Disorder-induced Limited Path Percolation (original) (raw)
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Percolation Transitions Are Not Always Sharpened by Making Networks Interdependent
Physical Review Letters, 2011
We study a model for coupled networks introduced recently by Buldyrev et al., Nature 464, 1025Nature 464, (2010, where each node has to be connected to others via two types of links to be viable. Removing a critical fraction of nodes leads to a percolation transition that has been claimed to be more abrupt than that for uncoupled networks. Indeed, it was found to be discontinuous in all cases studied. Using an efficient new algorithm we verify that the transition is discontinuous for coupled Erdös-Rényi networks, but find it to be continuous for fully interdependent diluted lattices. In 2 and 3 dimension, the order parameter exponent β is larger than in ordinary percolation, showing that the transition is less sharp, i.e. further from discontinuity, than for isolated networks. Consequences for spatially embedded networks are discussed.
Limited Path Percolation in Complex Networks
Physical Review Letters, 2007
We study the stability of network communication after removal of q = 1 − p links under the assumption that communication is effective only if the shortest path between nodes i and j after removal is shorter than aℓ ij (a ≥ 1) where ℓ ij is the shortest path before removal. For a large class of networks, we find a new percolation transition atp c = (κ o − 1) (1−a)/a , where κ o ≡ k 2 / k and k is the node degree. Belowp c , only a fraction N δ of the network nodes can communicate, where δ ≡ a(1 − | log p|/ log (κ o − 1)) < 1, while abovep c , order N nodes can communicate within the limited path length aℓ ij . Our analytical results are supported by simulations on Erdős-Rényi and scale-free network models. We expect our results to influence the design of networks, routing algorithms, and immunization strategies, where short paths are most relevant.
Effect of Disorder Strength on Optimal Paths in Complex Networks
We study the transition between the strong and weak disorder regimes in the scaling properties of the average optimal path ᐉ opt in a disordered Erdős-Rényi (ER) random network and scale-free (SF) network. Each link i is associated with a weight i ϵ exp͑ar i ͒, where r i is a random number taken from a uniform distribution between 0 and 1 and the parameter a controls the strength of the disorder. We find that for any finite a, there is a crossover network size N * ͑a͒ at which the transition occurs. For N Ӷ N * ͑a͒ the scaling behavior of ᐉ opt is in the strong disorder regime, with ᐉ opt ϳ N 1/3 for ER networks and for SF networks with ജ 4, and ᐉ opt ϳ N ͑−3͒/͑−1͒ for SF networks with 3 Ͻ Ͻ 4. For N ӷ N * ͑a͒ the scaling behavior is in the weak disorder regime, with ᐉ opt ϳ ln N for ER networks and SF networks with Ͼ 3. In order to study the transition we propose a measure which indicates how close or far the disordered network is from the limit of strong disorder. We propose a scaling ansatz for this measure and demonstrate its validity. We proceed to derive the scaling relation between N * ͑a͒ and a. We find that N * ͑a͒ ϳ a 3 for ER networks and for SF networks with ജ 4, and N * ͑a͒ ϳ a ͑−1͒/͑−3͒ for SF networks with 3 Ͻ Ͻ 4.
Inducing effect on the percolation transition in complex networks
Nature Communications, 2013
Percolation theory concerns the emergence of connected clusters that percolate through a networked system. Previous studies ignored the effect that a node outside the percolating cluster may actively induce its inside neighbours to exit the percolating cluster. Here we study this inducing effect on the classical site percolation and K-core percolation, showing that the inducing effect always causes a discontinuous percolation transition. We precisely predict the percolation threshold and core size for uncorrelated random networks with arbitrary degree distributions. For low-dimensional lattices the percolation threshold fluctuates considerably over realizations, yet we can still predict the core size once the percolation occurs. The core sizes of real-world networks can also be well predicted using degree distribution as the only input. Our work therefore provides a theoretical framework for quantitatively understanding discontinuous breakdown phenomena in various complex systems.
