Impact of Single Links in Competitive Percolation -- How complex networks grow under competition (original) (raw)

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.

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.

Explosive percolation in graphs

Journal of Physics: Conference Series, 2011

Percolation is perhaps the simplest example of a process exhibiting a phase transition and one of the most studied phenomena in statistical physics. The percolation transition is continuous if sites/bonds are occupied independently with the same probability. However, alternative rules for the occupation of sites/bonds might affect the order of the transition. A recent set of rules proposed by Achlioptas et al. [Science 323, 1453 (2009)], characterized by competitive link addition, was claimed to lead to a discontinuous connectedness transition, named "explosive percolation". In this work we survey a numerical study of the explosive percolation transition on various types of graphs, from lattices to scale-free networks, and show the consistency of these results with recent analytical work showing that the transition is actually continuous.

Explosive Percolation in Scale-Free Networks

Physical Review Letters, 2009

We study scale-free networks constructed via a cooperative Achlioptas growth process. Links between nodes are introduced in order to produce a scale-free graph with given exponent λ for the degree distribution, but the choice of each new link depends on the mass of the clusters that this link will merge. Networks constructed via this biased procedure show a percolation transition which strongly differs from the one observed in standard percolation, where links are introduced just randomly. The different growth process leads to a phase transition with a non-vanishing percolation threshold already for λ > λc ∼ 2.2. More interestingly, the transition is continuous when λ ≤ 3 but becomes discontinuous when λ > 3. This may have important consequences both for the structure of networks and for the dynamics of processes taking place on them.

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

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.

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...

Double percolation phase transition in clustered complex networks

We perform an extensive numerical study of the effects of clustering on the structural properties of complex networks. We observe that strong clustering in heterogeneous networks induces the emergence of a core-periphery organization that has a critical effect on their percolation properties. In such situation, we observe a novel double phase transition, with an intermediate phase where only the core of the network is percolated, and a final phase where the periphery percolates regardless of the core. Interestingly, strong clustering makes simultaneously the core more robust and the periphery more fragile. These phenomena are also found in real complex networks.

Infinite-order percolation and giant fluctuations in a protein interaction network

Physical Review E, 2002

We investigate a model protein interaction network whose links represent interactions between individual proteins. This network evolves by the functional duplication of proteins, supplemented by random link addition to account for mutations. When link addition is dominant, an infiniteorder percolation transition arises as a function of the addition rate. In the opposite limit of high duplication rate, the network exhibits giant structural fluctuations in different realizations. For biologically-relevant growth rates, the node degree distribution has an algebraic tail with a peculiar rate dependence for the associated exponent.

Agglomerative percolation in two dimensions

EPL (Europhysics Letters), 2012

We study a process termed agglomerative percolation (AP) in two dimensions. Instead of adding sites or bonds at random, in AP randomly chosen clusters are linked to all their neighbors. As a result the growth process involves a diverging length scale near a critical point. Picking target clusters with probability proportional to their mass leads to a runaway compact cluster. Choosing all clusters equally leads to a continuous transition in a new universality class for the square lattice, while the transition on the triangular lattice has the same critical exponents as ordinary percolation. PACS numbers: 64.60.ah, 68.43.Jk, 89.75.Da Percolation is a pervasive concept in statistical physics and an important branch of mathematics [1]. It typifies the emergence of long range connectivity in many systems such as the flow of liquids through porous media [2], transport in disordered media [3], spread of disease in populations [4], resilience of networks to attack [5], formation of gels [6] and even of social groups [7]