# Transition from localized to mean field behaviour of cascading failures in
the fiber bundle model on complex networks

Attia Batool  Gergő Pál  Zsuzsa Danku  Ferenc Kun

###### Abstract

We study the failure process of fiber bundles on complex networks focusing on
the effect of the degree of disorder of fibers’ strength on the transition
from localized to mean field behaviour. Starting from a regular square lattice
we apply the Watts-Strogatz rewiring technique to introduce long range random
connections in the load transmission network and analyze how the ultimate
strength of the bundle and the statistics of the size of failure cascades
change when the rewiring probability is gradually increased. Our calculations
revealed that the degree of strength disorder of nodes of the network has a
substantial effect on the localized to mean field transition. In particular,
we show that the transition sets on at a finite value of the rewiring
probability, which shifts to higher values as the degree of disorder is
reduced. The transition is limited to a well defined range of disorder, so
that there exists a threshold disorder of nodes’ strength below which the
randomization of the network structure does not provide any improvement
neither of the overall load bearing capacity nor of the cascade tolerance of
the system. At low strength disorder the fully random network is the most
stable one, while at high disorder best cascade tolerance is obtained at a
lower structural randomness. Based on the interplay of the network structure
and strength disorder we construct an analytical argument which provides a
reasonable description of the numerical findings.

###### keywords:

cascading failure , fiber bundle model , Watts-Strogatz rewiring , disorder of
failure strength

††journal: Chaos, Solitons, and Fractals

[unideb]organization=University of Debrecen, Faculty of Science and
Technology, Doctoral School of Physics, Department of Theoretical Physics,
addressline=P.O. Box 400, city=Debrecen, postcode=H-4002, country=Hungary
[atomki]organization=Institute of Nuclear Research (Atomki),
addressline=Poroszlay út 6/c, city=Debrecen, postcode=H-4026, country=Hungary

A transition from localized to mean field behaviour of the failure dynamics of
fiber bundles is revealed as the structure of the network of load transmitting
connections is varied.

The transition is limited to a well defined range of disorder of the failure
strength of nodes (fibers) of the network.

A threshold disorder is determined below which the randomization of the
network structure does not provide any improvement neither of the load bearing
capacity nor of the cascade tolerance of the system.

An optimal network structure is identified with the highest stability against
cascading failure.

##  1 Introduction

Load redistribution following local damage often drives the cascading failure
of connected elements in complex systems [1, 2]. From crackling bursts
accompanying materials breakdown, through the failure avalanches of
transportation and communication networks, to the cascading blackouts of
electric transmission grids, the statistical features of failure cascades, and
the overall performance of the damaged system strongly depend on the structure
of the underlying network of load transmitting connections between elements of
the system [3, 4, 5, 6, 7, 8, 9, 10]. The interplay of the cascading dynamics
and of the network topology has recently been studied using discrete models on
various types of complex networks. In these approaches either the nodes [5, 8]
or the links [11] of the network are assumed to undergo a degradation process
accompanied by a mechanism of load rearrangement on the intact elements which
can give rise to cascades of failure events. Among these approaches the so-
called fiber bundle model (FBM), widely used to study materials breakdown
phenomena [12, 13, 14], has proven very useful since it grasps the essential
mechanisms of the intermittent failure spreading yet being simple enough to
offer analytic solutions in certain limiting cases [15, 16, 12].

In the basic setup, an FBM is composed of a set of parallel fibers organized
on a regular lattice [13, 17]. Under a slowly increasing external load the
fibers fail irreversibly when the local load on them exceeds their strength
value, which is assumed to have a certain degree of randomness. Under the
constraint of load conservation, the load dropped by the failed fiber gets
redistributed over the remaining intact ones. Recently, two limiting cases of
load sharing have been subject to intensive investigations both with a high
practical relevance: in case of equal load sharing (ELS) all intact fibers
receive the same fraction of load irrespective of their distance from the
failed one, while for localized load sharing (LLS) only the intact nearest
neighbors share equally the load of the broken element [18, 19, 16, 12, 20].
In both cases the load increments can cause further breakings so that a single
broken fiber may trigger an entire cascade of failure events. Due to the
generality of this failure spreading mechanism, fibers of the model can easily
be replaced by roads carrying traffic [21, 22], flow channels [23], or
electric power stations [24, 2, 25, 26, 10] on a high voltage transmission
grid, making FBMs a basic modelling framework for cascading failure with
widespread applications on complex networks [14].

During the past decades it has been shown that for a broad class of the
distributions of fibers’ strength, FBMs exhibit universal behaviour with two
distinct universality classes according to the range of load redistribution:
for long range load sharing (ELS class) the size of failure cascades proved to
be power law distributed with a universal exponent $5/2$ [18, 16, 12] and the
bundle has a finite asymptotic strength in the limit of large system sizes
[27, 28, 13]. For equal load sharing conditions the fibers always keep the
same load, no stress concentration can arise, hence, ELS realizes the mean
field limit of FBMs. Under short range load sharing (LLS class) the
distribution of cascade sizes is a significantly steeper (non universal) power
law or exponential, and additionally, for large system sizes the ultimate
strength of the bundle tends to zero [29, 30, 31, 32]. Recently, LLS FBMS have
been analyzed on complex networks where fibers were assigned to the nodes and
localized load sharing was realized along the links of the network [33]. Based
on the statistics of cascade sizes and on the ultimate strength of the system,
it was demonstrated for scale-free, Erdős-Rényi (ER) and Watts-Strogatz (WS)
rewired networks that LLS FBMs on complex networks fall in the ELS
universality class [33]. Later on it was shown on a ring graph with two
nearest neighbor links that adding a single random load transmitting
connection to each fiber, the localized load sharing FBM exhibits ELS
behaviour in terms of the size distribution of failure cascades and global
strength [34]. However, all these studies of FBMs on complex networks were
limited to a high disorder of node strength and considered only networks with
a high degree of randomness in their structure even in the case when all
fibers had the same degree.

Here we present a detailed numerical and analytical study of the transition of
the failure process of the fiber bundle model from the LLS to the ELS
universality class when an initially regular lattice of load transmitting
connections is gradually randomized. Starting from a square lattice we apply
the Watts-Strogats rewiring technique [35, 36] to introduce long range random
connections and study how the critical load and strain of the bundle,
furthermore, the statistics of the size of failure cascades change when the
rewiring probability is gradually increased at different degrees of disorder
of the strength of nodes (fibers). Our calculations revealed that the degree
of strength disorder of nodes of the network has a substantial effect on the
transition. In particular, we show that the LLS-ELS transition sets on at a
finite value of the rewiring probability, which shifts to higher values as the
degree of disorder is reduced. The transition is limited to a well defined
range of disorder, i.e. there exists a threshold disorder of nodes’ strength
below which the randomization of the network structure does not provide any
improvement neither of the overall load bearing capacity nor of the cascade
tolerance of the system. Based on the interplay of the network structure and
strength disorder we construct an analytical argument which provides a
reasonable description of the numerical findings.

