# Spreading of Failures in Small-World Networks: A Connectivity-Dependent Load Sharing Fibre Bundle Model

- Institute of Mathematics, Czestochowa University of Technology, Czestochowa, Poland

A rich variety of multicomponent systems operating under parallel loading may be mapped on and then examined by employing a family of the Fiber Bundle Models. As an example, we consider a system composed of *N* immobile units located in nodes of a network *F* imposed uniformly on the units. Each unit, characterized by a load threshold *δ*, is classified as reliable or irreversibly failed, depending on whether *δ* is bigger, or respectively smaller, than the load felt by the unit. A pair of interdependent units is uniquely indicated by an edge of *F* and the loads that are transferred according to such a see-saw switch between the local and global sharing rules (sLGS), a set of nodes, that holds the reliable units, evolves as *p* that characterizes possible rearrangements of edges in *p*, where the mean highest load *N* limit *p* and both considered distributions of

## Introduction

Numerous systems, encountered in nature as well as in different areas of science and technology, are multicomponent, i.e., they are composed of a great number of functionally identical units. When loaded, the units process a given task in a fully parallel manner. It happens, however, that a unit becomes overloaded and fails. Its load has to be undertaken by other units, which in turn may trigger subsequent overloading followed by resulting failures. Such a chain of failures gradually degrades the system performance and leads to an avalanche of failures. It may even happen that the avalanche becomes self-sustained giving rise to a catastrophe which overwhelms all the units. Different factors characterize a given system. This is important to identify those working together that push the system toward the catastrophic avalanche.

The Fiber Bundle Model (FBM) is a particular case of a wide class of cascading processes on networks [1]. It offers a flexible approach to study how multicomponent systems evolve under varying load [2–8]. The flexibility refers to such aspects as: a) range and symmetry of interactions among units [9], b) rate of load’s variation, c) heterogeneity/uniformity of units [10, 11]), or d) varying quality of units [12, 13], to name a few. The aspects a) and b) especially refer to ingredients of the FBM that play a major role when a given system is mapped onto a bundle of interacting fibers [14]. Exemplary problems, from an ample set of systems expressed in the FBM framework, cover research fields that span from geophysics including earthquakes, snow or landslides, to technology with electrical and mechanical engineering systems.

In this context, we consider a toy model of failures spreading in a set of interconnected units. Our model consists of *N* units that reside at nodes of an undirected simple graph *F* is distributed identically on all reliable units. When *F* starts growing, some units begin to suffer from insufficient strength to bear the load and they fail. Their loads remain in the system and are shared either by the nearest neighboring units, if they are reliable, or by all other reliable units. If on a given node a failure emerges, this node is removed from the graph together with corresponding edges, i.e., *F*, an initially connected *F* pushes the set of reliable units to extinction. If the growth of *F* is sufficiently slow, then a distinct group of reliable units may be selected in the course of evolution:

This group, identified by nodes of *F* whereas intermediate graphs, induced exclusively by loads sharing processes to be precise, are omitted for the sake of simplicity. We use the subscripts “c” to mark that the load

Within this work we are interested in questions like: how small a group of units can be and/or to what extent we can apply the external load while still preventing the extinction of reliable units. Subsequently, we apply the FBM to study evolving failure on “small-world” networks that are omnipresent in life and technology. Specifically, we will focus on a family of random graphs generated by the Watts-Strogatz model [15]. The reason is that such graphs reveal short average path lengths and high clustering that are key features of social networks [16].

## Model Description

Take a locally overloaded system which detects a failure of a unit. In the first instance the system attempts to solve the problem locally by distributing the load among nearest neighbors of the failed unit. If such a neighborhood does not exist, the entire set of reliable units is engaged into sharing the load from the unit being lost. Such a mode of load transfer yields a significant impact on the system’s strength. Whenever an island of reliable units emerges during the evolution, its terminal load is shared globally by the system. This means that the net load transferred to reliable units that are located on the outer island’s perimeter is lower than it would be if the local load sharing (LLS) rule has been in operation. In consequence, the switching Local-Global-Sharing (sLGS) mitigates the expansion of a dominantly large cluster (DLC) of failed units and thus, the strength of the system becomes higher than that one corresponding to the LLS rule [5].

In the following, we consider an ensemble of units assigned to nodes of a graph *δ*. Units are not perfect and differ in their efficiency to sustain the load. Hence, the corresponding *δ*s are different. For the sake of simplicity we assume that

### Watts-Strogatz Model and Small-World Networks

There exists an ample set of papers that discuss the Watts-Strogatz model in details [17]. Hence, for the purpose of our model, it is sufficient to present the simplest exemplary graph and sketch how its modifications enable a smooth passage from an ordered network to disordered ones through a multitude of “small-world” graphs. One such passage is shown in Figure 1. The presented graphs are generated in two steps:

• a ring over *N* nodes is created and each node is connected with its *k* nearest neighbors, *k* is even.

