Explosive Percolation Transition Research Articles

Explosive rigidity percolation in kirigami

Controlling the connectivity and rigidity of kirigami, i.e. the process of cutting paper to deploy it into an articulated system, is critical in the manifestations of kirigami in art, science and technology, as it provides the resulting metamaterial with a range of mechanical and geometric properties. Here, we combine deterministic and stochastic approaches for the control of rigidity in kirigami using the power of k choices, an approach borrowed from the statistical mechanics of explosive percolation transitions. We show that several methods for rigidifying a kirigami system by incrementally changing either the connectivity or the rigidity of individual components allow us to control the nature of the explosive transition by a choice of selection rules. Our results suggest simple lessons for the design of mechanical metamaterials.

Random adsorption process of linear k-mers on square lattices under the Achlioptas process.

We study the explosive percolation with k-mer random sequential adsorption (RSA) process. We consider both the Achlioptas process (AP) and the inverse Achlioptas process (IAP), in which giant cluster formation is prohibited and accelerated, respectively. By employing finite-size scaling analysis, we confirm that the percolation transitions are continuous, and thus we calculate the percolation threshold and critical exponents. This allows us to determine the universality class of the k-mer explosive percolation transition. Interestingly, the numerical simulation suggests that the universality class of the explosive percolation transition with the AP alters when the k-mer size changes. In contrast, the universality class of the transition with the IAP is independent of k, but it differs from that of the RSA without the IAP.

Scaling behavior of information entropy in explosive percolation transitions.

An explosive percolation transition is the abrupt emergence of a giant cluster at a threshold caused by a suppression of the growth of large clusters. In this paper, we consider the information entropy of the cluster-size distribution, which is the probability distribution for the size of a randomly chosen cluster. It has been reported that information entropy does not reach its maximum at the threshold in explosive percolation models, a result seemingly contrary to other previous results that the cluster-size distribution shows power-law behavior and the cluster-size diversity (number of distinct cluster sizes) is maximum at the threshold. Here, we show that this phenomenon is due to the fact that the scaling form of the cluster-size distribution is given differently below and above the threshold. We also establish the scaling behaviors of the first and second derivatives of the information entropy near the threshold to explain why the first derivative has a negative minimum at the threshold and the second derivative diverges negatively (positively) at the left (right) limit of the threshold, as predicted through previous simulation.

Diverse types of percolation transitions

Percolation has long served as a model for diverse phenomena and systems. The percolation transition, that is, the formation of a giant cluster on a macroscopic scale, is known as one of the most robust continuous transitions. Recently, however, many abrupt percolation transitions have been observed in complex systems. To illustrate such phenomena, considerable effort has been made to introduce models and construct theoretical frameworks for explosive, discontinuous, and hybrid percolation transitions. Experimental results have also been reported. In this review article, we describe such percolation models, their critical behaviors and universal features, and real-world phenomena.

Publisher's Note: Explosive percolation transitions in growing networks [Phys. Rev. E 93, 032316 (2016)

This corrects the article DOI: 10.1103/PhysRevE.93.032316.

Explosive percolation transitions in growing networks.

Recent extensive studies of the explosive percolation (EP) model revealed that the EP transition is second order with an extremely small value of the critical exponent β associated with the order parameter. This result was obtained from static networks, in which the number of nodes in the system remains constant during the evolution of the network. However, explosive percolating behavior of the order parameter can be observed in social networks, which are often growing networks, where the number of nodes in the system increases as dynamics proceeds. However, extensive studies of the EP transition in such growing networks are still missing. Here we study the nature of the EP transition in growing networks by extending an existing growing network model to a general case in which m node candidates are picked up in the Achiloptas process. When m = 2, this model reduces to the existing model, which undergoes an infinite-order transition. We show that when m ≥ 3, the transition becomes second order due to the suppression effect against the growth of large clusters. Using the rate-equation approach and performing numerical simulations, we also show that the exponent β decreases algebraically with increasing m, whereas it does exponentially in a corresponding static random network model. Finally, we find that the hyperscaling relations hold but in different forms.

