AUTHOR=Domanski Zbigniew 

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

JOURNAL=Frontiers in Physics

VOLUME=8

YEAR=2020

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.552550

DOI=10.3389/fphy.2020.552550

ISSN=2296-424X

ABSTRACT=<p>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 <italic>N</italic> immobile units located in nodes of a network <inline-formula id="inf1"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">G</mml:mi></mml:math></inline-formula> and subjected to a growing external load <italic>F</italic> imposed uniformly on the units. Each unit, characterized by a load threshold <italic>δ</italic>, is classified as reliable or irreversibly failed, depending on whether <italic>δ</italic> is bigger, or respectively smaller, than the load felt by the unit. A pair of interdependent units is uniquely indicated by an edge of <inline-formula id="inf2"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">G</mml:mi></mml:math></inline-formula>. Initially all the units are reliable. When a unit fails, its load is distributed locally among interdependent neighbors if they are reliable, or is otherwise shared globally by all the reliable units. Because of the growing <italic>F</italic> 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 <inline-formula id="inf3"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="script">G</mml:mi><mml:mo>→</mml:mo><mml:mi mathvariant="normal">∅</mml:mi></mml:mrow></mml:math></inline-formula>. During the evolution, a subset <inline-formula id="inf4"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="script">G</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mo>⊂</mml:mo><mml:mi mathvariant="script">G</mml:mi></mml:mrow></mml:math></inline-formula> emerges that represents the limiting state of the system’s functionality when the smallest group of <inline-formula id="inf5"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> reliable units sustains the highest load <inline-formula id="inf6"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>. We concentrate on how the Fiber Bundle Model and switching Local-Global-Sharing conspire to drive the system toward <inline-formula id="inf7"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="script">G</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>. Specifically, we assume that <inline-formula id="inf8"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mi>δ</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:mrow><mml:mi mathvariant="script">G</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> are quenched-random quantities distributed uniformly over <inline-formula id="inf9"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mn>0,1</mml:mn></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> or governed by the Weibull distribution and networks <inline-formula id="inf10"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">G</mml:mi></mml:math></inline-formula> are the Watts-Strogatz “small-world” graphs with the rewiring probability <italic>p</italic> that characterizes possible rearrangements of edges in <inline-formula id="inf11"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">G</mml:mi></mml:math></inline-formula>. We have identified a range of values of <italic>p</italic>, where the mean highest load <inline-formula id="inf12"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:mo>〈</mml:mo><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow><mml:mo>〉</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mi>N</mml:mi></mml:mrow></mml:mrow></mml:math></inline-formula>, supported by reliable units, scales linearly with the average global-clustering coefficient of the host network. Similar scaling holds for <inline-formula id="inf13"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>〈</mml:mo><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow><mml:mo>〉</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> and <inline-formula id="inf14"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>〈</mml:mo><mml:mrow><mml:mrow><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:mrow><mml:mo>〉</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>. We have also found that in the large <italic>N</italic> limit <inline-formula id="inf15"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>f</mml:mi><mml:mi>c</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>→</mml:mo><mml:msubsup><mml:mi>f</mml:mi><mml:mi>c</mml:mi><mml:mi>∞</mml:mi></mml:msubsup><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math></inline-formula>, for all values of <italic>p</italic> and both considered distributions of <inline-formula id="inf16"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mi>δ</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:mrow><mml:mi mathvariant="script">G</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>. The symbol <inline-formula id="inf17"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>〈</mml:mo><mml:mo>…</mml:mo><mml:mo>〉</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula> represents averaging over <inline-formula id="inf18"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mi>δ</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:mrow><mml:mi mathvariant="script">G</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and a suitable ensemble of networks <inline-formula id="inf19"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>{</mml:mo><mml:mi mathvariant="script">G</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math></inline-formula>.</p>