Aging in transport processes on networks with stochastic cumulative damage

In this paper we explore the evolution of transport capacity on networks with stochastic incidence of damage and accumulation of faults in their connections. For each damaged configuration of the network, we analyze a Markovian random walker that hops over weighted links that quantify the capacity of transport of each connection. The weights of the links in the network evolve due to randomly occurring damage effects that reduce gradually the transport capacity of the structure. We introduce a global measure to determine the functionality of each configuration and how the system ages due to the accumulation of damage that cannot be repaired completely. Then, by assuming a minimum value of the functionality required for the system to be “alive,” we explore the statistics of the lifetimes for several realizations of this process in different types of networks. Finally, we analyze the characteristic longevity of such a system and its relation with the “complexity” of the network structure. One finding is that systems with greater complexity live longer. Our approach introduces a model of aging processes relating the reduction of functionality with the accumulation of “misrepairs” and the lifetime of a complex system.

Figure 1

Monte Carlo simulation of the reduction of functionality in networks with random faults in the lines. We generate a random hit (fault) in the lines at times $T=1,2,...$, and in each line we have a value $h_{ij}$ that depends on $T$. At the time $T$, the value $h_{ij}-1$ gives the number of random hits that the line $(i,j)$ has suffered (considering the initial values $h_{ij}=1$ at $T=0$). The probability to have a new fault in one of the lines is determined by Eq. (1). By using this algorithm, we implement Monte Carlo simulations to generate random faults in the network described at time $T=0$. The values in the color bar indicate $h_{ij}$ for all the lines, and the widths represent the values ${\mathrm{\Omega }}_{ij}\left(T\right)$ that describe their capacity of transport reduced with the increment of $h_{ij}$; in all the realizations we use the misrepair parameter $\alpha =2$ in Eq. (2). We depict the results of three independent realizations of this process, and we present the configurations of the system at times $T=0,25,50,100$ along with the global value $\mathcal{F}$ in Eq. (8) that determines the global functionality of the structure being $\mathcal{F}=1.0$ for the initial case without damage [in this initial configuration the value $\tau =12.1$ is calculated by using Eq. (4)]. The functionality $\mathcal{F}$ evolves with time and gradually is reduced with the faults in the lines.

Figure 2

Evolution of the functionality $\mathcal{F}\left(T\right)$ in connected networks with random faults in the lines. We implement the algorithm described in Eqs. (1) and (2) to simulate the reduction of the capacity of transport in different types of networks with $N=100$ nodes: (a) Barabási-Albert (BA) network, (b) Watts-Strogatz (WS) network with rewiring probability $p=0.1$, (c) Erdös-Rényi (ER) network at the percolation limit $p=log\left(N\right)/N$, and (d) a tree. We depict the numerical results of the functionality $\mathcal{F}\left(T\right)$ given by Eq. (8). We simulate 1000 realizations of this process for the misrepair parameters $\alpha =1, \alpha =2, \alpha =3, \alpha =4$ and show with wide dashed lines the results obtained for the average over realizations $\langle \mathcal{F}\left(t\right)\rangle$ for each $\alpha$. With different thin lines we depict the values of $\mathcal{F}\left(t\right)$ obtained in each realization.

Figure 3

Statistical analysis of the lifetime of networks under stochastic damage and misrepair. We consider stochastic faults in the lines, the misrepair parameter $\alpha =2$ in Eq. (2) and the threshold value ${\mathcal{F}}^{★}=0.2$ to define the limit of functionality in Eq. (9). (a) Lifetime $T^{★}$ of 1000 realizations of the process in several types of networks with $N=100$ nodes: Barabási-Albert (BA), Watts-Strogatz (WS) with rewiring probability $p=0.1$, Erdös-Rényi (ER), and a tree. (b) Frequencies $f\left(T^{★}\right)$ of the times $T^{★}$ obtained in ${10}^5$ realizations. In panel (c) we express the frequencies of the times $T^{★}$ in terms of their average over realizations $\langle T^{★}\rangle$.

Figure 4

Ensemble average $〈T^{★}〉$ of the times $T^{★}$ in connected Watts-Strogatz networks with rewiring probability $p$. The results for each network were calculated from ${10}^5$ realizations with misrepair parameter $\alpha =2$ and considering ${\mathcal{F}}^{★}=0.2$. The error bars were obtained from the standard deviation of the data ${\sigma }_{T^{★}}$.

Figure 5

Ensemble average $〈T^{★}〉$ as a function of $N$ for interacting cycles with (a) $J=1$ and (b) $J=2$. The results for each network were calculated from ${10}^4$ realizations with a misrepair parameter $\alpha =2$ and a functionality threshold ${\mathcal{F}}^{★}=0.2$. The error bars were obtained with the standard deviation of the data ${\sigma }_{T^{★}}$. Dashed lines represent the linear fit $〈T^{★}〉=a+bN$. For the ring with $J=1$, we obtain the values $a=10.069, b=1.557$ with a correlation coefficient $r=0.999960$, and we have $a=85.654, b=10.0324, r=0.999964$ for $J=2$.

Figure 6

Nonisomorphic connected graphs with $N=6$ nodes. We sort the structures considering the average of the times $T^{★}$ from ${10}^5$ realizations of the cumulative damage process with the misrepair parameter $\alpha =2$ and threshold functionality ${\mathcal{F}}^{★}=0.75$. The average times $〈T^{★}〉$ and the respective ${\sigma }_{T^{★}}$ for each configuration are presented in Fig. 7. The catalog of graphs was obtained from Ref. [29].

Figure 7

Average lifetime $\langle T^{★}\rangle$ of nonisomorphic connected networks with $N=6$ nodes. We analyze the times $T^{★}$ for the networks presented in Fig. 6 for ${10}^5$ Monte Carlo realizations of the cumulative damage process with $\alpha =2$ and ${\mathcal{F}}^{★}=0.75$. The error bars represent the values of the standard deviation ${\sigma }_{T^{★}}$ of the times $T^{★}$. The different colors codified in the color bar show the number of edges $\left|\mathcal{E}\right|$ in each network.