Anomalous critical and supercritical phenomena in explosive percolation

Explosive Percolation describes the abrupt onset of large-scale connectivity that results from a simple random process designed to delay the onset of the transition on an underlying random network or lattice. Explosive percolation transitions exhibit an array of novel universality classes and supercritical behaviors including a stochastic sequence of discontinuous transitions, multiple giant components, and lack of self-averaging. Many mechanisms that give rise to explosive percolation have been discovered, including overtaking, correlated percolation, and evolution on hierarchical lattices. Many connections to real-world systems, ranging from social networks to nanotubes, have been identified and explosive percolation is an emerging paradigm for modeling these systems as well as the consequences of small interventions intended to delay phase transitions. This review aims to synthesize existing results on explosive percolation and to identify fruitful directions for future research.

Inverting the Achlioptas rule for explosive percolation.

In the usual Achlioptas processes the smallest clusters of a few randomly chosen ones are selected to merge together at each step. The resulting aggregation process leads to the delayed birth of a giant cluster and the so-called explosive percolation transition showing a set of anomalous features. We explore a process with the opposite selection rule, in which the biggest clusters of the randomly chosen ones merge together. We develop a theory of this kind of percolation based on the Smoluchowsky equation, find the percolation threshold, and describe the scaling properties of this continuous transition, namely, the critical exponents and amplitudes, and scaling functions. We show that, qualitatively, this transition is similar to the ordinary percolation one, though occurring in less connected systems.

Solution of the explosive percolation quest. II. Infinite-order transition produced by the initial distributions of clusters.

We describe the effect of power-law initial distributions of clusters on ordinary percolation and its generalizations, specifically, models of explosive percolation processes based on local optimization. These aggregation processes were shown to exhibit continuous phase transitions if the evolution starts from a set of disconnected nodes. Since the critical exponents of the order parameter in explosive percolation transitions turned out to be very small, these transitions were first believed to be discontinuous. In this article we analyze the evolution starting from clusters of nodes whose sizes are distributed according to a power law. We show that these initial distributions change dramatically the position and order of the phase transitions in these problems. We find a particular initial power-law distribution producing a peculiar effect on explosive percolation, namely, before the emergence of the percolation cluster, the system is in a "critical phase" with an infinite generalized susceptibility. This critical phase is absent in ordinary percolation models with any power-law initial conditions. The transition from the critical phase is an infinite-order phase transition, which resembles the scenario of the Berezinskii-Kosterlitz-Thouless phase transition. We obtain the critical singularity of susceptibility at this peculiar infinite-order transition in explosive percolation. It turns out that susceptibility in this situation does not obey the Curie-Weiss law.

Critical exponents of the explosive percolation transition.

In a new type of percolation phase transition, which was observed in a set of nonequilibrium models, each new connection between vertices is chosen from a number of possibilities by an Achlioptas-like algorithm. This causes preferential merging of small components and delays the emergence of the percolation cluster. First simulations led to a conclusion that a percolation cluster in this irreversible process is born discontinuously, by a discontinuous phase transition, which results in the term "explosive percolation transition." We have shown that this transition is actually continuous (second order) though with an anomalously small critical exponent of the percolation cluster. Here we propose an efficient numerical method enabling us to find the critical exponents and other characteristics of this second-order transition for a representative set of explosive percolation models with different number of choices. The method is based on gluing together the numerical solutions of evolution equations for the cluster size distribution and power-law asymptotics. For each of the models, with high precision, we obtain critical exponents and the critical point.

Growth dominates choice in network percolation

The onset of large-scale connectivity in a network (i.e., percolation) often has a major impact on the function of the system. Traditionally, graph percolation is analyzed by adding edges to a fixed set of initially isolated nodes. Several years ago, it was shown that adding nodes as well as edges to the graph can yield an infinite order transition, which is much smoother than the traditional second-order transition. More recently, it was shown that adding edges via a competitive process to a fixed set of initially isolated nodes can lead to a delayed, extremely abrupt percolation transition with a significant jump in large but finite systems. Here we analyze a process that combines both node arrival and edge competition. If started from a small collection of seed nodes, we show that the impact of node arrival dominates: although we can significantly delay percolation, the transition is of infinite order. Thus, node arrival can mitigate the trade-off between delay and abruptness that is characteristic of explosive percolation transitions. This realization may inspire new design rules where network growth can temper the effects of delay, creating opportunities for network intervention and control.