##  2 Fiber bundle model on a rewired square lattice

To study cascading failures we consider a bundle of parallel fibers which are
assigned to the nodes of a complex network. The bundle is subject to a slowly
increasing external mechanical load parallel to the fibers’ direction. To
connect the model to the mechanics of materials, we assume that the fibers are
linearly elastic up to a threshold load $\sigma_{th}$ where they break
irreversibly. The Young modulus $E$ of fibers has a fixed value $E=1$,
however, their local strength $\sigma_{th}$ is a random variable sampled from
a probability density function $p(\sigma_{th})$. When a fiber fails its load
has to be overtaken by the remaining intact fibers. We assume localized load
sharing (LLS), i.e. load is redistributed along the links of the underlying
load transmission networks. In the following details of the model construction
are presented:

Figure 1: Demonstration of the model construction. $(a)$ The network of load
transmitting connections is obtained by rewiring a two-dimensional regular
square lattice with periodic boundary connections in both directions. The
rewiring introduces long range randomized connections, which broadens the
degree distribution of the network while keeping the average degree fixed.
$(b)$ Fibers of the bundle are assigned to the nodes of the network oriented
perpendicular to the plane of the original lattice.

To generate the network of connections along which load is redistributed over
fibers, we start from a regular square lattice of side length $l$ with
$N=l^{2}$ fibers, and apply the Watts-Strogatz rewiring technique to randomize
the connections [35, 36]. The fibers are assigned to the nodes oriented
perpendicular to the plane of the lattice. Figure 1 provides an overview of
the model construction. On the square lattice with periodic boundary condition
in both directions, all fibers (nodes) are connected to their four nearest
neighbors, hence, initially the degree distribution $\rho(k)$ of fibers has
the simple form

$\displaystyle\rho(k)=\left\\{\begin{array}[]{lcl}1&\mbox{for}&k=4,\\\\[5.69054pt]
0&\mbox{otherwise.}&\end{array}\right.$ (3)

As to the next, each of the $L=2N$ initially existing connections is rewired
with a probability $p$ which spans the interval $0\leq p\leq 1$. For both ends
of a rewired link a new fiber is selected randomly in the bundle with the
constraint that neither multiple links nor loops are allowed between fibers
(see Fig. 1 for illustration). As a consequence, long range randomized
connections are introduced in the bundle and the degree distributions
$\rho(k)$ broadens while the average degree of nodes $\left<k\right>$, i.e.
the average number of interacting partners of fibers, remains the same
$\left<k\right>=4$. The degree distribution $\rho(k)$ of the network is
presented in Fig. 2 for several values of the rewiring probability $p$. For
large values of $p$ in the vicinity of 1, isolated fibers and small clusters
of a few fibers may occur due to rewiring. In order to exclude their effect,
after the rewiring process we identify all clusters of nodes of the bundle and
keep the largest one for further calculations.

Figure 2: Degree distribution $\rho(k)$ of the network of fibers’ connections
at several rewiring probabilities $p$. As $p$ increases the distribution
$\rho(k)$ gets broader, however, the value of the average number of
connections is preserved $\left<k\right>=4$.

The load bearing capacity of nodes, i.e. the threshold load $\sigma_{th}$
where fibers fail, is a random variable which is sampled from a Weibull
distribution

$p(\sigma_{th})=m\frac{\sigma_{th}^{m-1}}{\lambda^{m}}e^{-\left(\sigma_{th}/\lambda\right)^{m}}$
(4)

over the interval $0\leq\sigma_{th}<+\infty$. Here the parameter $\lambda$
sets the scale of strength values, while $m$ controlls the shape of the
distribution. The choice of the distribution Eq. (4) has two motivations:
$(i)$ the failure behaviour of FBMs with such a fast decaying strength
distribution, shows a high degree of robustness which has been well understood
both in the equal load sharing and localized load sharing limits on regular
square lattices [37, 13]. $(ii)$ Varying the Weibull shape parameter $m$ in
the range $m\geq 1$, the degree of strength disorder can be controlled in the
sense that increasing $m$ reduces the width of the distribution making the
response of the bundle more brittle. This feature of the distribution is
illustrated in Fig. 3 for several $m$ values. It is an important
characteristics of our model that the strength $\sigma_{th}^{i}$ and the
degree $k_{i}$ of fibers (nodes) ($i=1,\ldots,N$) are uncorrelated.

Figure 3: Weibull distribution of failure thresholds $p(\sigma_{th})$ of
fibers (nodes) of the network at different values of the exponent $m$. As the
exponent $m$ increases the distribution gets narrower.

As the load gradually increases on the bundle, initially all fibers keep the
same load, hence, the weakest fiber with the lowest breaking threshold breaks
first. We assume that fibers have a nearest neighbor interaction so that the
load dropped by a broken fiber is equally shared by its intact nearest
neighbors on the underlying network. As a consequence, the updated load of the
neighboring fibers may exceed their local breaking threshold resulting in
additional breakings which are then followed again by load redistribution. As
a results of subsequent breaking and load redistribution steps a single fiber
failure can trigger an entire cascade of failures, which stops when all the
fibers receiving load in a load redistribution step, can sustain the elevated
load. This so called localized load sharing has the consequence that fibers
breaking in an avalanche form a connected cluster on the underlying network in
such a way that on the intact fibers along the cluster perimeter a large
amount of load can accumulate.

The system has two sources of disorder, i.e. the stochastic strength of fibers
and the randomness of the underlying network of connections, which are both
quenched. The interplay of the two gives rise to an inhomogeneous stress field
on the fibers, which evolves as the failure of the system proceeds. If fiber
$i$ of load $\sigma_{i}$ fails, then its $n_{i}$ intact nearest neighbors all
receive the load increment $\Delta\sigma_{i}=\sigma_{i}/n_{i}$, so that the
load $\sigma_{j}$ of a neighboring fiber $j$ will have the updated value

$\sigma_{j}\to\sigma_{j}+\Delta\sigma_{i}.$ (5)

It follows that $n_{i}\leq k_{i}$, where $k_{i}$ is the initial degree of the
node $i$. During an avalanche the external load is kept constant so that the
failure spreading is solely driven by the redistribution of load through the
transmission network.

After a cascade stops the external load is further increased to provoke the
breaking of a single element: the load $\sigma_{i}$ of each intact fiber is
incremented by the same amount $\delta\sigma$

$\sigma_{i}\to\sigma_{i}+\delta\sigma,$ (6)

where $\delta\sigma$ is determined as the smallest difference between the load
$\sigma_{i}$ and strength $\sigma_{th}^{i}$ of intact fibers
$\delta\sigma=\min\limits_{i}(\sigma_{th}^{i}-\sigma_{i})$. Ultimate failure
of the system occurs when a load increment triggers a catastrophic cascade
breaking all the intact fibers.

