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

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Mathematical Preliminaries

_{ℓ}(x). It is customary to use f

_{X}(x) to denote the pdf of X when more than one random variable is involved. In this case, however, there is no risk of confusion, and thus, we replace X with the support scale, which is relevant in our analysis. The integral of the pdf, i.e., the cumulative distribution function (cdf), will be denoted by:

_{ℓ}(x) gives the probability that the system has not “failed” when the variable X has reached the level x. The survival function is equal to one for x = x

_{min}and tends to zero for x → ∞. A characteristic property of the survival function is that its integral equals the expectation of X, i.e.,

_{ℓ}(x)/dx = −f

_{ℓ}(x), which follows from (1) and integration by parts.

- In the case of fracture strength, R
_{ℓ}(x) is the probability that the system has not failed if the external load takes values that do not exceed x. - In the case of earthquake return times, the survival function is the probability that there are no successive earthquakes separated by a time interval less than or equal to x.
- In the case of annual rainfall maxima, R
_{ℓ}(x) gives the probability that the maximum rainfall over the support area for the duration of one year does not exceed the value x.

_{ℓ}(x) is the conditional probability that the system will fail at time x* > x within the infinitesimal time window x < x* ≤ x + δx, given that the system has not failed during the interval [0, x]. Let A denote the event that the system has not failed at x and B the event that the system fails at x < x* ≤ x + δx. Then, h

_{ℓ}(x) = Prob(B|A) = Prob(B ∩ A)/Prob(A), which leads to the following:

## 2. Weakest-Link Scaling Theory

_{N}consisting of N links with support ℓ

_{0}. The WLS premise is then expressed as follows:

_{N}in the latter with ℓ

_{0}, implies the following scaling property:

#### 2.1. Systems Composed of Interacting Links

_{i}

_{;}

_{ℓ}

_{0}(x) = Prob(Λ

_{i}) is the probability that the i-th link survives, the system’s survival probability under WLS conditions is given by:

_{2}|Λ

_{1}) may in general depend on the state of Λ

_{1}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:

_{1}> 0 is a scaling factor. Since ${R}_{{\ell}_{0}}(\xb7)$ is a decreasing function, λ

_{1}> 1 implies that the conditional survival probability of Λ

_{2}is reduced compared to the probability ${R}_{{\ell}_{0}}(x)$.

_{i}can be expressed as:

_{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.

_{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].

#### 2.2. Hazard Function for Links with Variable Parameters

_{min}, θ

_{max}], we can define an effective link hazard rate ${h}_{{\ell}_{0}}^{*}(x)$ by:

_{min}, θ

_{max}], such that:

#### 2.3. Observed Distributions for Non-ergodic Conditions

_{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.

_{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}= 10

^{6}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.,

^{−}

^{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 = 10

^{3}and 10

^{5}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.

## 3. Exponential and Non-Exponential Link Survival Functions

#### 3.1. Weibull Link Ansatz

_{s})

^{m}, where x

_{s}is the scale parameter and m is the shape parameter or Weibull modulus. More generally, the exponential dependence ${R}_{{\ell}_{n}}(x)={\mathrm{e}}^{-\varphi}{}^{(x)}$, implies the self-similarity relation:

_{c}, where x

_{c}is a threshold above which the algebraic dependence of ${F}_{{\ell}_{0}}(x)$ ceases. This argument leads to the Weibull expression; however, it does not clarify the tail behavior of the distribution function for $x\gg {x}_{c}$.

#### 3.2. Non-Exponential Link Ansatz

_{x→∞}ϕ(x) → ∞ and (iii) ϕ(x) is a monotonically increasing function of x. Condition (iii) is necessary to ensure that $\mathrm{d}{R}_{{\ell}_{0}}(x)/\mathrm{d}x\le 0$, because the latter is given by:

## 4. Finite-Size System with Gamma Link Distribution

#### 4.1. The Gamma Distribution

_{s}and m are respectively the scale and shape parameters, γ(z, m) is the lower incomplete gamma function defined by means of the integral:

_{z→∞}γ(z, m). The pdf of the gamma distribution is given by:

#### 4.2. The Modified Gamma Distribution

^{−}

^{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.,

_{γ}(x) is given by (24) and ${g}_{{\ell}_{N}}(x)$ is a modulation function given by:

#### 4.3. Motivation and Properties

_{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].

_{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).

## 5. Finite-Sized System with κ-Weibull Distribution

#### 5.1. The κ-exponential and κ-logarithm Functions

#### 5.2. The κ-Weibull Function

#### 5.3. Motivation and Properties

_{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.

_{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}.