• for every node with uniform independent probability *p*, each edge is rewired to a node that is selected uniformly at random while avoiding loops and edge duplication.

**FIGURE 1**. Exemplary “small-world” networks generated by the Watts-Strogatz model with mean node degree *p* of rewiring.

These steps are illustrated in **Figure 1**, e.g., the first step corresponds to the graph with

Among the different characteristics of a network, one is particularly important in the view of our study, namely the global clustering coefficient C defined as:

where nodes of a triangle form a 3-clique, and a connected triple is a tree.

### Applying External Load

We have assumed the external load *F* is distributed identically on all reliable units. Consider a load *i*-th unit, out of M reliable ones that are present at a given stage of evolution. This

We consider a configuration *F* all reliable units keep their states unchanged. When no reliable unit exists the corresponding configuration is the empty graph

In order to identify

From this, we derive the stopping rule:

where

### Load Sharing Rule

The load transfer requires a rule that indicates how a load released by a failure is shared by other reliable units. We define our rule in a following way: the reliable network neighbors are obliged to equally share the load if they are accessible and all the reliable units acquire the load in the contrary case.

From this definition’s point of view, our rule “dynamically” switches between two rules, which are known in the FBM framework as global load sharing (GLS) and LLS. These rules correspond to two extremal ranges of load transfer. In the GLS rule, a load originating from a failed unit is transferred equally to all the reliable units and thus, the range of transfer is maximal. The LLS rule, in turn, engages only the nearest neighbors of a node that fails, so the range of load transfer is minimal. As a consequence, the load distributed according to the GLS rule is the least harmful for the system, whereas the LLS represents the most damaging method of the load distribution.

In simulations, we call this rule the sLGS and assume that the load transfer is an almost instantaneous process that happens simultaneously. We can mathematically express the sLGS in a framework for cascading processes on networks [18]. For this purpose, let *i* and *j* are interdependent and let *i* at the stage *i* to a reliable unit *j* reads

where *i*.

Equation 4 has a structure that resembles schemes of load transfer known from the literature. Namely the mixed-mode load sharing (MMLS) [19] and the heterogeneous load sharing (HLS) [20] merge together the LLS and the GLS in order to study a crossover behavior in FBM on regular lattices. The MMLS employs a constant quota *q* to split each transferred load into two streams: a portion *q* of the load goes to nearest neighbors under the LLS rule and the remaining portion is transferred according to the GLS rule. Thereby, the MMLS folds the LLS and the GLS in a manner that both rules are simultaneously activated in each failure. This is in contrast to the HLS, which in turn assigns units to two groups in order to discriminate between units located in “rigid nodes” and those residing in a “flexible” fraction of the support. If the “rigid” unit fails then the GLS transfers its load whereas the LLS governs the transfer from the “flexible” unit. The MMLS and the HLS are static, i.e., the corresponding values of *q* and sets of nodes at which *q*-weighted sharing rules operate are chosen and fixed prior to loadings. We also want to mention the modified LLS rule [21]. By employing the scheme

It is worth noting that rules, similar to the sLGS have been applied recently in such contexts as a strategy for stopping failure cascades [22] or clogging in multichannel supply systems [23].

### A Range of Possible Applications

The above-described load sharing rule operating among units interconnected through a small world network may serve as a toy model of cascading failures in economy or technology. A general scenario we have in mind concerns a default initiated by an unsupported on-site demand that spreads through the system in a form of a contagion from the defaulter, either to units which are closely associated or to other ones. Clearly, when a unit switches into default this affects other units. Depending on the context, units could be: a) institutions, as, e.g., banks belonging to an interbank network, b) workers with beneficial loans from a company, borrowers in micro financial markets or c) elements of power grids, especially of small scale smart grids. With this same spirit a load could be seen as a demand, e.g., for liquidity or electric power. Below we list some basic facts that are relevant to our model.

#### Interbank Market

Undirected graphs are suitable to modeling interbank networks, especially in the context of a financial contagion [24, 25]. Among representations which are convenient and applied in studies, a possible one connects a pair of banks by an undirected edge whenever there exists an interbank liability or claim [25]. When an ensemble of interdependent banks is mapped onto a graph, one can analyze its static and dynamic properties. A class of small world graphs certainly is relevant in this context. It was shown, e.g., that the interbank market of

#### Microeconomy

Many companies offer beneficial loans to its employees. Specifically, to those suffering financial troubles. These employees-debtors, being colleagues and friends, are frequently mutual guarantors and can thus be considered as members of a resulting social network.