Simulations of the failure dynamics were performed starting from a square
lattice of size $l=400$ with $N=160.000$ fibers using periodic boundary
condition in both directions, which results in $L=320.000$ load transmitting
connections. To controll the degree of strength disorder, the scale parameter
of the Weibull distribution was fixed to $\lambda=1$, while the shape
parameter $m$ was varied in the range $1\leq m\leq 22$. For the rewiring
probability $p$ we considered 30 different values in the interval $0\leq p\leq
1$. At each parameter set averages were calculated over 2000 samples.

##  3 Macroscopic response of fiber bundles on complex networks

Figure 4: Constitutive curves $\sigma(\varepsilon)$ of the bundle at different
rewiring probabilities $p$ compared to the mean field solution of the model
Eq. (8) indicated by the dotted line for the Weibull parameter $m=1$. The two
axis are rescaled by the mean field critical load $\sigma_{c}^{ELS}$ and
strain $\varepsilon_{c}^{ELS}$. The inset demonstrates the entire
$\sigma(\varepsilon)$ curve of Eq. (8) including the definition of the
critical load $\sigma_{c}^{ELS}$ and strain $\varepsilon_{c}^{ELS}$.

The macroscopic response of the bundle can be characterized by determining the
relation $\sigma(\varepsilon)$ of its stress $\sigma$ and strain
$\varepsilon$. In the limit of equal load sharing, where all fibers interact
with each other and keep the same load, this constitutive relation can be
obtained analytically as

$\sigma=E\varepsilon\left[1-P(E\varepsilon)\right],$ (7)

where $P(x)$ denotes the cumulative distribution of failure thresholds [17,
13]. Since at a given strain $\varepsilon$ all the fibers keep the same load
$E\varepsilon$, the total load on the system is the product of the load of
single fibers and of the fraction of intact fibers $1-P(E\varepsilon)$.
Substituting the Weibull distribution of thresholds Eq. (4) we obtain the mean
field constitutive equation of our model

$\sigma(\varepsilon)=\displaystyle{E\varepsilon
e^{-\left(\frac{E\varepsilon}{\lambda}\right)^{m}}},$ (8)

which is presented by the inset of Fig. 4 for $m=1$. In a load controlled
experiment the constitutive curve can only be realized up to the maximum,
where immediate failure occurs in the form of a catastrophic cascade. Hence,
the value $\sigma_{c}$ and the position $\varepsilon_{c}$ of the maximum
define the critical load and strain of the bundle, respectively.

In computer simulations of finite bundles of localized load sharing, the
stress $\sigma$ and strain $\varepsilon$ of the system

Figure 5: Inset: Average critical strain $\left<\varepsilon_{c}\right>$ and
load $\left<\sigma_{c}\right>$ of the network scaled with their mean field
counterparts $\varepsilon_{c}^{ELS}$ and $\sigma_{c}^{ELS}$ as function of the
rewiring probability $p$ for the Weibull parameter $m=1$ of the failure
thresholds. Main panel: The strength values $\left<\varepsilon_{c}\right>$ and
$\left<\sigma_{c}\right>$ obtained for $p=1$ as function of the average number
of neighbors in the range $2\leq\left<k\right>\leq 30$. The legend is provided
in the inset for both figures.

can be obtained on any network topology by summing up the load $\sigma_{i}$ of
fibers (nodes) $F=\sum_{i=1}^{N}\sigma_{i}$ and dividing it by the bundle size
$\sigma=F/N$ and by the total number of intact fibers
$\varepsilon=F/N_{intact}$, respectively. It has been shown by means of
computer simulations that for localized load sharing on regular lattices the
constitutive curve of the bundle follows the mean field solution Eq. (7) [38,
19]. However, the response becomes more brittle in the sense that the
$\sigma(\varepsilon)$ curve stops earlier at lower $\sigma_{c}$ and
$\varepsilon_{c}$ closer to the initial linear regime. Figure 4 demonstrates
that this behaviour remains valid for all the networks considered, however, as
the rewiring probability $p$ increases, the constitutive curves reach to
higher $\sigma_{c}$ and $\varepsilon_{c}$ values.

To have a more transparent view on the effect of the network structure on the
strength of the bundle, we determined the average value of the critical load
$\left<\sigma_{c}\right>$ and strain $\left<\varepsilon_{c}\right>$ as
function of the rewiring probability $p$. It can be observed in the inset of
Fig. 5 for the Weibull parameter $m=1$ that for small values of $p$ the
randomized contacts hardly have any effect on the strength of the bundle so
that both the critical load $\left<\sigma_{c}\right>$ and strain
$\left<\varepsilon_{c}\right>$ retain their original values characteristic for
the square lattice at $p=0$. However, when the rewiring probability exceeds a
threshold value $0.01\lesssim p$ the strength starts to increase and saturates
to a limit value for completely randomized networks $p\to 1$. Note that in the
figure the strength values are scaled with their mean field (equal load
sharing) counterparts

$\displaystyle\varepsilon_{c}^{ELS}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{E}\left(\frac{1}{m}\right)^{1/m},$ (9)
$\displaystyle\sigma_{c}^{ELS}$ $\displaystyle=$
$\displaystyle\lambda\left(\frac{1}{m}\right)^{1/m}e^{-1/m},$ (10)

obtained as the position and value of the maximum of the $\sigma(\varepsilon)$
curve of Eq. (8). This comparison shows that as random connections start to
dominate the load transmission among fibers, the strength of the bundle
approaches the equal load sharing limit but saturates at a lower value. The
reason is that the average number of interacting partners of fibers is fixed
to $\left<k\right>=4$ which still gives rise to a significant stress
concentration on the network, and hence, reduces the fracture strength of the
bundle compared to the mean field limit. To support this argument we performed
computer simulations on random graphs corresponding to the $p=1$ limit of our
system with the same number of nodes $N$ as the original square lattice
varying the average number of neighbors $\left<k\right>$ in a broad range.
Figure 5 demonstrates that increasing the average number of interacting
partners $\left<k\right>$ of fibers in a completely random network $p=1$ the
strength values $\left<\sigma_{c}\right>$ and $\left<\varepsilon_{c}\right>$
converge to their mean field counterparts as expected.