## 6. Conclusions

**PACS classifications:**02.50.-r; 02.50.Ey; 02.60.Ed; 89.20.-a; 89.60.-k

## Author Contributions

## Conflicts of Interest

## References

- Sornette, D. Critical Phenomena in Natural Sciences; Springer: Berlin, Germany, 2004. [Google Scholar]
- Gumbel, E.J. Les valeurs extrêmes des distributions statistiques. Annales de l’Institut Henri Poincaré
**1935**, 5, 115–158. [Google Scholar] - Weibull, W. A statistical distribution function of wide applicability. J. Appl. Mech.
**1951**, 18, 293–297. [Google Scholar] - Eliazar, I.; Klafter, J. Randomized central limit theorems: A unified theory. Phys. Rev. E
**2010**, 82, 021122. [Google Scholar] - Barlow, R.E.; Proschan, F. Mathmatical Theory of Reliability; SIAM: Philadelphia, PA, USA, 1996. [Google Scholar]
- Hristopulos, D.T.; Uesaka, T. Structural disorder effects on the tensile strength distribution of heterogeneous brittle materials with emphasis on fiber networks. Phys. Rev. B
**2004**, 70, 064108. [Google Scholar] - Pang, S.D.; Bažant, Z.; Le, J.L. Statistics of strength of ceramics: Finite weakest-link model and necessity of zero threshold. Int. J. Fract.
**2008**, 154, 131–145. [Google Scholar] - Amaral, P.M.; Fernandes, J.C.; Rosa, L.G. Weibull statistical analysis of granite bending strength. Rock Mech. Rock Eng.
**2008**, 41, 917–928. [Google Scholar] - Hagiwara, Y. Probability of earthquake occurrence as obtained from a Weibull distribution analysis of crustal strain. Tectonophysics
**1974**, 23, 313–318. [Google Scholar] - Rikitake, T. Recurrence of great earthquakes at subduction zones. Tectonophysics
**1976**, 35, 335–362. [Google Scholar] - Rikitake, T. Assessment of earthquake hazard in the Tokyo area, Japan. Tectonophysics
**1991**, 199, 121–131. [Google Scholar] - Holliday, J.R.; Rundle, J.B.; Turcotte, D.L.; Klein, W.; Tiampo, K.F.; Donnellan, A. Space-Time Clustering and Correlations of Major Earthquakes. Phys. Rev. Lett.
**2006**, 97, 238501. [Google Scholar] - Abaimov, S.G.; Turcotte, D.L.; Rundle, J.B. Recurrence-time and frequency-slip statistics of slip events on the creeping section of the San Andreas fault in central California. Geophys. J. Int.
**2007**, 170, 1289–1299. [Google Scholar] - Abaimov, S.G.; Turcotte, D.; Shcherbakov, R.; Rundle, J.B.; Yakovlev, G.; Goltz, C.; Newman, W.I. Earthquakes: Recurrence and Interoccurrence Times. Pure Appl. Geophys.
**2008**, 165, 777–795. [Google Scholar] - Hristopulos, D.T.; Mouslopoulou, V. Strength statistics and the distribution of earthquake interevent times. Physica A
**2013**, 392, 485–496. [Google Scholar] - Conradsen, K.; Nielsen, L.; Prahm, L. Review of Weibull statistics for estimation of wind speed distributions. J. Clim. Appl. Meteorol.
**1984**, 23, 1173–1183. [Google Scholar] - Van den Brink, H.; Können, G. The statistical distribution of meteorological outliers. Geophys. Res. Lett.
**2008**, 35. [Google Scholar] [CrossRef] - Beck, C.; Cohen, E. Superstatistics. Physica A
**2003**, 322, 267–275. [Google Scholar] - Sornette, D.; Knopoff, L. The paradox of the expected time until the next earthquake. Bull. Seismol. Soc. Am.
**1997**, 87, 789–798. [Google Scholar] - Chakrabarti, B.K.; Benguigui, L.G. Statistical Physics of Fracture and Breakdown in Disordered Systems; Clarendon Press: Oxford, UK, 1997. [Google Scholar]
- Curtin, W.A. Size Scaling of Strength in Heterogeneous Materials. Phys. Rev. Lett.
**1998**, 80, 1445–1448. [Google Scholar] - Alava, M.J.; Phani, K.V.V.N.; Zapperi, S. Statistical models of fracture. Adv. Phys.
**2006**, 55, 349–476. [Google Scholar] - Alava, M.J.; Phani, K.V.V.N.; Zapperi, S. Size effects in statistical fracture. J. Phys. D
**2009**, 42, 214012. [Google Scholar] - Ditlevsen, O.D.; Madsen, H.O. Structural Reliability Methods; Wiley: Chichester, UK and New York, NY, USA, 1996. [Google Scholar]
- Mouslopoulou, V.; Hristopulos, D.T. Patterns of tectonic fault interactions captured through geostatistical analysis of microearthquakes. J. Geophys. Res. Solid Earth.
**2011**, 116. [Google Scholar] [CrossRef] - Cohen, E. Superstatistics. Physica D
**2004**, 193, 35–52. [Google Scholar] - Hristopulos, D.T.; Petrakis, M.; Kaniadakis, G. Finite-size Effects on Return Interval Distributions for Weakest-link-scaling Systems. Phys. Rev. E
**2014**, 89, 052142. [Google Scholar] - Bažant, Z.P.; Le, J.L.; Bažant, M.Z. Scaling of strength and lifetime probability distributions of quasi-brittle structures based on atomistic fracture mechanics. Proc. Natl. Acad. Sci. USA.
**2009**, 1061, 11484–11489. [Google Scholar] - Daniels, H.E. The statistical theory of the strength of bundles of threads. Proc. R. Soc. A
**1945**, 183, 405–435. [Google Scholar] - Smith, R.L.; Phoenix, S.L. Asymptotic distributions for the failure of fibrous materials under series-parallel structure and equal load sharing. J. Appl. Mech.
**1981**, 48, 75–82. [Google Scholar] - Kaniadakis, G. Statistical mechanics in the context of special relativity II. Phys. Rev. E
**2005**, 72, 036108. [Google Scholar] - Clementi, F.; Di Matteo, T.; Gallegati, M.; Kaniadakis, G. The κ-generalized distribution: A new descriptive model for the size distribution of incomes. Physica
**2008**, 387, 3201–3208. [Google Scholar] - Kaniadakis, G. Maximum Entropy Principle and power-law tailed distributions. Eur. Phys. J. B
**2009**, 70, 3–13. [Google Scholar] - Clementi, F.; Gallegati, M.; Kaniadakis, G. A κ-generalized statistical mechanics approach to income analysis, 2009; arXiv:0902.0075.
- Kaniadakis, G. Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions. Entropy
**2013**, 15, 3983–4010. [Google Scholar] - Bažant, Z.P.; Pang, S.D. Activation energy based extreme value statistics and size effect in brittle and quasi-brittle fracture. J. Mech. Phys. Solids.
**2007**, 55, 91–131. [Google Scholar] - Tsubo, Y.; Isomura, Y.; Fukai, T. Power-Law Inter-Spike Interval Distributions Infer a Conditional Maximization of Entropy in Cortical Neurons. PLoS Comput. Biol.
**2012**, 8, e1002461. [Google Scholar] - Manzato, C.; Shekhawat, A.; Nukala, P.K.V.V.; Alava, M.J.; Sethna, J.P.; Zapperi, S. Fracture Strength of Disordered Media: Universality, Interactions, and Tail Asymptotics. Phys. Rev. Lett.
**2012**, 108, 065504. [Google Scholar]

