Investigation of the rupture of a synthetic tape within the framework of the percolation theory

Within the framework of the percolation theory (bond problem), a new model of breaking a complex synthetic tape is proposed as a continuous-type phase transition when the state jump is zero. The percolation threshold and accompanying characteristics are calculated for the model of rupture of a synthetic reinforced tape when flowing along the first and second neighbours. The knots of the tape form a strip of a square lattice, the width of which is fixed. All nodes are intact and cannot be damaged, links (tape threads) can be intact and broken (blocked). The dependences of the percolation threshold in the bond problem and the relative deviation of the threshold from the ribbon length are calculated. It is proved that for the simplest model of one-dimensional percolation with percolation along the nearest neighbours (the problem of nodes), the percolation threshold in the thermodynamic limit is equal to unity. It is shown that, with an accuracy of 10%, the percolation threshold for a sufficiently long ribbon is equal to unity. This indicates that the system is quasi-one-dimensional. Thus, using the method of computer simulation, the percolation threshold, root-mean-square and relative threshold deviations were calculated. The critical susceptibility index was also calculated. In contrast to the usual percolation problem, in the proposed model it makes sense to consider only the region above the percolation threshold. The proposed model can be generalized to the case when nodes are also damaged (blocked), then we come to a mixed percolation model, which is supposed to be considered in the future.

Entanglement generation in a quantum network at distance-independent rate

We develop a protocol for entanglement generation in the quantum internet that allows a repeater node to use n-qubit Greenberger-Horne-Zeilinger (GHZ) projective measurements that can fuse n successfully-entangled links, i.e., two-qubit entangled Bell pairs shared across n network edges, incident at that node. Implementing n-fusion, for n≥3, is in principle not much harder than 2-fusions (Bell-basis measurements) in solid-state qubit memories. If we allow even 3-fusions at the nodes, we find---by developing a connection to a modified version of the site-bond percolation problem---that despite lossy (hence probabilistic) link-level entanglement generation, and probabilistic success of the fusion measurements at nodes, one can generate entanglement between end parties Alice and Bob at a rate that stays constant as the distance between them increases. We prove that this powerful network property is not possible to attain with any quantum networking protocol built with Bell measurements and multiplexing alone. We also design a two-party quantum key distribution protocol that converts the entangled states shared between two nodes into a shared secret, at a key generation rate that is independent of the distance between the two parties.

Algorithm for Identification of Infinite Clusters Based on Minimal Finite Automaton

We propose a finite automaton based algorithm for identification of infinite clusters in a 2D rectangular lattice with L=X×Y cells. The algorithm counts infinite clusters and finds one path per infinite cluster in a single pass of the finite automaton. The finite automaton is minimal according to the number of states among all the automata that perform such task. The correctness and efficiency of the algorithm are demonstrated on a planar percolation problem. The algorithm has a computational complexity of O(L) and could be appropriate for efficient data flow implementation.

Robustness and modular structure in networks

Complex networks have recently attracted much interest due to their prevalence in nature and our daily lives (Vespignani, 2009; Newman, 2010). A critical property of a network is its resilience to random breakdown and failure (Albert et al., 2000; Cohen et al., 2000; Callaway et al., 2000; Cohen et al., 2001), typically studied as a percolation problem (Stauffer & Aharony, 1994; Achlioptas et al., 2009; Chen & D'Souza, 2011) or by modeling cascading failures (Motter, 2004; Buldyrev et al., 2010; Brummitt, et al. 2012). Many complex systems, from power grids and the Internet to the brain and society (Colizza et al., 2007; Vespignani, 2011; Balcan & Vespignani, 2011), can be modeled using modular networks comprised of small, densely connected groups of nodes (Girvan & Newman, 2002). These modules often overlap, with network elements belonging to multiple modules (Palla et al. 2005; Ahn et al. 2010). Yet existing work on robustness has not considered the role of overlapping, modular structure. Here we study the robustness of these systems to the failure of elements. We show analytically and empirically that it is possible for the modules themselves to become uncoupled or non-overlapping well before the network disintegrates. If overlapping modular organization plays a role in overall functionality, networks may be far more vulnerable than predicted by conventional percolation theory.