Weakest-Link Scaling and Extreme Events in Finite-Sized Systems

1. Introduction

2. Weakest-Link Scaling Theory

2.1. Systems Composed of Interacting Links

Above, we assumed an identical survival function for all of the links, a condition that is plausible in a mean-field framework. Herein, we consider systems in which the link properties vary. This is an interesting possibility, which can lead to new probability models for reliability engineering. In general, if R_i;ℓ0(x) = Prob(Λ_i) is the probability that the i-th link survives, the system’s survival probability under WLS conditions is given by:

$$R_{{\ell }_N}(x)=\mathrm{Prob}\left({\mathrm{\Lambda }}_1\cap {\mathrm{\Lambda }}_2\dots \cap {\mathrm{\Lambda }}_N\right).$$

Using the law of joint probability it follows that:

$$R_{{\ell }_N}(x)=\mathrm{Prob}({\mathrm{\Lambda }}_1)\mathrm{Prob}({\mathrm{\Lambda }}_2|{\mathrm{\Lambda }}_1)\dots \mathrm{Prob}({\mathrm{\Lambda }}_{{\ell }_N}|{\mathrm{\Lambda }}_1,\dots ,{\mathrm{\Lambda }}_{N-}{}_1).$$

The conditional probability Prob(Λ₂|Λ₁) may in general depend on the state of Λ₁ and the degree of interaction between the two links. This agrees with the observation of complex interaction patterns in geological fault systems, if the links are identified with single faults or fault groups [25]. One possibility for expressing this dependence is by means of the following ansatz:

$$\mathrm{Prob}\left({\mathrm{\Lambda }}_2|{\mathrm{\Lambda }}_1\right)=R_{{\ell }_0}({\lambda }_{1}x),$$

where λ₁ > 0 is a scaling factor. Since $R_{{\ell }_0}(·)$ is a decreasing function, λ₁ > 1 implies that the conditional survival probability of Λ₂ is reduced compared to the probability $R_{{\ell }_0}(x)$.

More generally, for an N-link system, the survival function of the link Λ_i can be expressed as:

$$\mathrm{Prob}\left({\mathrm{\Lambda }}_{{\ell }_i}|{\mathrm{\Lambda }}_1,\dots ,{\mathrm{\Lambda }}_{i-}{}_1\right)=R_{{\ell }_0}\left({\lambda }_{i-}{}_{1}x\right),i=2,\dots N.$$

In light of the above, the system survival function is given by the following equation:

$$R_{{\ell }_N}(x,\overrightarrow{\lambda })=R_{{\ell }_0}(x){\displaystyle \prod _{i=2}^N{R}_{{\ell }_0}({\lambda }_{i-1}x)},$$

(13)

which is determined by the coefficient vector $\overrightarrow{\lambda }={\left({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_{N-1}\right)}^T$. The above system survival function is given by a product rule in agreement with the survival function (4) of the homogeneous-link system. In addition, all terms of the product involve the same function $R_{{\ell }_0}(·)$, albeit the argument of the latter is modified by the respective coefficients λ_i, i = 1, …, N − 1. Hence, (13) maintains the product rule and the same functional form of the link survival function, which are central features of WLS, but it also captures interactions between the links via the coefficients $\overrightarrow{\lambda }$. We illustrate the behavior of the system survival function for a fictitious two-link system in Figure 1, where $R_{{\ell }_2}(x,\lambda )$ is compared with $R_{{\ell }_2}(x,1)={[R_{{\ell }_0}(x)]}^2$: the former is the survival function of the correlated system, whereas the latter is the standard WLS survival function. For the Weibull survival function, λ effectively renormalizes the scale parameter x_s to $x_s^{\prime }={[2/(1+{\lambda }^m)]}^{1/m}{x}_s$. Hence, the ansatz (13) essentially accounts for the correlations by renormalizing a parameter of the “free distribution”, which is an idea with deep roots in statistical physics. Figure 1 demonstrates that values λ > 1 (λ < 1) lead to lower (higher) survival probability for the same x than the uniform-link WLS function.

Figure 1. Survival function $R_{{\ell }_2}(x,\lambda )$ for a system comprising two links that follow the Weibull survival function $R_{{\ell }_0}(x)$ with x_s = 1 and m = 2.5. Continuous surface: $R_{{\ell }_2}(x,\lambda )$ for different values of λ. Mesh: $R_{{\ell }_2}(x,1)$.

