Spreading of Failures in Small-World Networks: A Connectivity-Dependent Load Sharing Fibre Bundle Model - Domanski, Zbigniew (Frontiers in Physics, 2020)
A rich variety of multicomponent systems operating under parallel loading may be mapped on and then examined by employing a family of the Fibre Bundle Models (FBM). 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 d, is classified as reliable or irreversibly failed, depending on whether d 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 neighbours 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 see-saw switch between the local and global sharing rules (sLGS), a set of nodes, that holds the reliable units, evolves as G -> 0. During the evolution, a subset Gc of G emerges that represents the limiting state of the system's functionality when the smallest group of Nc reliable units sustains the highest load Fc. We concentrate on how the FBM and sLGS conspire to drive the system towards Gc. Specifically, we assume that load thresholds 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 fc(N)=