Continuous Percolation Transition Research Articles

Aftermath epidemics: Percolation on the sites visited by generalized random walks.

We study percolation on the sites of a finite lattice visited by a generalized random walk of finite length with periodic boundary conditions. More precisely, consider Levy flights and walks with finite jumps of length >1 [like Knight's move random walks (RWs) in two dimensions and generalized Knight's move RWs in 3D]. In these walks, the visited sites do not form (as in ordinary RWs) a single connected cluster, and thus percolation on them is nontrivial. The model essentially mimics the spreading of an epidemic in a population weakened by the passage of some devastating agent-like diseases in the wake of a passing army or of a hurricane. Using the density of visited sites (or the number of steps in the walk) as a control parameter, we find a true continuous percolation transition in all cases except for the 2D Knight's move RWs and Levy flights with Levy parameter σ≥2. For 3D generalized Knight's move RWs, the model is in the universality class of pacman percolation, and all critical exponents seem to be simple rationals, in particular, β=1. For 2D Levy flights with 0<σ<2, scale invariance is broken even at the critical point, which leads at least to very large corrections in finite-size scaling, and even very large simulations were unable to unambiguously determine the critical exponents.

Induced diffusion percolation model: Examining the role of superactive nodes in the diffusion of innovations

Building on a recent study on scientific collaboration networks, we propose an induced diffusion percolation model that brings superactive nodes into focus. Defined as active nodes surrounded by at least k active or superactive neighbors, superactive nodes play a key role in innovation diffusion by inducing their neighbors to adopt an innovation. We investigate the induced diffusion percolation model using the modified Newman–Ziff algorithm on two-dimensional lattices (square and triangular lattices) and regular random networks with and without clustering. The induction by superactive nodes leads to a first-order percolation phase transition in two-dimensional lattices and a double transition – a continuous percolation transition followed by a discontinuous jump of the order parameter of the largest cluster’s strength – in regular random networks. Whereas clustering increases the percolation threshold in the classical percolation model on regular random networks, it decreases the critical initial activation probability that triggers a discontinuous jump of the induced activation.

Aggregation of self-propelled particles with sensitivity to local order

We study a system of self-propelled particles (SPPs) in which individual particles are allowed to switch between a fast aligning and a slow nonaligning state depending upon the degree of the alignment in the neighborhood. The switching is modeled using a threshold for the local order parameter. This additional attribute gives rise to a mixed phase, in contrast to the ordered phases found in clean SPP systems. As the threshold is increased from zero, we find the sudden appearance of clusters of nonaligners. Clusters of nonaligners coexist with moving clusters of aligners with continual coalescence and fragmentation. The behavior of the system with respect to the clustering of nonaligners appears to be very different for values of low and high global densities. In the low density regime, for an optimal value of the threshold, the largest cluster of nonaligners grows in size up to a maximum that varies logarithmically with the total number of particles. However, on further increasing the threshold the size decreases. In contrast, for the high density regime, an initial abrupt rise is followed by the appearance of a giant cluster of nonaligners. The latter growth can be characterized as a continuous percolation transition. In addition, we find that the speed differences between aligners and nonaligners is necessary for the segregation of aligners and nonaligners.

Explosive dismantling of two-dimensional random lattices under betweenness centrality attacks

In the present paper, we study the robustness of two-dimensional random lattices (Delaunay triangulations) under attacks based on betweenness centrality. Together with the standard definition of this centrality measure, we employ a range-limited approximation known as ℓ-betweenness, where paths having more than ℓ steps are ignored. For finite ℓ, the attacks produce continuous percolation transitions that belong to the universality class of random percolation. On the other hand, the attack under the full range betweenness induces a discontinuous transition that, in the thermodynamic limit, occurs after removing a sub-extensive amount of nodes. This behavior is recovered for ℓ-betweenness if the cutoff is allowed to scale with the linear length of the network faster than ℓ∼L0.91. Our results suggest that betweenness centrality encodes information on network robustness at all scales, and thus cannot be approximated using finite-ranged calculations without losing attack efficiency.