##  4 Size distribution of failure cascades

The microscopic mechanism of the failure of the system is the cascading
failure of nodes (fibers) triggered by single

Figure 6: Spreading of a failure cascade on the network of load sharing
connections at $p=0.5$ with Weibull distributed failure thresholds at $m=1$.
The cascade starts and ends with a single breaking indicated by the arrows.
All fibers of the network which receive load from broken fibers are indicated
by black circles, and among them the ones which break as a consequence of load
sharing are highlighted by colors different from black. For clarity, the
circles representing broken fibers have also a larger size. Fibers breaking in
the same load redistribution step have the same color. The cascade in the
example has the size $\Delta=87$, which was generated in 24 load
redistribution steps. For clarity, nodes of the network which do not
participate in the cascade are not shown.

breaking events as a consequence of external load increments. Our simulations
revealed that the structure of the load transmission network plays an
essential role in the growth of avalanches which in turn also determines the
macroscopic behaviour of the bundle. A cascade always starts from a single
failing node and spreads over the transmission network, which is demonstrated
in Fig. 6 for a network at the rewiring probability $p=0.5$ with the Weibull
shape parameter $m=1$ of strength values. In the figure all fibers (nodes)
which receive load from breaking fibers are indicated by black color, and
among them those ones which suffer breaking are highlighted by colors
different from black. Fibers breaking as a consequence of the same load
redistribution step, are represented by the same color. Starting from the
externally imposed fiber breaking in the top of the figure, one can easily
follow the development of the cascade through the consecutive colors. Note
that the cascade forms a connected cluster of broken fibers on the network,
however, fibers breaking in the same sub-cascade can be far from each other.
The number of fibers breaking in the cascade defines the cascade size
$\Delta$.

Figure 7: The size distribution of failure cascades $p(\Delta)$ for several
values of the rewiring probability $p$ at the Weibull parameter $m=1$ of the
strength of fibers. For low $p$ values a crossover occurs between two power
law regimes of different exponents. The two straight lines represent power
laws of exponents $2.3$ and $3.4$, characteristic for $p=1$ and $p=0$,
respectively. The vertical dashed line highlights the crossover burst size
$\Delta_{c}$ estimated for $p=0.01$.

To characterize the statistics of failure cascades we determined the
probability distribution of their size $p(\Delta)$, which proved to have a
strong dependence on the network topology of load transmitting connections. It
can be observed in Fig. 7 for the Weibull parameter $m=1$ that on the regular
square lattice ($p=0$) where strong spatial localization of load occurs around
failed regions, the distribution $p(\Delta)$ can be approximated as a power
law

$p(\Delta)\sim\Delta^{-\tau},$ (11)

which is followed by a finite size cutoff. The value of the exponent is rather
high $\tau=3.4\pm 0.1$ in agreement with former studies of the fiber bundle
model [31, 38, 19]. The rapidly decreasing distribution and the low cutoff
burst size $\Delta_{max}$ clearly show that cascades are typically small
compared to the system size $N$. Due to the strong localization large
avalanches would lead to immediate collapse of the bundle on the regular
lattice. However, increasing the fraction of long range connections by
increasing the rewiring probability $p$, the stress localization gets
gradually reduced, hence, the system can tolerate larger and larger failure
cascades without suffering catastrophic collapse. As a consequence, the cutoff
burst size $\Delta_{max}$ increases and the distribution $p(\Delta)$ exhibits
a crossover to a second power law regime with a lower exponent. The smaller
value of $\tau$ shows the growing fraction of large size cascades in the
failure dynamics of the system (see Fig. 7).

Figure 8: The power law exponent $\tau$ of the probability distribution of the
cascade size $p(\Delta)$ as a function of the rewiring probability $p$ (left
vertical axis). The average value of the largest cascade
$\left<\Delta_{max}\right>$ scaled with the corresponding ELS result
$\left<\Delta_{max}^{ELS}\right>$ is presented on the right vertical axis as a
function of $p$. The horizontal dashed line represents the mean field value of
the cascade size exponent $\tau^{ELS}=5/2$. All the results were obtained at
the Weibull parameter $m=1$.

Of course, the crossover burst size $\Delta_{c}$, which separates the two
power law regimes, depends on the rewiring probability, i.e. $\Delta_{c}$
gradually shifts to smaller values with increasing $p$ in such a way that in
the limit $p\to 1$ practically a single power law remains with a significantly
lower exponent than that of the original square lattice at $p=0$. To
characterize this evolution of $p(\Delta)$ we determined the average size of
the largest avalanche $\left<\Delta_{max}\right>$ and the power law exponent
$\tau$ of the regime of large avalanches as function of the rewiring
probability $p$. It can be observed in Fig. 8 for the Weibull parameter $m=1$
that up to the rewiring probability $p_{l}\approx 0.01$ the cutoff cascade
size $\left<\Delta_{max}\right>$ is nearly constant, although the exponent
$\tau$ suffered some change. This behaviour implies that the small fraction of
randomized contacts has a minor effect on the cascading failure dynamics in
this parameter range $p\lesssim 0.01$. However, above this threshold
probability a rapid change of the cascade size distribution sets on indicated
by the steep increase of $\left<\Delta_{max}\right>$ and decrease of the
exponent $\tau$. For high values of the rewiring probability $p\to 1$, the
exponent $\tau$ converges to a constant $\tau\approx 2.3$, which falls very
close to the mean field burst size exponent of FBMs $\tau^{ELS}=5/2$ [18, 12].
The result indicates that on sufficiently randomized load transmission
networks the statistics of failure cascades of the localized load sharing FBM
becomes equivalent to the mean field universality class of the system in
agreement with the behaviour of the macroscopic strength of the bundle. The
result is consistent with Ref. [33] where FBMs were analyzed on Watts-Strogatz
networks in the range of high rewiring probabilities $p\geq 0.2$ recovering
the mean field behaviour.

It is interesting to note that in Fig. 8 the average largest cascade size
$\left<\Delta_{max}\right>$ has a maximum around the rewiring probability
$p^{*}\approx 0.1$ which practically coincides with the position of the
minimum value of the exponent $\tau$. This behaviour indicates that there
exists a network topology determined by $p^{*}$ where the network can tolerate
the largest cascades with a considerable frequency. The reason is that
increasing the rewiring probability the growing randomness of the network
increases the perimeter of the failed clusters, hence, reducing the load
concentration on it. This mechanism stabilizes the system in the sense that
cascades can reach larger sizes without becoming instable destroying the
system. However, at higher $p$ a counter effect occurs that low degree fibers
appear on the network with a growing fraction, which increases the load
concentration in their vicinity and makes the system more vulnerable to
cascades. The value $p^{*}$ provides the optimum for the cascade tolerance of
the system.

Figure 9: The average critical load $\left<\sigma_{c}\right>$ $(a)$ and the
average size of the largest cascade $\left<\Delta_{max}\right>$ $(b)$ scaled
with their mean field values $\sigma_{c}^{ELS}$ and
$\left<\Delta_{max}^{ELS}\right>$ as function of the rewiring probability $p$
for several values of the Weibull parameter $m$. In $(b)$ the position of the
local maximum of $\left<\Delta_{max}\right>$ determines the rewiring
probability $p^{*}$ where the network tolerates the largest failure cascades.
$(c)$ The relative scatter of the size of the largest cascade, i.e. the
standard deviation $\sigma_{\Delta_{max}}$ divided by the average
$\left<\Delta_{max}\right>$ of the largest cascade size. The dashed lines
serve to guide the eye indicating the position of the maximum of the curves
for the Weibull parameters $m=1,3,10,15$. For $(a,b,c)$ the legend is provided
in $(b)$. $(d)$ The rewiring probability $p_{l}$ of the onset of the
transition from LLS to ELS behaviour obtained as the position of the maximum
of the curves in $(c)$ (numerical) and as the solution of Eq. (16)
(analytical).

