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

Abstract

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 G and subjected to a growing external load 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 G. 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 F and the loads that are transferred according to such a seesaw switch between the local and global sharing rules (sLGS), a set of nodes, that holds the reliable units, evolves as G → ∅. During the evolution, a subset G c ⊂ G emerges that represents the limiting state of the system's functionality when the smallest group of n c reliable units sustains the highest load F c. We concentrate on how the Fiber Bundle Model and switching Local-Global-Sharing conspire to drive the system toward G c. Specifically, we assume that {δ} G are quenched-random quantities distributed uniformly over (0, 1) or governed by the Weibull distribution and networks G are the Watts-Strogatz "small-world" graphs with the rewiring probability p that characterizes possible rearrangements of edges in G. We have identified a range of values of p, where the mean highest load f c (N) 〈F c 〉/N, supported by reliable units, scales linearly with the average globalclustering coefficient of the host network. Similar scaling holds for 〈n c 〉 and 〈F c /n c 〉. We have also found that in the large N limit f c (N) → f ∞ c > 0, for all values of p and both considered distributions of {δ} G. The symbol 〈. .. 〉 represents averaging over {δ} G and a suitable ensemble of networks {G}.

Scaling laws of failure dynamics on complex networks

“…The FBM is one of the most important modelling approaches to the cascading failure dynamics, which was originally developed to study the gradual fracturing of heterogeneous materials. However, due to the generality of its cascading mechanism, FBMs have gained applications in diverse fields including the modelling of cascading blackouts of power grids [22][23][24] , or the breakdown of urban traffic networks [25][26][27] , and flow channels 26 .…”

A neural network regression model for estimating the lifespan of a Fibre Bundle

“…Other theoretical works have focused on understanding brittle to quasi-brittle transitions when disordered materials are loaded [14]. The fundamentals of FBM were mapped to study questions related to microeconomics, inter-bank networks and power grid using graph theory and random networks [15]. Predicting imminent failures of Fibre Bundles and maximising their strength is an active area of research.…”

Survivability of Suddenly Loaded Arrays of Micropillars

“…It turns out that the ultimate strength of the system crucially depends on how such a transfer happens [ 15 , 16 , 17 ], what the topology of interconnections among components is [ 18 ], what the loading conditions are [ 19 , 20 ], and whether the loading is applied step-wise, suddenly [ 21 ] or cyclically [ 22 , 23 ]. This means that a given ensemble of interconnected components, working together along with an established rule of load transfer between failed and intact components, may either sustain an externally applied load or become entirely destroyed if an unsuitable rule governs the load transfer or interconnections switch to unfavourable topology [ 18 , 24 ].…”