The proposed system survival function (13) is quite general, since it allows for different functional forms of the link survival function. In (13), it is possible to consider correlated random variables λ_i as coefficients. In this case, the random variable X becomes doubly stochastic and is described by superstatistics [18,26]. We expect that the variability of link parameters is a plausible scenario in heterogeneous physical or engineered systems. For example, a theoretical framework has been proposed that links the statistics of earthquake recurrence times to fracture strength in the Earth’s crust [15,27]. Since the crust is a heterogeneous material decorated with flaws (faults) of widely varying sizes and orientations, it is reasonable to expect that the fracture strength parameters exhibit local random fluctuations in addition to systematic changes with depth [25].

2.3. Observed Distributions for Non-ergodic Conditions

In the case of natural phenomena, however, the available sample is often finite, and the ergodic condition may not be justifiable. For example, an earthquake catalog for a particular area covers only a small portion of the area’s geological history. This means that the full ensemble, involving both the distribution of the scale parameters and the distribution of x values for fixed parameter values, is not adequately sampled during the observation process. If g(θ) has a narrow mode (e.g., Weibull with m > 1), the estimated pdf ĝ(θ) will have a peak in the interval [θ_min, θ_max] near the mode of g(θ). For disperse parameter distributions (e.g., Weibull with m ≤ 1), ĝ(θ) may exhibit more complicated behavior due to insufficient sampling.

In order to illustrate the effect of the sample size and the link parameter’s variability on the observed link probability distribution, we conduct a numerical experiment. We simulate a random variable X that represents individual links, assuming that X follows the Weibull distribution $f_{{\ell }_0}(x;m,x_s)$ with a fixed shape parameter m and uniformly distributed scale parameter x_s. The distribution of x_s is g(x_s) = (x_s;max − x_s;min)⁻¹ if x_s ∈ [x_s;min, x_s;max] and g(x_s) = 0 otherwise. We generate a large set containing n_total = 10⁶ random numbers drawn from Weibull distributions with different values of x_s. The latter correspond to different links drawn from the uniform distribution g(x_s). We define the mixing ratio as the number of sampled links over the number of samples per link, i.e.,

$$\rho =\frac{N}{n_s},N=\#\text{number of links},n_s=\#\text{samples per link}$$

The ratio ρ determines the mixing of random values from distributions with different scale parameters. Small values of ρ denote that a small number of links are sampled compared to the samples taken from each link, whereas high values of ρ denote the opposite.

In order to investigate different mixing scenarios, we consider two values of ρ equal to 0.1 and 10⁻⁴. The lower limit x_s;min of the scale parameter distribution is set to one, whereas the width δ = x_s;max − x_s;min of the uniform distribution is set to δ = 1 and 100. From the initial set of n_total random numbers, we randomly select n = 10³ and 10⁵ samples, for each of the four scenarios obtained by the different combinations of ρ and δ. The different values of n allow investigating the sample size effect. For each sample, we calculate the maximum likelihood estimates of the optimal-fit Weibull parameters, which are listed in Table 1. For m = 0.7, the sample-based estimates of the Weibull modulus are close to the true value, especially in the case of the narrower distribution of the scale parameter. For m = 2.5, the estimates of the Weibull modulus are close to the true value only for the narrow scale parameter distribution.

Table 1. Estimates of Weibull distribution parameters for the simulated data of Figure 2.

We also show the estimated $\widehat{\mathrm{\Phi }}(x;\ell )$ derived from the empirical (sample-based) cumulative distribution function in Figure 2. For reference purposes, if the data follow the Weibull distribution, $\widehat{\mathrm{\Phi }}(x;\ell )$ is a straight line with slope $\widehat{m}$ and intercept $\widehat{m}\ln {\widehat{x}}_s$. As shown in Figure 2(c), for m < 1 and large n, the Weibull distribution is obtained. For smaller n (Figure 2(a)), deviations from the Weibull emerge. For m > 1 and small n, the data cannot be described with a Weibull distribution (Figure 2(b)). For a larger sample size (Figure 2(d)), $\widehat{\mathrm{\Phi }}(x;\ell )$ demonstrates Weibull tails with different slopes. Overall, for m = 0.7 (in general, for m < 1), the distribution of sampled values maintains, albeit approximately, the Weibull characteristic shape and the original modulus. For m > 1, the Weibull scaling is maintained only for a narrow scale parameter distribution.