##  5 Effect of the degree of threshold disorder on the LLS-ELS transition

We carried out a large amount of simulations of the failure process of the
network of fibers at several values of the Weibull exponent $m$ varying the
degree of strength disorder in a broad range. These calculations revealed that
the LLS to ELS transition has a high complexity as the network structure is
gradually changed where the degree of strength disorder of nodes plays a
crucial role. Figure 9$(a)$ demonstrates for the average critical load
$\left<\sigma_{c}\right>$ that as the strength disorder is reduced by
increasing $m$, the onset of the transition, i.e. the rewiring probability
$p_{l}$ where the first significant deviation occurs from the LLS result of
the regular lattice, shifts to higher values. For instance, for $m=7$ the
transition starts at about $p_{l}\approx 0.1$, which is an order of magnitude
higher than the corresponding value $p_{l}\approx 0.01$ obtained at $m=1$. The
transition is completed at a rewiring probability $p_{u}$ beyond which the
bundle strength practically does not change. With decreasing strength disorder
the value of $p_{u}$ also increases and tends towards 1 in such a way that the
transition regime shrinks. Note that the asymptotic strength
$\left<\sigma_{c}\right>(p=1)$ decreases with increasing $m$ compared to its
mean field counterpart $\sigma_{c}^{ELS}$, indicating that at lower threshold
disorder randomization of the network structure provides less and less
improvement of the overall load bearing capacity of the system. It is
interesting to note that at the lowest disorders considered, starting from
$m=10$, the $\left<\sigma_{c}\right>(p)$ curves proved to be non-monotonous,
i.e. for $m=10,15$ the onset of the increase of the ultimate strength of the
system is preceded by a local minimum. Additionally, for $m=22$ the limit
value of the strength attained at $p\to 1$ falls below the strength of the
original square lattice. The result implies that when the strength of nodes is
sampled from a sufficiently narrow interval the rewiring process gives rise to
a reduction of the bundle strength at any rewiring probability.

The analysis of the statistics of cascade size revealed a similar effect of
the strength disorder of fibers on the LLS-ELS transition of the failure
process: It can be observed in Fig. 9$(b)$ for the average size of the largest
cascade $\left<\Delta_{max}\right>$ that as the degree of threshold disorder
gets reduced with increasing $m$ the value of $\left<\Delta_{max}\right>$
remains constant keeping its $p=0$ value for a broader and broader range of
the rewiring probability $p$. The estimated lower bounds $p_{l}$ of the
transition regime are consistent in Figs. 9$(a)$ and 9$(b)$ for the
macroscopic and microscopic quantities showing that $p_{l}$ increases with
decreasing threshold disorder. Note that the $\left<\Delta_{max}\right>(p)$
curves rise sharper than the bundle strength $\left<\sigma_{c}\right>(p)$
making the transition regime more transparent. It can be expected that at the
onset of the LLS-ELS transition the value of $\Delta_{max}$ has large
fluctuations. To quantify this Fig. 9$(c)$ presents the relative scatter of
$\Delta_{max}$, i.e. the ratio of its standard deviation
$\sigma_{\Delta_{max}}$ and average $\left<\Delta_{max}\right>$. For each
degree of disorder $m$, a sharp maximum can be observed whose position
provides a good measure of $p_{l}$. The vertical dashed lines highlight for a
few Weibull exponents $m$ that indeed the maximum of the relative scatter of
$\Delta_{max}$ well coincides with the onset of the sharp rise of
$\left<\Delta_{max}\right>(p)$ in Fig. 9$(b)$.

It is important to note that as the strength disorder decreases the position
$p^{*}$ of the maximum of $\left<\Delta_{max}\right>$ where the network
tolerates the largest avalanches, shifts to higher values. Additionally, the
maximum gradually decreases and eventually disappears around $m\approx 4$,
where the $\left<\Delta_{max}(p)\right>$ curves become monotonous. It follows
that for threshold disorder in the range $m>4$ the fully random graph provides
the highest tolerance of cascades. In agreement with the behaviour of the
ultimate strength of the bundle, at lower strength disorder of the nodes, the
randomization of the network structure provides less and less improvement
compared to the LLS limit of regular lattices.

For each degree of disorder $m$ the size distribution of cascades $p(\Delta)$
goes over the same evolution as for $m=1$ in Fig. 7: below the rewiring
probability $p_{l}$ the cascade size distribution practically remains the same
as on the original regular lattice at $p=0$. The second power law regime with
a lower exponent emerges for networks with $p\geq p_{l}$ accompanied by the
growth of the cutoff cascade size $\left<\Delta_{max}\right>$ and by the
gradual decrease of the crossover cascade size $\Delta_{c}$. When $p$ exceeds
$p_{u}$, the transition gets completed and a single power law remains of
$p(\Delta)$.

###  5.1 Rewiring probability of the onset of the transition

In order to understand how the transition emerges from LLS to ELS with the
rewiring probability $p$ at different degrees of disorder $m$, we construct an
analytical argument based on the changing structure of the underlying load
transmission network. On the original square lattice of fibers, localized load
sharing dynamics leads to early failure of the entire bundle due to the strong
stress concentration on the perimeter of failed clusters [39, 31, 38]. In the
last stable configuration of the bundle failed clusters are very small
compared to the system size so that the majority of fibers break in the last
catastrophic cascade. Adding randomized long range connections leads to the
reduction of the local stress concentration by increasing the perimeter of the
growing failed cluster. As a consequence, the system can tolerate larger
cascades and has a higher overall load bearing capacity.

At a given value of $p$ the average number of rewired connections can be
estimated as $2Np$ since each of the initial $2N$ links is rewired with
probability $p$. When $p$ is low only a very small fraction of fibers is
affected in the bundle by the rewiring either by having a removed nearest
neighbor link or by getting a newly established long range contact. So it is
reasonable to assume that at low $p$ values the majority of spreading cascades
have a high chance to avoid fibers with rewired links so that cascades remain
small following the same statistics as on the original square lattice at
$p=0$. Those cascades which involve fibers with rewired connections may grow
to larger sizes resulting in a statistics different from the one of the small
avalanches. This mechanism leads to the emergence of a crossover in the
distribution of the size of cascades presented in Fig. 7.