Understanding of Explosive Percolation Transitions in a Unified Framework

Understanding of Explosive Percolation Transitions in a Unified Framework

Numerical simulations of the phase transition property of the explosive percolation model on Erds Rnyi random network

Based on the modified Newman and Ziff algorithm combined with the finite-size scaling theory, in this present work we analytically study the phase transition property of the explosive percolation model induced by Achlioptas process on the Erds Rnyi random network via numerical simulations for the basic percolation quantities including the order parameter, the average cluster size, the moments, the standard deviation and the cluster heterogeneity. It is explicitly shown that all these relevant quantities display a typical power-law scaling behavior, which is the characteristic of continuous phase transition at the percolation threshold despite the fact that the order parameter presents a certain feature of discontinuous transition at the same time. Strictly, the explosive percolation transition during the Erds Rnyi random network is a singular transition, which means that it is neither a standard discontinuous phase transition nor the continuous transition in the regular random percolation model.

Suppression Effect on Explosive Percolation

Percolation transitions (PTs) of networks, leading to the formation of a macroscopic cluster, are conventionally considered to be continuous transitions. However, a modified version of the classical random graph model was introduced in which the growth of clusters was suppressed, and a PT occurs explosively at a delayed transition point. Whether the explosive PT is indeed discontinuous or continuous becomes controversial. Here, we show that the behavior of the explosive PT depends on detailed dynamic rules. Thus, when dynamic rules are designed to suppress the growth of all clusters, the discontinuity of the order parameter tends to a finite value as the system size increases, indicating that the explosive PT could be discontinuous.

Continuity of the explosive percolation transition

The explosive percolation problem on the complete graph is investigated via extensive numerical simulations. We obtain the cluster-size distribution at the moment when the cluster size heterogeneity becomes maximum. The distribution is found to be well described by the power-law form with the decay exponent τ=2.06(2), followed by a hump. We then use the finite-size scaling method to make all the distributions at various system sizes up to N=2(37) collapse perfectly onto a scaling curve characterized solely by the single exponent τ. We also observe that the instant of that collapse converges to a well-defined percolation threshold from below as N→∞. Based on these observations, we show that the explosive percolation transition in the model should be continuous, contrary to the widely spread belief of its discontinuity.

Explosive percolation in graphs

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 Synchronization Transitions in Scale-Free Networks

Explosive collective phenomena have attracted much attention since the discovery of an explosive percolation transition. In this Letter, we demonstrate how an explosive transition shows up in the synchronization of scale-free networks by incorporating a microscopic correlation between the structural and the dynamical properties of the system. The characteristics of the explosive transition are analytically studied in a star graph reproducing the results obtained in synthetic networks. Our findings represent the first abrupt synchronization transition in complex networks and provide a deeper understanding of the microscopic roots of explosive critical phenomena.

Criterion for explosive percolation transitions on complex networks

In a recent Letter, Friedman and Landsberg discussed the underlying mechanism of explosive phase transitions on complex networks [Phys. Rev. Lett. 103, 255701 (2009).]. This Brief Report presents a modest though more insightful extension of their arguments. We discuss the implications of their results on the cluster-size distribution and deduce that, under general conditions, the percolation transition will be explosive if the mean number of nodes per cluster diverges in the thermodynamic limit and prior to the transition threshold. In other words, if upon increase of the network size n the amount of clusters in the network does not grow proportionally to n, the percolation transition is explosive. Simulations and analytical calculations on various models support our findings.

Watersheds and Explosive percolation

The recent work by Achlioptas, D'Souza, and Spencer opened up the possibility of obtaining a discontinuous (explosive) percolation transition by changing the stochastic rule of bond occupation. Despite the active research on this subject, several questions still remain open about the leading mechanism and the properties of the system. We review the largest cluster and the Gaussian models recently introduced. We show that, to obtain a discontinuous transition it is solely necessary to control the size of the largest cluster, suppressing the growth of a cluster di_ering significantly, in size, from the average one. As expected for a discontinuous transition, a Gaussian cluster-size distribution and compact clusters are obtained. The surface of the clusters is fractal, with the same fractal dimension of the watershed line.

Explosive Percolation Transition is Actually Continuous

Recently a discontinuous percolation transition was reported in a new "explosive percolation" problem for irreversible systems [D. Achlioptas, R. M. D'Souza, and J. Spencer, Science 323, 1453 (2009)] in striking contrast to ordinary percolation. We consider a representative model which shows that the explosive percolation transition is actually a continuous, second order phase transition though with a uniquely small critical exponent of the percolation cluster size. We describe the unusual scaling properties of this transition and find its critical exponents and dimensions.