#### Power Grids

The small world topology is frequently reported as present in power grid networks [26–28]. This is equally true for large scale installations involving nationwide power systems in the US or Europe as well as for medium or small power grids [29, 30]. Particularly, in smart grids of renewable energy sources, such as small-scale photovoltaic systems or small-wind turbines [31, 32], the small world topology is beneficial. For example, networks with small world connectivity can significantly enhance their robustness against different attack by simultaneous increase of the rewiring probability and average degree [33].

## Results and Discussion

In order to acquire data necessary to build reliable empirical distributions, we have adopted two computational schemes that correspond to small and large numbers of units. In the first scheme, for each *N*, an ensemble of *N* regime, respectively.

We use both computational schemes for uniformly distributed load thresholds. In simulations with the Weibull distribution we consider *ρ* we conduct simulation following the first computational scheme. In the large *N* limit, we restrict ourselves to distributions with

Subsequently, when averaging a quantity *Y* over either

### Maximal Supported Load and Minimal Number of Reliable Units

Following the described computational schemes, we have collected large data sets containing detailed information about how the maximal load, together with the minimal number of units, vary when we pass through all pairs

The gathered data turn out to be skewed independently of what distribution governs *N* and

where *μ*, *σ* and *α* are the location, scale and shape parameters, respectively.

**FIGURE 2**. Calculated distributions of *N* corresponds to a population of 2,500 load thresholds

We have rigorously examined the data sets employing a number of goodness of fit tests, including the Cramer-von Mises and Anderson-Darling tests [35] and have accepted *p*-values than the Weibull p.d.

We have also estimated values of the parameters *μ*, *σ* and *α*. The gathered data yield estimate functional dependences of *μ*, *σ* and *α* on model parameters *ρ*. As an example, consider empirical p.d. of

We have directly written that *α* are functions of *N* and *p* whereas parameters characterizing distributions of *μ* varies with *p*, while keeping constant values of *N*. The resulting fitting function turns out to be a polynomial of the third order in *p*.

**FIGURE 3**. Estimated functional dependence of *p*: *b* =

Since the location parameter *p* and *ρ* are displayed in Figure 12.

**FIGURE 4**. Scaled mean critical number of reliable units

It should be pointed out that when *p* is growing, the resulting networks become more and more disordered and the probability that a given node has a low degree increases. Hence, the sLSG activates all the reliable units more frequently than it happens in networks generated with a small value of *p*.

### Large N Limit

Even though the applications mentioned in Section 2.4 refer to networks composed of about

It is known that the LLS model on a complex network behaves similarly to the GLS model giving rise to a non-vanishing critical strength *N* limit [5]. Formally, the family of Wats-Strogatz graphs covers the spectrum of networks ranging from the locally regular

Based on results of simulations of large-*N* systems, we have found that: i) *p* and

while for the Weibull p.d. the best fit reads

where the subscripts *u* and *w* stand for the uniform and Weibull distributions, respectively. The estimated system’s strength *α* are displayed in Figure 5. Correspondingly, for the Weibull p.d.

**FIGURE 5**. Left panel: The logarithmic size dependence of system strength for networks that: are ordered (*α* (Inset) computed for different values of *p* according to the best fit given in Eq. 7. Black marks represent the best fit to data of which some examples are shown in the left panel. The error bars indicate

**FIGURE 6**. The system’s strength scaled by size in the large *N* limit. Inset: the logarithmic dependence of scaled system strength for:

**FIGURE 7**. Amplitudes *N* limit for systems with

These plots illustrate a variety of ways in which *p* the ultimate system strength rapidly decreases, attains its minimum and then increases. Until

A deep minimum of *p* increases, the arguments (i) and (ii) come into the picture. First, for *p* roughly bigger than 0.1 enter a scenario characterized by the argument (i). The sLGS differs from the LLS, however. As it was already stated in the beginning of Section 2, whenever an island of reliable units appears its terminal load is transferred by the sLGS to all other reliable units and not to the closest ones. This inhibits the DLC growth and increases the system strength correspondingly.

Within our numerical approach, it is difficult to precisely estimate

### Internal vs. External Load From a Reliable-Unit’s Point of View

When considering its future reliability, a prospective unit behaves as an outer observer whose forecast is limited to the external load *F*. When entering the system, the unit is confronted with an internal-load impact. It is thus worth discussing to what extent these two points of view differ.

We have assumed that during the evolution, the external load *F* is distributed identically on reliable units and is growing stepwise along the rule that was discussed in SubSection 2.3.