To estimate the crossover burst size $\Delta_{c}$, it is instructive to
determine the probability that a randomly selected node of the network is
affected by rewiring. The probability that none of the 4 nearest neighbor
connections is rewired for a fiber is $(1-p)^{4}$, while the probability that
it does not get connected to any new fiber can be estimated as
$\exp{\left(-4p\right)}$ for large $N$ [40]. Hence, the probability $p_{r}$
that a node is affected by rewiring can be cast into the form

$p_{r}=1-(1-p)^{4}e^{-4p}.$ (12)

We assume that crossover occurs at a cascade size $\Delta_{c}$ above which
cascades involve on average at least one fiber affected by the rewiring
process. Hence, the relation

$\Delta_{c}p_{r}(p)\approx 1$ (13)

follows between $\Delta_{c}$ and the rewiring probability $p$, from which we
obtain

$\Delta_{c}\approx\frac{1}{p_{r}(p)}.$ (14)

Note that for $p\to 0$ the crossover avalanche size diverges
$\Delta_{c}\to\infty$, while it tends to 1 for $p\to 1$.

It follows from the above arguments that at very low rewiring probabilities
$p\ll 1$ the crossover cascade size is larger than the average largest
avalanche $\left<\Delta_{max}\right>(p=0)$ on the original regular lattice at
$p=0$. It has the consequence that the dynamics and statistics of stable
cascades is practically not affected by the rewiring in this $p$ range, no
crossover occurs, so that the distributions of the cascade size remain
practically the same as on the square lattice (see Fig. 7). Crossover of the
distributions emerges for those $p$ values where the condition

$\left<\Delta_{max}\right>(p=0,m)>\Delta_{c}(p)$ (15)

holds. Note that $\left<\Delta_{max}\right>$ of the original lattice $p=0$
also depends on the strength disorder $m$. This first occurs at the lower
bound $p_{l}$ of the transition regime

$\left<\Delta_{max}\right>(p=0,m)=\Delta_{c}(p_{l})$ (16)

from which $p_{l}$ can be obtained as a function of the degree of strength
disorder $p_{l}=p_{l}(m)$. In the regime $p>p_{l}(m)$ the crossover cascade
size separating the two power law regimes of different exponents can be
approximated by Eq. (14). The smallest possible value of $\Delta_{c}$ we could
identify in our numerical measurements is $\Delta_{c}\approx 1-3$, from which
the upper bound of the crossover $p_{u}$ can be determined. Further increasing
$p$ above $p_{u}$ no qualitative change of the failure process occurs so that
the statistics of the size of cascades remains the same.

It follows from the above arguments that the dependence of the LLS-ELS
transition on the degree of strength disorder of the fibers (nodes) originates
from the disorder dependence of the cascade activity of the system on the
unperturbed regular lattice. For each Weibull exponent $m$ we estimated
numerically the crossover point $\Delta_{c}$ of the cascade size distributions
$p(\Delta)$ by determining the value of $\Delta$ where the two fitted straight
lines of the two power law regimes of the distributions cross each other at
each rewiring probability $p$. It can be observed in Fig. 10 that the
analytical curve of $\Delta_{c}(p)$ obtained from Eq. (14) underestimates the
numerical values, however, its functional form provides a reasonable
description of the numerical findings. In Figure 10 the rewiring probability
of the lower $p_{l}$ and upper $p_{u}$ bounds of the transition at a given
disorder $m$ can be identified as the $p$ values where the crossover first
occurs, and where $\Delta_{c}$ becomes constant, respectively. To obtain a
more precise estimate of the transition regime, we solved numerically Eq. (16)
for $p_{l}$ substituting the value of the average largest cascade size
$\left<\Delta_{max}\right>(p=0)$ at each $m$. This semi-analytical value of
$p_{l}$ is compared in Fig. 9$(d)$ to the numerical one obtained as the
position of the maximum of the relative scatter of $\Delta_{max}$ in Fig. 9
$(c)$. The analytical results again underestimate the numerical ones but they
have the same functional form. The value of $p_{u}$ we can estimate from the
numerical results falls between 0.2 and 1.

Figure 10: The symbols represent the crossover cascade size $\Delta_{c}$
obtained from the size distribution of cascades for several Weibull exponents
$m$. The analytic prediction (continuous curve) somewhat underestimates
$\Delta_{c}$, however, its functional form provides a reasonable description
of the numerical findings.

###  5.2 Failure triggered by low degree fibers

Our simulations revealed that the competition of two mechanisms determine the
behaviour of the loaded network both on the micro- and macro-scales. At higher
rewiring probabilities $p$ the growing randomness of the network reduces the
local load concentration in the system. This mechanism can substantially
increase the cascade tolerance and the overall load bearing capacity of the
network especially at high disorder of the strength of nodes, see Fig.
9$(a,b)$. However, as the degree distribution $\rho(k)$ broadens with
increasing rewiring probability $p$, low degree nodes appear which have the
counter effect of increasing the local load concentration when they fail. This
effect becomes crucial at low node strength disorder, where the failure of a
low degree node can easily trigger a catastrophic avalanche of failure events.
In order to quantify this mechanism we characterize the degree of strength
disorder of nodes by estimating the average of the smallest
$\left<\sigma_{th}^{min}\right>$ and largest $\left<\sigma_{th}^{max}\right>$
failure thresholds in the bundle. Among $N$ independent random numbers sampled
from the same probability distribution $P$, the average of the smallest and
largest values can be obtained as

$\left<\sigma_{th}^{min}\right>=P^{-1}\left(\frac{1}{N+1}\right),\quad\mbox{and}\qquad\left<\sigma_{th}^{max}\right>=P^{-1}\left(1-\frac{1}{N+1}\right),$
(17)

where $P^{-1}$ denotes the inverse of the cumulative distribution [41].
Substituting the Weibull distribution Eq. (4), the limit thresholds in a
bundle of $N$ fibers can be cast into the form

$\left<\sigma_{th}^{min}\right>=\lambda\left(\frac{1}{N}\right)^{1/m},\quad\mbox{and}\qquad\left<\sigma_{th}^{max}\right>=\lambda\left(\ln
N\right)^{1/m}.$ (18)

The ratio $r$ of the two values provides a measure of the degree of threshold
disorder of the nodes

$r=\left<\sigma_{th}^{max}\right>/\left<\sigma_{th}^{min}\right>=\left(N\ln
N\right)^{1/m}.$ (19)

For the stability of the bundle the worst case is when a node of degree $k=1$
has the smallest failure threshold $\sigma_{th}^{min}$, since at failure it
will double the load on its neighbor right at the beginning of the failure
process. This load sharing will definitely result in failure of the neighbor
if the elevated load $2\sigma_{th}^{min}$ is greater than the largest
threshold $\sigma_{th}^{max}$ in the bundle so that the condition follows

$2\sigma_{th}^{min}>\sigma_{th}^{max}.$ (20)

Since this secondary failure event gives rise to a large load increment on its
own neighbors, it is reasonable to assume that the cascade does not stop
anymore and it becomes catastrophic. Equation (20) implies that this mechanism
determines the response of the bundle only when the strength distribution is
sufficiently narrow $r<2$, and the rewiring probability $p$ is sufficiently
high to have a finite fraction of nodes of degree $k=1$. Using the expression
of $r$ Eq. (19), the condition Eq. (20) can be cast into a condition for the
Weibull shape parameter