Discontinuous percolation transitions in growing networks

Growing networks are ubiquitous in the real world, ranging from co-authorship socio-networks to protein interaction bio-networks. It is conventionally known that the giant cluster in such growing networks emerges continuously with infinite-order critical behavior. In this study, we show that when the growth of large clusters is suppressed with global information, the continuous percolation transition changes to a discontinuous transition with an abrupt jump of the order parameter at a delayed transition point p c. Moreover, a second-order-type critical behavior appears in a wide region of the link occupation probability before the system explodes, in which while the largest cluster has not grown to the extensive size of the system yet, the mean cluster size diverges. Far below p c, the property of the infinite-order transition still remains. Accordingly, the features of infinite-order, second-order, and first-order transitions all occur in a single framework when the infinite-order transition is suppressed. We present a simple argument to explain the underlying mechanism of these abnormal transition behaviors. Finally, we show that this result is universal by examining percolation transitions of a protein-interacting-network model.

Newman-Ziff algorithm for the bootstrap percolation: Application to the Archimedean lattices

We propose very efficient algorithms for the bootstrap percolation and the diffusion percolation models by extending the Newman-Ziff algorithm of the classical percolation (M.E.J. Newman and R.M. Ziff (2000) [27]). Using these algorithms and the finite-size-scaling, we calculated with high precision the percolation threshold and critical exponents in the eleven two-dimensional Archimedean lattices. We present the condition for the continuous percolation transition in the bootstrap percolation and the diffusion percolation, and show that they have the same critical exponents as the classical percolation within error bars in two dimensions. We conclude that the bootstrap percolation and the diffusion percolation almost certainly belong to the same universality class as the classical percolation.

Random growth lattice filling model of percolation: a crossover from continuous to discontinuous transition

A random growth lattice filling model of percolation with a touch and stop growth rule is developed and studied numerically on a two dimensional square lattice. Nucleation centers are continuously added one at a time to the empty lattice sites and clusters are grown from these nucleation centers with a growth probability g. For a given g (), the system passes through a critical point during the growth process where the transition from a disconnected to a connected phase occurs. The model is found to exhibit second order continuous percolation transitions as ordinary percolation for whereas for it exhibits weak first order discontinuous percolation transitions. The continuous transitions are characterized by estimating the values of the critical exponents associated with the order parameter fluctuation and the fractal dimension of the spanning cluster over the whole range of g. The discontinuous transitions, however, are characterized by a compact spanning cluster, lattice size independent fluctuation of the order parameter per lattice, departure from power law scaling in the cluster size distribution and weak bimodal distribution of the order parameter. The nature of transitions are further confirmed by studying the Binder cumulant. Instead of a sharp tricritical point, a tricritical region is found to occur for 0.5 < g < 0.8 within which the values of the critical exponents change continuously until the crossover from continuous to discontinuous transition is completed.

Four Types of Percolation Transitions in the Cluster Aggregation Network Model**Supported by the National Natural Science Foundation of China under Grant Nos 11575036 and 11505016.

We study the percolation transition in a one-species cluster aggregation network model, in which the parameter α describes the suppression on the cluster sizes. It is found that the model can exhibit four types of percolation transitions, two continuous percolation transitions and two discontinuous ones. Continuous and discontinuous percolation transitions can be distinguished from each other by the largest single jump. Two types of continuous percolation transitions show different behaviors in the time gap. Two types of discontinuous percolation transitions are different in the time evolution of the cluster size distribution. Moreover, we also find that the time gap may also be a measure to distinguish different discontinuous percolations in this model.

Percolation on networks with weak and heterogeneous dependency.

In real networks, the dependency between nodes is ubiquitous; however, the dependency is not always complete and homogeneous. In this paper, we propose a percolation model with weak and heterogeneous dependency; i.e., dependency strengths could be different between different nodes. We find that the heterogeneous dependency strength will make the system more robust, and for various distributions of dependency strengths both continuous and discontinuous percolation transitions can be found. For Erdős-Rényi networks, we prove that the crossing point of the continuous and discontinuous percolation transitions is dependent on the first five moments of the dependency strength distribution. This indicates that the discontinuous percolation transition on networks with dependency is determined not only by the dependency strength but also by its distribution. Furthermore, in the area of the continuous percolation transition, we also find that the critical point depends on the first and second moments of the dependency strength distribution. To validate the theoretical analysis, cases with two different dependency strengths and Gaussian distribution of dependency strengths are presented as examples.

Cascading failures in coupled networks with both inner-dependency and inter-dependency links.

