Critical percolation in self-organized media: A case study on random directed networks (original) (raw)

Self-organized critical directed percolation

Physica A: Statistical and Theoretical Physics, 1996

We introduce and study a dynamic transport model exhibiting Self-Organized Criticality. The novel concepts of our model are the probabilistic propagation of activity and unbiased random repartition of energy among the active site and its nearest neighbors. For space dimensionality d ≥ 2 we argue that the model is related to d + 1 dimensional directed percolation, with time interpreted as the preferred direction. 05.40+j, 64.60Ht, 64.60Ak, 05.70Ln

Dynamic Networks and Directed Percolation

We introduce a model for dynamic networks, where the links or the strengths of the links change over time. We solve the model by mapping dynamic networks to the problem of directed percolation, where the direction corresponds to the time evolution of the network. We show that the dynamic network undergoes a percolation phase transition at a critical concentration pc, that decreases with the rate r at which the network links are changed. The behavior near criticality is universal and independent of r. We find that for dynamic random networks fundamental laws are changed. i) The size of the giant component at criticality scales with the network size N for all values of r, rather than as N 2/3 in static networks. ii) In the presence of a broad distribution of disorder, the optimal path length between two nodes in a dynamic network scales as N 1/2 , compared to N 1/3 in a static network.

Percolation in directed scale-free networks

Physical Review E, 2002

Many complex networks in nature have directed links, a property that affects the network's navigability and large-scale topology. Here we study the percolation properties of such directed scale-free networks with correlated in and out degree distributions. We derive a phase diagram that indicates the existence of three regimes, determined by the values of the degree exponents. In the first regime we regain the known directed percolation mean field exponents. In contrast, the second and third regimes are characterized by anomalous exponents, which we calculate analytically. In the third regime the network is resilient to random dilution, i.e., the percolation threshold is pc → 1. 02.50.Cw, 05.40.a, 05.50.+q, 64.60.Ak

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.

Weakly explosive percolation in directed networks

Physical Review E, 2013

Percolation, the formation of a macroscopic connected component, is a key feature in the description of complex networks. The dynamical properties of a variety of systems can be understood in terms of percolation, including the robustness of power grids and information networks, the spreading of epidemics and forest fires, and the stability of gene regulatory networks. Recent studies have shown that if network edges are added "competitively" in undirected networks, the onset of percolation is abrupt or "explosive." The unusual qualitative features of this phase transition have been the subject of much recent attention. Here we generalize this previously studied network growth process from undirected networks to directed networks and use finite-size scaling theory to find several scaling exponents. We find that this process is also characterized by a very rapid growth in the giant component, but that this growth is not as sudden as in undirected networks.

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.

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.

Network Anatomy Controlling Abrupt-like Percolation Transition

Scientific Reports, 2017

We virtually dissect complex networks in order to understand their internal structure, just as doctors do with the bodies of animals. Our novel method classifies network links into four categories: bone, fat, cartilage, and muscle, based on network connectivity. We derive an efficient percolation strategy from this new viewpoint of network anatomy, which enables abrupt-like percolation transition through removal of a small amount of cartilage links, which play a crucial role in network connectivity. Furthermore, we find nontrivial scaling laws in the relationships between four types of links in each cluster and evaluate power exponents, which characterize network structures as seen in the real largescale network of trading business firms and in the Erdős-Rényi network. Finally, we observe changes in the transition point for random bond percolation process, demonstrating that the addition of muscle links enhances network robustness, while fat links are irrelevant. These findings aid in controlling the percolation transition for an arbitrary network. Different networks, such as human and business relationship networks, and power networks, are everywhere in our world 1-4. Most complex networks in social systems can be categorized as having scale-free and small-world properties 5, 6. Many methods quantifying such inhomogeneous networks have been proposed from various fields including biology, information science and physics 4, 6-9. It is important to understand how a network can be made robust under attack, including methods for intentional removal of nodes and links because such networks form the basis of the society and economy 10-14. Thus, it is necessary to determine what elements contribute to reinforcing network connectivity, and to find practical ways to enhance robustness of the system. Percolation theory has been studied in the fields of mathematics and physics to clarify macroscopic connectivity from a microscopic viewpoint 4, 15-20. Specifically, the percolation transition properties of complex networks have been attracting the attention of many scientists since the proposal of the scale-free network model 5. It has been reported that a scale-free network is fragile against targeted attacks, but robust against random failures 10. Furthermore, recent percolation models have been extended to explain a discontinuous percolation transition (DPT) 2, 18, 19, 21, 22. It has been suggested that a real power network carries the risk of massive blackouts due to cascading failures in a multi-layered network 2 , and discontinuity in the explosive percolation model has attracted great interest in recent years due to its simple yet diverse characteristics 23-25. Classification of nodes and links, such as community extraction, is also an important field of study, and comprehensive graphical expressions have become available for this application 26-30. In the theory of percolation transition for square lattices, there are studies classifying "backbone links" based on significant contribution to overall connectivity 31-33 , however, such classifications have not been yet introduced to percolation study in complex networks. In this study, we further generalize the anatomical concept of a "backbone" by introducing a novel "network anatomy", which virtually dissects any given complex network and classifies its links based on their contribution to network connectivity. In the Method section, all network links are classified into four categories: bone, fat, cartilage, and muscle links, as an analogy for the anatomy of animal bodies. In the Result 1 section, we show that a percolation strategy assembled from these link categories enables the abrupt-like transition in a large-scale real network, as well as artificial networks. Nontrivial scaling laws are observed among the four classified link types and scaling exponents that characterize a network are shown in the Result 2 section. In the Result 3 section, we observe shifts of the percolation transition point caused by doping fat and muscle links to clarify the functional

Inducing Self-Organized Criticality in a network toy model by neighborhood assortativity

2016

Complex networks are a recent type of frameworks used to study complex systems with many interacting elements, such as Self-Organized Criticality (SOC). The network node's tendency to link to other nodes of similar type is characterized by assortative mixing. Real networks exhibit assortative mixing by vertex degree, however typical random network models, such as Erdos-Renyi or Barabasi-Albert, show no assortative arrangements. In this paper we introduce the neighborhood assortativity notion, as the tendency of a node to belong to a community (its neighborhood) showing an average property similar to its own. Imposing neighborhood assortative mixing by degree in a network toy model, SOC dynamics can be found. The long-range correlations resulting from the criticality have been characterized by means of fluctuation analysis and show an anticorrelation in the node's activity. The model contains only one parameter and its statistics plots for different values of the parameter ca...

Self-Organized-Critical network dynamics

AIP Conference Proceedings, 2005

We propose a simple model that aims at describing, in a stylized manner, how local breakdowns due to imbalances or congestion propagate in real dynamical networks. The model converges to a self-organized critical stationary state in which the network shapes itself as a consequence of avalanches of rewiring processes. Depending on the model's specification, we obtain either single-scale or scale-free networks. We characterize in detail the relation between the statistical properties of the network and the nature of the critical state, by computing the critical exponents. The model also displays a nontrivial, sudden collapse to a complete network.