We find that the time of major disconnection is well described with a linear dependence on $\mu$. Furthermore, we propose a lower bound for ${\mu }^{*}$ by measuring the strength of the dynamics leading to abrupt collapses.

Figure 1

(top) Link failing time and (bottom) failing order for the two different regimes for a single sample. For $\mu <{\mu }^{*}(\mu =100)$ after the failure of a minor fraction of links, one devastating avalanche destroys the entire network at once (top left) with the damage propagating radially outwards (bottom left). By contrast, for $\mu >{\mu }^{*}(\mu ={10}^5)$ each avalanche only destroys a minor fraction of links and no long-range correlation in failing time (top right) and order (bottom right) can be observed. $t_{\min }$ and $t_{\max }$ denote the time of the first and the last failed link, respectively. Results were obtained on a square lattice of $L=512$, with dissipation rate $\varepsilon =0.01$.

Figure 2

Abrupt collapse for low failing thresholds. For $\mu <{\mu }^{*}(\mu =100)$ due to a single devastating avalanche, the fraction of nodes belonging to the largest connected component decreases from $S\approx 1$ to $S\approx 0$ discontinuously. The time $t_c$ when this avalanche occurs is approximately Gaussian distributed (inset) with a mean and standard deviation which decrease with $L$. For $L=\{512,1024,2048,4096\}$ the number of samples is $\{1000,250,125,25\}$. $\varepsilon =0.01$.

Figure 3

Avalanche damage distribution. Damage probability distributions of prior avalanches (symbols) with $d\ge 1$ and damage due to the devastating avalanche (dashed lines) for $\mu <{\mu }^{*}(\mu =100)$. Note the gap between the damage of prior avalanches and the devastating avalanche. For $L=\{512,1024,2048,4096\}$ the number of samples is $\{1000,250,125,25\}$. $\varepsilon =0.01$.

Figure 4

Gradual destruction for high failing thresholds. For $\mu >{\mu }^{*}(\mu ={10}^5)$ the fraction of nodes belonging to the largest cluster $S$ decreases continuously in time. $t_c$ is the time at which the biggest jump $\Delta S$ due to a single avalanche occurs and follows approximately a Gaussian (inset) with a mean and standard deviation which decrease with $L$. For $L=\{512,1024,2048,4096\}$ the number of samples is $\{1000,250,125,25\}$. $\varepsilon =0.01$.

Figure 5

S versus fraction of failed links. Fraction of nodes belonging to the largest cluster $S$ versus fraction of failed links $1-p$ for $\mu >{\mu }^{*}(\mu ={10}^5)$ with dissipation $\varepsilon =0.01$. Inset: Scaling of the effective critical thresholds $p_{c,\text{eff}}$ with $\nu =4/3$ leads to $p_c=0.476\pm 0.002$ in the thermodynamic limit. For $L=\{512,1024,2048,4096\}$ the number of samples is $\{1000,250,125,25\}$. $\varepsilon =0.01$.

Figure 6

Universality class of random percolation. Finite-size scaling of the second moment of the cluster size distribution $\chi$ and fraction of total nodes belonging to the largest cluster $S$ for $\mu >{\mu }^{*}(\mu ={10}^5)$. Our simulation data agree with the scaling behavior known from random percolation $\chi L^{-\frac{\gamma }{\nu }}\sim L^{\frac{1}{\nu }}|p-p_c|/p_c$ with $\chi \sim |p-p_c{|}^{-\gamma }$ and $S{L}^{\frac{\beta }{\nu }}\sim L^{\frac{1}{\nu }}|p-p_c|/p_c$ with $S\sim {(p-p_c)}^{\beta }$ for $p>p_c$, respectively, $S\sim {(p_c-p)}^{\beta -\gamma }$ for $p<p_c$. For $L=\{2048,4096\}$ the number of samples is $\{125,25\}$. $\varepsilon =0.01$.

Figure 7

Extrapolation to find the critical threshold. Extrapolation of different estimators of ${\mu }_{\mathrm{eff}}^{*}$ for $\varepsilon =0.01$ and $\eta =0.34\pm 0.01$. Linear regressions (lines) for $L=\{1024,2058,4096\}$ (solid symbols). Significant deviation is observed for small system sizes $L=\{128,256,512\}$ (open symbols). Error bars are below $0.6\%$. Inset: Values of ${\mu }^{*}$ for $\varepsilon =\{0.01,0.02\}$ and a guide to the eyes (solid line) with the upper limit of dissipation $\varepsilon =0.30\pm 0.01$ where one only observes a devastating avalanche for $\mu =0$. For $L=\{128,256,512,1024,2048,4096\}$, measurements were performed in steps of $\Delta \mu =10$ with the number of samples being $\{3\times {10}^4,3\times {10}^4,7500,8000,1200,500\}$.