Having initially

if

This iterative chain involves successive patterns of local load

Now, consider *t* of the evolution. Let us choose one of them, say the *i*-th unit. This means that *i*. When *t*. Clearly, internal-load distributions are subject to non-trivial variations that can be observed during the evolution.

It is important to make a distinction between impacts of external and internal loads on units. To obtain a closer look at these different impacts, we compare

**FIGURE 8**. Mean critical load per reliable unit: white plot marks - internal load

Analyzing computed values, we detect that the mean internal load prevails over the mean external one for all values of *p*. In networks with *δ*s.

### Small-World Properties at Critical Configuration

When the sLGS rule is in operation, a load is assigned according to accessibility of reliable units, i.e., either locally or globally. If the hosting network reveals a relatively strong local connectivity, then the sLGS looks like the LLS.

A lasting presence of reliable nearest-neighbours depends on a connectivity of an underlying network. Independently of the value of rewiring probability *p*, random graphs generated by the Watts-Strogatz model preserve the number of edges and mean-node degree. This means that when *p* grows, we pass from ordered to disordered networks, keeping the numbers of nodes and edges unchanged. For intermediate values of *p*, the resulting networks turn out to be locally clustered, whereas randomly rewired edges reduce the mean path lengths. Thus, there exists a range of *p*, where networks belonging to

We thus expect that the “small-world” properties would mark their presence in data sets related to *C*, defined by the Eq. 2 and computed for corresponding networks, we notice that for a given value of *N*, formula

best fits the quantity *N*, namely:

**TABLE 1**. Estimated coefficients in Eq. 9: *δ*s.

Figure 9 displays respective relations for systems with uniformly distributed *N* limit, the relation (10) is valid for uniformly distributed

with

**FIGURE 9**. **(A)** Calculated mean empirical global clustering coefficient *p* for employed sets of 400 Watts-Strogatz graphs, each with connectivity **(B**–**D)** refer to:

**TABLE 2**. Estimated coefficients in Eq. 10:

**FIGURE 10**. Mean strength of the system vs*.* mean global clustering coefficient for growing number of units. The linear scaling (11) between the ultimate strength

When sets *N*, as it is shown in Figure 12. This indicates that when *ρ* grows an ascending degree of order among load thresholds homogenizes the system and suppresses the linear relation between network’s clustering and system’s strength.

**FIGURE 11**. Upper panels: Mean system’s strength per reliable unit at critical configuration: **(left)** as function of *p* with solid lines drawn according to 9 and **(right)** on a linear scale for the respective mean *N*, as functions of mean empirical clustering coefficient calculated for corresponding networks. Results were obtained from

**FIGURE 12**. Scaled mean critical quantities: *ρ* and system size *N* as functions of *p*. The solid lines represent Eq. 9 with coefficients estimated from data. The sample size equals

It should be noticed that the expressibility of

## Summary

We have investigated the evolution of failure among units that live at nodes of “small-world” networks and are exposed to a growing load. By introducing the sLGS rule of load transfer, which switches between the LLS and GLS rules depending on the accessibility of local interdependent nodes, we were able to mimic unit failures, and thus follow the evolution of the system toward the limit of its functionality. In particular, by employing the Watts-Strogatz random graphs to simulate the networks, we have collected data sufficient to form empirical distributions of the maximal load *p* with which the links among interdependent nodes are modified.

The simulations show that if *p* is within the range of values given in Table 2 then

We are conscious of the fact that our simplified model of failure evolution involves some less realistic assumptions. Among the most serious is that we have considered each link between a pair of units as a reciprocally profitable relation. The other less strict assumption is that we allow the load thresholds be identically distributed. Our model can be tailored to fit a particular realistic scenario, e.g., by employing directed graphs, we would prevent some less reliable units from being interdependent.

## Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the author, without undue reservation.

## Author Contributions

ZD performed the research and wrote the manuscript.

## Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: failure evolution, fiber bundle model, switchable load sharing, simulations, small-world network, statistics

Citation: Domanski Z (2020) Spreading of Failures in Small-World Networks: A Connectivity-Dependent Load Sharing Fibre Bundle Model. *Front. Phys.* 8:552550. doi: .3389/fphy.2020.552550

Received: 16 April 2020; Accepted: 18 September 2020;

Published: 13 October 2020.

Edited by:

Ferenc Kun, University of Debrecen, HungaryReviewed by:

Soumyajyoti Biswas, SRM University, IndiaBikas K. Chakrabarti, Saha Institute of Nuclear Physics (SINP), India

Zoltan Neda, Babes-Bolyai University, Romania

Copyright © 2020 Domanski. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Zbigniew Domanski, zbigniew.domanski@im.pcz.pl