Figure 2. Sample estimates $\widehat{\mathrm{\Phi }}(x;\ell )$ of data sets with sizes n = 10³ (top) and n = 10⁵ (bottom). The samples are randomly drawn from a total of n_total = 10⁶ simulated values generated from Weibull distributions with fixed m = 0.7 (left column) and m = 2.5 (right column). The x_s are uniformly distributed in the interval [1, 1 + δ] with δ = 1, 100. The mixing ratio that determines the number of x_s values (links) over the number of random values per link is ρ = 0.1, 10⁻⁴.

3. Exponential and Non-Exponential Link Survival Functions

4. Finite-Size System with Gamma Link Distribution

4.2. The Modified Gamma Distribution

We use the WLS scaling relation (5) and a gamma link probability function to define the modified gamma distribution, which applies to the entire system as follows:

$$F_{{\ell }_N}(x)=1-{[1-F_{\gamma }(x)]}^N=1-{[R_{{\ell }_0}(x)]}^N.$$

(26)

The quantile function (inverse survival function) of the modified gamma distribution is:

$$T_{{\ell }_N}(r)=x_s{\gamma }^{-1}[\mathrm{\Gamma }(m)(1-r^{1/N}),m],$$

(27)

where γ⁻¹(·) is the inverse of the lower incomplete gamma function. The quantile function of the modified gamma distribution is plotted in Figure 3. The median of the modified gamma distribution is $x_{\mathrm{med}}=T_{{\ell }_N}(1/2,m)$, i.e.,

$$x_{\mathrm{med}}=x_s{\gamma }^{-1}\left[\mathrm{\Gamma }(m)\left(1-\frac{1}{\sqrt[N]{2}}\right),m\right].$$

(28)

Figure 3. Quantile function for the modified gamma distribution (27) with m = 2.5 (a) and m = 0.7 (b) for different link sizes N (shown with different line types).

Based on (5) and assuming uniform link parameters, Equation (9) leads to the following modified gamma pdf for the N-link system:

$$f_{{\ell }_N}(x)=N\frac{e^{-x/x_s}}{\mathrm{\Gamma }(m)x_s}{\left(\frac{x}{x_s}\right)}^{m-1}{\left[1-\frac{\gamma (x/x_s,m)}{\mathrm{\Gamma }(m)}\right]}^{N-1}.$$

(29)

Equation (29) can also be expressed as:

$$f_{{\ell }_N}(x)=f_{\gamma }(x)g_{{\ell }_N}(x),$$

where f_γ(x) is given by (24) and $g_{{\ell }_N}(x)$ is a modulation function given by:

$$g_{{\ell }_N}(x)=N{[1-F_{\gamma }(x)]}^{N-1}=N{[R_{{\ell }_0}(x)]}^{N-1}.$$

(30)

The hazard rate of the modified gamma distribution is given according to (8) by:

$$h_{{\ell }_N}(x)=\frac{f_{{\ell }_N}(x)}{R_{{\ell }_N}(x)}=N{h}_{{\ell }_0}(x),$$

(31)

4.3. Motivation and Properties

The gamma distribution is used in reliability engineering to model extreme events, such as the lifetime of devices, the intensity of rainfall and earthquake return times [1]. The gamma distribution is also known in statistical physics as the stretched exponential distribution for m < 1, whereas it defaults to the exponential distribution for m = 1.

The modified gamma survival function is constructed assuming an WLS system that comprises N links that have an identical link failure probability that follows the gamma cdf. The modulation function $g_{{\ell }_N}(x)$ given by (30) takes values between N (at x = 0) and zero (at x → ∞). For m > 1 and increasing N, $g_{{\ell }_N}(x)$ shifts the mode of the pdf to lower values and reduces the width of the peak, as shown in Figure 4(a). Increasing N also suppresses the right tail of the pdf. The gamma distribution is obtained for N = 1, as it is easily verified from (29). For m < 1, as N increases, the $g_{{\ell }_N}(x)$ increases the values of $f_{{\ell }_N}(x)$ as x → 0 and reduces the right tail, as shown in Figure 4(b). For m < 1, $f_{{\ell }_N}(x)\to \infty$ as x → 0.

Figure 4. Modified gamma distribution pdf $f_{{\ell }_N}(x)$ with m = 2.5 (a) and m = 0.7 (b) for different link sizes N (shown with different line types).

The gamma hazard rate is plotted in Figure 5; it is an increasing function for x < x_s, whereas at $x\gg 1$, it is dominated by the exponential dependence, which leads to an asymptotically constant hazard function. In contrast, the Weibull distribution for m > 1 has a hazard rate that increases with x [13,19].