Percolation phenomena in networks under random dynamics
2012
Abstract We show that the probability of source routing success in dynamic networks, where the link up-down dynamics is governed by a time-varying stochastic process, exhibit critical phase-transition (percolation) phenomena as a function of the end-to-end message latency per unit path length. We evaluate the probability of routing success on dynamic network (1D and 2D) lattices with links going up and down as per an arbitrary binary-valued stationary random process (such as a Markov process), in a source-routing framework.
Limited Percolation on Complex Networks
We study the stability of network communication after removal of q = 1 − p links under the assumption that communication is effective only if the shortest path between nodes i and j after removal is shorter than aℓij(a ≥ 1) where ℓij is the shortest path before removal. For a large class of networks, we find a new percolation transition atpc = (κo − 1) (1−a)/a , where κo ≡ k 2 / k and k is the node degree. Abovepc, order N nodes can communicate within the limited path length aℓij, while belowpc, N δ (δ < 1) nodes can communicate. Our analytical results are supported by simulations. We expect our results to influence network design, routing algorithms, and immunization strategies, where short paths are most relevant.
Simultaneous first- and second-order percolation transitions in interdependent networks
Physical Review E, 2014
In a system of interdependent networks, an initial failure of nodes invokes a cascade of iterative failures that may lead to a total collapse of the whole system in a form of an abrupt first order transition. When the fraction of initial failed nodes 1 − p reaches criticality, p = p c , the abrupt collapse occurs by spontaneous cascading failures. At this stage, the giant component decreases slowly in a plateau form and the number of iterations in the cascade, τ, diverges. The origin of this plateau and its increasing with the size of the system remained unclear. Here we find that simultaneously with the abrupt first order transition a spontaneous second order percolation occurs during the cascade of iterative failures. This sheds light on the origin of the plateau and on how its length scales with the size of the system. Understanding the critical nature of the dynamical process of cascading failures may be useful for designing strategies for preventing and mitigating catastrophic collapses.
We study the cascading failures in a system composed of two interdependent square lattice networks A and B placed on the same Cartesian plane, where each node in network A depends on a node in network B randomly chosen within a certain distance r from the corresponding node in network A and vice versa. Our results suggest that percolation for small r below r max % 8 (lattice units) is a second-order transition, and for larger r is a first-order transition. For r < r max , the critical threshold increases linearly with r from 0.593 at r ¼ 0 and reaches a maximum, 0.738 for r ¼ r max , and then gradually decreases to 0.683 for r ¼ 1. Our analytical considerations are in good agreement with simulations. Our study suggests that interdependent infrastructures embedded in Euclidean space become most vulnerable when the distance between interdependent nodes is in the intermediate range, which is much smaller than the size of the system.
Universal Behavior of Optimal Paths in Weighted Networks with General Disorder
We study the statistics of the optimal path in both random and scale-free networks, where weights w are taken from a general distribution Pw. We find that different types of disorder lead to the same universal behavior. Specifically, we find that a single parameter (S AL ÿ1= for d-dimensional lattices, and S AN ÿ1=3 for random networks) determines the distributions of the optimal path length, including both strong and weak disorder regimes. Here is the percolation connectivity exponent, and A depends on the percolation threshold and Pw. We show that for a uniform Pw, Poisson or Gaussian, the crossover from weak to strong does not occur, and only weak disorder exists.
The dynamic nature of percolation on networks with triadic interactions
Nature Communications
Percolation establishes the connectivity of complex networks and is one of the most fundamental critical phenomena for the study of complex systems. On simple networks, percolation displays a second-order phase transition; on multiplex networks, the percolation transition can become discontinuous. However, little is known about percolation in networks with higher-order interactions. Here, we show that percolation can be turned into a fully fledged dynamical process when higher-order interactions are taken into account. By introducing signed triadic interactions, in which a node can regulate the interactions between two other nodes, we define triadic percolation. We uncover that in this paradigmatic model the connectivity of the network changes in time and that the order parameter undergoes a period doubling and a route to chaos. We provide a general theory for triadic percolation which accurately predicts the full phase diagram on random graphs as confirmed by extensive numerical si...