Optimal Path Cracks in Complex Networks

As we can see in nature, we are surrounded by materials from which their structures have some kind of disorder that may be related to microscopic voids, detachments and even structural composition, which makes them susceptible to fractures that depend on the degree of disorder significantly wherein the material is given. For this reason optimal paths play a fundamental role in physical numerous applications ranging from random polymers to brittle fracture, from the flow through porous media to information propagation. Following the study of optimal path cracks (OPC) on regular square lattices, here we investigate the same problem in Complex Networks. In this problem we associate to each site a determined energy. The optimum path is defined as the one among all possible paths that crosses the system which has the minimum cost, namely the sum of the energies along the path. Once the optimum path is determined, at each step, one blocks its site with highest energy, and then a new optimal path is calculated. This procedure is repeated until there is a set of blocked sites forming a macroscopic fracture which connects the opposite sides of the system. As observed in the work on regular square lattices, the fractured system studied here also presents different behaviors depending on the level of disorder, namely weak, moderated and strong disorder intensities. In the regime of weak and moderated disorder, while the density of removed sites in the system does not depend of the size L in the case of regular lattices, in the regime of high disorder the density becomes substantially dependent on L. We did the same type of study for Complex Networks. In this case, each new site is connected with previous ones. As in the previous work, we observe that the density of removed sites presents a similar behavior. Moreover, a new result is obtained, i.e., we analyze the dependency of the disorder with the attachment parameter m.