Figure 5. Hazard rate $h_{{\ell }_0}(x)$ of the gamma probability distribution for m = 2.5 (a) and m = 0.7 (b).

In Figure 6 we plot the cdf and Φ(x; ℓ_N) of the modified gamma distribution for m = 0.7, m = 2.5 and for different values of N. We compare these functions to the respective functions for the Weibull distribution. For m > 1, the Φ(x; ℓ_N) curves are straight lines on the Weibull plot for ln(x/x_s) ≤ a_c, where a_c ≈ −1, but become concave as ln(x/x_s) increases further. This means that the right tail of the modified gamma distribution is heavier than the respective Weibull tail. In contrast, for m < 1, the Φ(x; ℓ_N) curve becomes convex upwards for ln(x/x_s) > a_c, and the right tail of the modified gamma distribution is lighter than the respective Weibull tail. The Φ(x; ℓ_N) curves of the modified gamma distribution for different N with fixed m collapse onto a single curve by parallel shifting, in agreement with the scaling relation (12).

Figure 6. Cumulative distribution function $F_{{\ell }_N}(x)$ (left) and associated Φ(x; ℓ_N) (right) for the modified gamma distribution with m = 2.5 (top) and m = 0.7 (bottom) and different values of N (shown by different line types), as well as the respective Weibull functions with the corresponding Weibull modulus.

5. Finite-Sized System with κ-Weibull Distribution

5.3. Motivation and Properties

The κ-Weibull is motivated by the non-exponential link ansatz discussed in Section 3.2. In particular, we assume that ϕ(x) is the following monotonically increasing function with link-size dependence:

$$\varphi (x)=\frac{1}{N}{(x/x_s)}^m.$$

Then, the link survival function is given according to (20) by the function:

$$R_{{\ell }_0}(x)=\sqrt{1+\frac{1}{N^2}{\left(\frac{x}{x_s}\right)}^{2m}}-\frac{1}{N}{\left(\frac{x}{x_s}\right)}^m.$$

The κ-Weibull system survival function (35) is then obtained from the above link survival function and the WLS principle (5).

Based on the properties of the κ-exponential, the survival function (35) corresponds to an extension of the Weibull survival function. The term κ-Weibull is due to the presence of the κ-exponential. This distribution has been used in economics to model the distribution of income [32,34] and in statistical physics as a model of earthquake recurrence times [27]. Other potential applications include the fracture strength of quasi-brittle materials, which exhibit deviations from Weibull scaling [7,28], and reliability models that involve the Pareto distribution.

The κ-exponential function can describe heavy-tailed processes, because for κ > 0, it decays asymptotically as a power-law function, i.e.,

$${\exp }_{\kappa }(z)\underset{z\to \pm \infty }{\sim }{|2\kappa z|}^{\pm 1/\kappa }.$$

(41)

The κ-Weibull pdf also exhibits a power-law tail, which is inherited by the κ-exponential and is relevant in applications where such heavy tails are observed. In addition, if we define the κ-Weibull plot by means of the function Φ′_κ(x) = ln ln_κ(1/R_κ(x)), it follows that Φ′_κ(x) = m ln (x/x_s). Hence, Φ′_κ(x) is independent of κ and regains the logarithmic scaling of the double logarithm of the inverse survival function. This means that the κ-Weibull plot, in which Φ′_κ(x) is plotted instead of Φ_κ(x), can be used to visually detect the κ-Weibull scaling.

Plots of the κ-Weibull pdf for x_s = 1 and two values of m (m < 1 and m > 1) are shown in Figure 7. The plots also include the Weibull pdf (κ = 0) for comparison. For both m, lower N lead to a heavier right tail. For m = 2.5 the mode of the pdf moves to the right as N increases (for m = 0.7 the mode is at zero independently of N, since the distribution is zero-modal for m ≤ 1) [34]. To the right of the mode, higher N correspond, at first, to higher pdf values. This is reversed at a crossover point beyond which the lower-N pdfs exhibit slower power-law decay, i.e., lim_x→∞ f_κ(x) ∝ x^−α, where α = 1 + m/κ [34]. The crossover point occurs at ≈ 1.5x_s for m = 2.5, whereas for m = 0.7 at ≈ 5 x_s.

Figure 7. Semi-logarithmic plots of κ-Weibull pdfs for x_s = 1, different values of N (shown by different line types), and Weibull modulus equal to (a) m = 2.5 (b) m = 0.7.

6. Conclusions

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