$m>\frac{\ln\left(N\ln N\right)}{\ln 2},$ (21)

which yields $m>20.9$ for the setup of our fiber bundle. It follows that the
highest $m$ value we considered $m=22$ fulfills the condition so that the
ultimate failure of the network at this $m$ should be dominantly triggered by
fibers of degree 1 at sufficiently high rewiring probabilities. To support the
above arguments Fig. 11 presents the average number of intact neighbors
$\left<k_{t}\right>$ of fibers the failure of which initiated the catastrophic
cascade along with its original degree $\left<k_{t}^{orig}\right>$. It can be
observed that both quantities monotonically decrease with increasing rewiring
probability $p$, i.e. as the degree distribution of nodes gets broader with
increasing $p$ the triggering node has a lower and lower degree. At high
strength disorder (low Weibull exponent $m$) the original
$\left<k_{t}^{orig}\right>$ and final degrees $\left<k_{t}\right>$ of
triggering fibers have a larger difference, however, as the strength disorder
gets reduced with increasing $m$ both curves shift to lower values in such a
way that their difference gets also smaller. It is important to emphasize that
the $\left<k_{t}^{orig}\right>$ and $\left<k_{t}\right>$ curves tend to 1,
which confirms that at low strength

Figure 11: The average number of intact neighbors $\left<k_{t}\right>$ (empty
symbols) and the average of the original degree $\left<k_{t}^{orig}\right>$
(filled symbols) of the fiber the failure of which triggers the final
breakdown of the network of fibers. Results are presented for three values of
the Weibull exponent $m$.

disorder the lowest degree nodes make the system vulnerable to cascading
failure triggering the catastrophic breakdown of the system. The most
remarkable outcome of these calculations is that the LLS-ELS transition is
limited to a disorder range of the strength of nodes. At too low disorder
rewiring makes the system more vulnerable to cascades which prevents any
improvement of the strength and cascade tolerance of the system. For Weibull
exponents fulfilling the condition Eq. (21) no LLS-ELS transition emerges.

##  6 Discussion and conclusions

We presented a theoretical study of the evolution of the failure dynamics of
the fiber bundle model as the underlying network of load transmitting
connections is gradually changed from a regular lattice to a random network. A
complex network of fibers was constructed by randomizing a regular square
lattice using the Watts-Strogatz rewiring technique. Fibers assigned to the
nodes of the network are assumed to have a finite load bearing capacity which
is a random variable. Initially all fibers are intact and their state is
switched to failed when the local load on them exceeds their strength. The
system was subject to a slowly increasing external load by adding the same
load increment to each intact fiber in such a way that the failure of a single
fiber is provoked. The load of failed fibers is transmitted to their intact
nearest neighbors which may trigger an entire cascade of failure events under
the constraint of load conservation.

Gradually increasing the rewiring probability, we showed that the changing
network structure gives rise to a transition from the localized to the mean
field behaviour of failure processes accompanied by a complex evolution both
on the macro- and micro-scales. The first deviations from the LLS behaviour of
the regular lattice appear at a threshold probability $p_{l}$ where the
transition sets on and it gets completed by reaching the upper bound $p_{u}$,
beyond which no further change occurs in the system. In the transition regime
the probability distribution of the size of failure cascades exhibits a
crossover between two power laws of different exponents. On the macroscopic
scale the $\sigma(\varepsilon)$ curve of LLS bundles follow the mean field
solution of the model at all rewiring probabilities but with a lower strength.
The critical load and strain where ultimate failure occurs increase with the
rewiring probability and tend towards limits which fall close to their mean
field values.

We demonstrated that the degree of disorder of the strength of fibers has a
substantial effect on the transition: as the disorder gets reduced the
transition regime shrinks and shifts to higher rewiring probabilities. Most
notably the LLS-ELS transition is limited to a well-defined range of disorder
of the strength of nodes. In particular, there exists a threshold amount of
node strength disorder below which the randomization of the network of load
transmitting connections does not provide any improvement neither of the
overall load bearing capacity nor of the cascade tolerance of the system.
Computer simulations revealed that at low strength disorder the fully random
network is the most stable one, while at high disorder best cascade tolerance
is obtained at a lower structural randomness.

Based on the interplay of the network structure and node strength disorder we
constructed an analytical argument which provided a reasonable description of
the numerical findings. These calculations revealed that two competing
mechanisms determine the response of the network: the rewiring of the
underlying lattice introduces long range random connections in the load
transmission network which reduce the load concentration around failed
regions, and in turn allow the system to tolerate larger cascades without
becoming instable. At low rewiring probabilities due to the low fraction of
long range contacts, small sized cascades remain practically unaffected.
However, beyond a characteristic size, cascades involve more and more fibers
with rewired contacts which increase their stability. This mechanism leads to
the increase of the cutoff cascade size and the emergence of a second power
law regime of the distributions with a lower exponent than for the small
cascades. However, as the rewiring probability increases the degree
distribution broadens which increases the fraction of low degree nodes. This
gives rise to the counter effect that in the vicinity of failing low degree
nodes a large load concentration emerges, which can trigger catastrophic
cascades. As a consequence, at high rewiring probabilities the network becomes
vulnerable to early cascades when the strength of nodes is sampled from a
narrow distribution.

The failure mechanism of the fiber bundle model we focused on is quite generic
with four key elements: $(I)$ the total load on the system is increased by
adding the same load increments to all the intact elements; $(II)$ nodes fail
irreversibly such that they are removed from the bundle together with their
links; $(III)$ failed nodes transfer their load to their intact nearest
neighbors through their links; $(IV)$ the load on the system is conserved
during the spreading of failure cascades. The fiber bundle model has been used
to study the emergence of cascading breakdown of roads carrying traffic, flow
channels, and power grids. Due to the minimum amount of additional assumptions
we made, our results should be relevant for these modelling approaches as
well.

## Acknowledgments

The work is supported by the EFOP-3.6.1-16-2016-00022 project. The project is
co-financed by the European Union and the European Social Fund. This research
was supported by the National Research, Development and Innovation Fund of
Hungary, financed under the K-16 funding scheme Project no. K 119967. The
research was supported by the Thematic Excellence Programme (TKP2020-NKA-04)
of the Ministry for Innovation and Technology in Hungary.