Figure 8

Devastating avalanche in the worst case scenario. First three steps of the devastating avalanche when initially all nodes have mass $z_i=z_c^{\prime }<z_c$, where $z_c^{\prime }$ is arbitrarily close to $z_c$, and links fail after the first transport $\left(\mu =0\right)$. Every node that receives mass topples and every link used is removed.

Figure 9

Predicting a lower bound for the critical threshold. Our measure $\Delta$ for superlinear growth versus $\mu$ for $L=512$, averaged over ${10}^5$ samples. With a statistical test $(Z$ test) which assumes with the so-called null hypothesis $H_0$ that $\Delta$ is Gaussian distributed with mean ${\Delta }_0=0$ and standard deviation equal to the average standard error of the data ${\sigma }_0=0.0154$, one expects $\Delta \ge 0.0358$ (see dashed line) only with probability $0.01$. For $\mu \le 130$ we detect that there is significant superlinear growth in power ($\Delta =0.0658$ for $\mu =130$ and probability $<{10}^{-5}$ that $\Delta \ge 0.0658$ under $H_0$). Inset: Power ${\Psi }_l$ of devastating avalanche versus step $l$ for $\mu =50$ averaged over ${10}^5$ samples. Tangent to ${\Psi }_l$ at the first inflection point $l=87$ (dashed), fit in the region $74\le l\le 100$ (vertical dotted lines). Independently of $\mu$, we define $\Delta$ as the difference at $l=L/2$ between ${\Psi }_l$ and its tangent, where the tangent is incident to ${\Psi }_l$ at $l=87$. $\varepsilon =0.01$.

Figure 10

Time $t_c$ of the biggest jump $\Delta S$ due to a single avalanche. $t_c$ is well fit by a linear relation to $\mu$, except for very low $\mu$ (see the amplification in the inset at the upper left). Lower-right inset: Not detectable by eye, the residuals of linear fits of $t_c$ of a particular system size ($L=\{512,2048\}$ as examples) reveal that the function $t_c\left(\mu \right)$ is superlinear below the transition failing threshold ${\mu }_{\mathrm{eff}}^{*}$ and sublinear above. Vertical lines bound the estimated range of ${\mu }_{\mathrm{eff}}^{*}$ (compare Fig. 7) which includes the inflection point (linear extrapolations from the estimation range help to verify that by eye). $t_c$ was measured at $\mu =\{0,5,...,295,300,350,...,950,1000\}$ and for $L=\{128,256,512,1024,2048\}$; the number of samples is $\{1600,800,400,200,100\}$. $\varepsilon =0.01$.

Figure 11

Influence of the toppling threshold $z_c$. The critical failing threshold ${\mu }^{*}$ increases with $z_c$. Here we show one estimate of ${\mu }_{\mathrm{eff}}^{*}$ (2D wrapping, compare Fig. 7) versus $z_c$. To find ${\mu }_{\mathrm{eff}}^{*}$, 1000 samples were each measured in steps of $0.5\le \Delta \mu \le 50$ depending on $\mu$. Error bars are less than $2\%$. Lower-right inset: Finite size scaling of exponent $b$ of the power-law fit ${\mu }_{\mathrm{eff}}^{*}\sim z_c^b$ for $0.4\le z_c\le 6$. Error bars are less than the size of the symbols. With different estimations of ${\mu }_{\mathrm{eff}}^{*}$ we predict ${\mu }^{*}\sim {z_c}^{b_c}$ with $b_c=1.8\pm 0.1$, under the assumption $(b-b_c)\sim L^{-\zeta }$ and $0.1<\zeta <1$. In the plot $\zeta =0.5$. Upper-left inset: $t_c$ versus $z_c$ for $\mu =500$ and $L=\{128,1024\}$ (number of samples is $\{100,50\}$, error bars less than $1\%$). $t_c$ is nonsensitive to changes in $z_c$ for $z_c⪅5$, i.e., $z_c\ll \mu$. $\varepsilon =0.01$.

Figure 12

Dispersed failing thresholds can prevent an abrupt collapse. Behavior of $S$ versus time for Gaussian distributed failing thresholds with mean $\mu =100$ and standard deviations $\sigma =\{1,2,10\}$ for $L=4096$ (averaged over 25 samples). An abrupt collapse is still seen for $\sigma =1$ but not for $\sigma =\{2,10\}$. $\varepsilon =0.01$.