**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: ${R}_{{\ell}_{2}}(x,1)$.

**Figure 2.**Sample estimates $\widehat{\mathrm{\Phi}}(x;\ell )$ of data sets with sizes n = 10

^{3}(top) and n = 10

^{5}(bottom). The samples are randomly drawn from a total of n

_{total}= 10

^{6}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}.

**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).

**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).

**Figure 5.**Hazard rate ${h}_{{\ell}_{0}}(x)$ of the gamma probability distribution for m = 2.5 (a) and m = 0.7 (b).

**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.

**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.

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

m = 0.7 | m = 2.5 | |||||
---|---|---|---|---|---|---|

N | δ | ρ | ${\widehat{x}}_{s}$ | $\widehat{m}$ | ${\widehat{x}}_{s}$ | $\widehat{m}$ |

10^{3} | 1 | 10^{−4} | 1.39 | 0.688 | 1.42 | 2.14 |

10^{−1} | 1.55 | 0.694 | 1.54 | 2.25 | ||

100 | 10^{−4} | 45.6 | 0.592 | 51.8 | 1.23 | |

10^{−1} | 43.6 | 0.634 | 49.6 | 1.38 | ||

10^{5} | 1 | 10^{−4} | 1.40 | 0.689 | 1.42 | 2.19 |

10^{−1} | 1.50 | 0.691 | 1.52 | 2.23 | ||

100 | 10^{−4} | 57.0 | 0.679 | 59.3 | 1.94 | |

10^{−1} | 42.9 | 0.611 | 48.4 | 1.30 |

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hristopulos, D.T.; Petrakis, M.P.; Kaniadakis, G.
Weakest-Link Scaling and Extreme Events in Finite-Sized Systems. *Entropy* **2015**, *17*, 1103-1122.
https://doi.org/10.3390/e17031103

**AMA Style**

Hristopulos DT, Petrakis MP, Kaniadakis G.
Weakest-Link Scaling and Extreme Events in Finite-Sized Systems. *Entropy*. 2015; 17(3):1103-1122.
https://doi.org/10.3390/e17031103

**Chicago/Turabian Style**

Hristopulos, Dionissios T., Manolis P. Petrakis, and Giorgio Kaniadakis.
2015. "Weakest-Link Scaling and Extreme Events in Finite-Sized Systems" *Entropy* 17, no. 3: 1103-1122.
https://doi.org/10.3390/e17031103