## References

  * [1] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D. Hwang, Complex networks: Structure and dynamics, Phys. Rep. 424 (2006) 175.
  * [2] I. Dobson, B. A. Carreras, V. E. Lynch, D. E. Newman, Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization, Chaos 17 (2) (2007) 026103.
  * [3] B. Dou, X. Wang, S. Zhang, Robustness of networks against cascading failures, Physica A 389 (11) (2010) 2310–2317.
  * [4] J. Borge-Holthoefer, R. A. Baños, S. González-Bailón, Y. Moreno, Cascading behaviour in complex socio-technical networks, Journal of Complex Networks 1 (1) (2013) 3–24.
  * [5] S. Siddique, V. Volovoi, Failure mechanisms of load-sharing complex systems, Phys. Rev. E 89 (2014) 012816.
  * [6] J. Gleeson, R. Durrett, Temporal profiles of avalanches on networks, Nat Commun 8 (2017) 1227.
  * [7] N. Jung, Q. A. Le, K.-E. Lee, J. W. Lee, Avalanche size distribution of an integrate-and-fire neural model on complex networks, Chaos 30 (6) (2020) 063118\.
  * [8] J. Awrejcewicz, M. A. F. Sanjuán, Introduction to focus issue: Recent advances in modeling complex systems: Theory and applications, Chaos 31 (7) (2021) 070401\.
  * [9] B. Ouyang, Z. Teng, Q. Tang, Dynamics in local influence cascading models, Chaos, Solitons & Fractals 93 (2016) 182–186.
  * [10] M. Frasca, L. V. Gambuzza, Control of cascading failures in dynamical models of power grids, Chaos, Solitons & Fractals 153 (2021) 111460.
  * [11] D. V. Stäger, N. A. M. Araújo, H. J. Herrmann, Usage leading to an abrupt collapse of connectivity, Phys. Rev. E 90 (2014) 042148.
  * [12] R. C. Hidalgo, F. Kun, K. Kovács, I. Pagonabarraga, Avalanche dynamics of fiber bundle models, Phys. Rev. E 80 (2009) 051108.
  * [13] A. Hansen, P. Hemmer, S. Pradhan, The Fiber Bundle Model: Modeling Failure in Materials, Statistical Physics of Fracture and Breakdown, Wiley, 2015.
  * [14] B. K. Chakrabarti, S. Biswas, S. Pradhan, Cooperative dynamics in the fiber bundle model, Frontiers in Physics 8 (2021) 664.
  * [15] Y. Moreno, J. B. Gómez, A. F. Pacheco, Instability of scale-free networks under node-breaking avalanches, Europhysics Letters 58 (4) (2002) 630–636.
  * [16] Y. Moreno, J. B. Gomez, A. F. Pacheco, Fracture and Second-Order phase transitions, Phys. Rev. Lett. 85 (14) (2000) 2865–2868.
  * [17] F. Kun, F. Raischel, R. C. Hidalgo, H. J. Herrmann, Extensions of fiber bundle models, in: P. Bhattacharyya, B. K. Chakrabarti (Eds.), Modelling Critical and Catastrophic Phenomena in Geoscience: A Statistical Physics Approach, Lecture Notes in Physics, Springer-Verlag Berlin Heidelberg New York, 2006, pp. 57–92.
  * [18] M. Kloster, A. Hansen, P. C. Hemmer, Burst avalanches in solvable models of fibrous materials, Phys. Rev. E 56 (1997) 2615–2625.
  * [19] R. C. Hidalgo, Y. Moreno, F. Kun, H. J. Herrmann, Fracture model with variable range of interaction, Phys. Rev. E 65 (2002) 046148.
  * [20] Z. Danku, F. Kun, Temporal and spacial evolution of bursts in creep rupture, Phys. Rev. Lett. 111 (2013) 084302.
  * [21] B. K. Chakrabarti, A fiber bundle model of traffic jams, Physica A 372 (2006) 162–166.
  * [22] J.-F. Zheng, Z.-Y. Gao, X.-M. Zhao, B.-B. Fu, Extended fiber bundle model for traffic jams on scale-free networks, Int. J. Mod. Phys. C 19 (11) (2008) 1727–1735.
  * [23] C. Barré, J. Talbot, Cascading blockages in channel bundles, Phys. Rev. E 92 (2015) 052141.
  * [24] O. Yağan, Robustness of power systems under a democratic-fiber-bundle-like model, Phys. Rev. E 91 (2015) 062811.
  * [25] J. M. Reynolds-Barredo, D. E. Newman, B. A. Carreras, I. Dobson, The interplay of network structure and dispatch solutions in power grid cascading failures, Chaos 26 (11) (2016) 113111.
  * [26] S. Biswas, L. Goehring, Load dependence of power outage statistics, Europhysics Letters 126 (4) (2019) 44002.
  * [27] R. L. Smith, The Asymptotic Distribution of the Strength of a Series-Parallel System with Equal Load-Sharing, Ann. Probab. 10 (1) (1982) 137–171.
  * [28] L. N. McCartney, R. L. Smith, Statistical Theory of the Strength of Fiber Bundles, J. Appl. Mech 50 (1983) 601–608.
  * [29] P. C. Hemmer, A. Hansen, The distribution of simultaneous fiber failures in fiber bundles, J. Appl. Mech. 59 (1992) 909–914.
  * [30] A. Hansen, P. C. Hemmer, Burst avalanches in bundles of fibers: Local versus global Load-Sharing, Phys. Lett. A 184 (1994) 394–396.
  * [31] F. Raischel, F. Kun, H. J. Herrmann, Local load sharing fiber bundles with a lower cutoff of strength disorder, Phys. Rev. E 74 (2006) 035104.
  * [32] K. Kovács, R. C. Hidalgo, I. Pagonabarraga, F. Kun, Brittle-to-ductile transition in a fiber bundle with strong heterogeneity, Phys. Rev. E 87 (2013) 042816.
  * [33] D.-H.Kim, B. J. Kim, H. Jeong, Universality class of the fiber bundle model on complex networks, Phys. Rev. Lett. 94 (2005) 025501.
  * [34] U. Divakaran, A. Dutta, Fibers on a graph with local load sharing, Int. J. Mod. Phys. C 18 (2007) 6.
  * [35] D. Watts, S. Strogatz, Collective dynamics of ‘small-world’ networks, Nature 393 (1998) 440–442.
  * [36] D. J. Watts, Networks, dynamics, and the Small‐World phenomenon, American Journal of Sociology 105 (2) (1999) 493–527.
  * [37] J. V. Andersen, D. Sornette, K. Leung, Tricritical behaviour in rupture induced by disorder, Phys. Rev. Lett. 78 (1997) 2140–2143.
  * [38] F. Kun, S. Zapperi, H. J. Herrmann, Damage in fiber bundle models, Eur. Phys. J. B 17 (2000) 269.
  * [39] S. Zapperi, P. Ray, H. E. Stanley, A. Vespignani, Analysis of damage clusters in fracture processes, Physica A 270 (1999) 57.
  * [40] M. E. J. Newman, The structure and function of complex networks, SIAM Review 45 (2003) 167–256.
  * [41] A. Hansen, S. Roux, Statistical toolbox for damage and fracture, in: D. Krajcinovic, J. v. Mier (Eds.), Damage and Fracture of Disordered Materials, no. 410 in CISM Courses and Lectures, Springer Verlag, 2000, pp. 17–101.