We study the percolation in coupled networks with both inner-dependency and inter-dependency links, where the inner- and inter-dependency links represent the dependencies between nodes in the same or different networks, respectively. We find that when most of dependency links are inner- or inter-ones, the coupled networks system is fragile and makes a discontinuous percolation transition. However, when the numbers of two types of dependency links are close to each other, the system is robust and makes a continuous percolation transition. This indicates that the high density of dependency links could not always lead to a discontinuous percolation transition as the previous studies. More interestingly, although the robustness of the system can be optimized by adjusting the ratio of the two types of dependency links, there exists a critical average degree of the networks for coupled random networks, below which the crossover of the two types of percolation transitions disappears, and the system will always demonstrate a discontinuous percolation transition. We also develop an approach to analyze this model, which is agreement with the simulation results well.

Percolation in real interdependent networks

The function of a real network depends not only on the reliability of its own components, but is affected also by the simultaneous operation of other real networks coupled with it. Robustness of systems composed of interdependent network layers has been extensively studied in recent years. However, the theoretical frameworks developed so far apply only to special models in the limit of infinite sizes. These methods are therefore of little help in practical contexts, given that real interconnected networks have finite size and their structures are generally not compatible with those of graph toy models. Here, we introduce a theoretical method that takes as inputs the adjacency matrices of the layers to draw the entire phase diagram for the interconnected network, without the need of actually simulating any percolation process. We demonstrate that percolation transitions in arbitrary interdependent networks can be understood by decomposing these system into uncoupled graphs: the intersection among the layers, and the remainders of the layers. When the intersection dominates the remainders, an interconnected network undergoes a continuous percolation transition. Conversely, if the intersection is dominated by the contribution of the remainders, the transition becomes abrupt even in systems of finite size. We provide examples of real systems that have developed interdependent networks sharing a core of "high quality" edges to prevent catastrophic failures.

Given enough choice, simple local rules percolate discontinuously

There is still much to discover about the mechanisms and nature of discontinuous percolation transitions. Much of the past work considers graph evolution algorithms known as Achlioptas processes in which a single edge is added to the graph from a set of $k$ randomly chosen candidate edges at each timestep until a giant component emerges. Several Achlioptas processes seem to yield a discontinuous percolation transition, but it was proven by Riordan and Warnke that the transition must be continuous in the thermodynamic limit. However, they also proved that if the number $k(n)$ of candidate edges increases with the number of nodes, then the percolation transition may be discontinuous. Here we attempt to find the simplest such process which yields a discontinuous transition in the thermodynamic limit. We introduce a process which considers only the degree of candidate edges and not component size. We calculate the critical point $t_{c}=(1-\theta(\frac{1}{k}))n$ and rigorously show that the critical window is of size $O(\frac{n}{k(n)})$. If $k(n)$ grows very slowly, for example $k(n)=\log n$, the critical window is barely sublinear and hence the phase transition is discontinuous but appears continuous in finite systems. We also present arguments that Achlioptas processes with bounded size rules will always have continuous percolation transitions even with infinite choice.

Phase transitions in supercritical explosive percolation

Percolation describes the sudden emergence of large-scale connectivity as edges are added to a lattice or random network. In the Bohman-Frieze-Wormald model (BFW) of percolation, edges sampled from a random graph are considered individually and either added to the graph or rejected provided that the fraction of accepted edges is never smaller than a decreasing function with asymptotic value of α, a constant. The BFW process has been studied as a model system for investigating the underlying mechanisms leading to discontinuous phase transitions in percolation. Here we focus on the regime αε[0.6,0.95] where it is known that only one giant component, denoted C(1) , initially appears at the discontinuous phase transition. We show that at some point in the supercritical regime C(1) stops growing and eventually a second giant component, denoted C(2), emerges in a continuous percolation transition. The delay between the emergence of C(1) and C(2) and their asymptotic sizes both depend on the value of α and we establish by several techniques that there exists a bifurcation point α(c)=0.763±0.002. For αε[0.6,α(c)), C(1) stops growing the instant it emerges and the delay between the emergence of C(1) and C(2) decreases with increasing α. For αε(α(c),0.95], in contrast, C(1) continues growing into the supercritical regime and the delay between the emergence of C(1) and C(2) increases with increasing α. As we show, α(c) marks the minimal delay possible between the emergence of C(1) and C(2) (i.e., the smallest edge density for which C(2) can exist). We also establish many features of the continuous percolation of C(2) including scaling exponents and relations.

Continuous Percolation Transition in Random Cluster Growth Model

A random cluster growth model is developed in terms of two parameters, the initial seed concen- tration ρ and the growth probability g of individual clusters. The model is studied on a two dimensional square lattice. For every ρ value, a critical value of g = gc is determined at which a percolation transition is observed. A scaling theory for this model is developed and numerically verified. The scaling functions are found to scale with ρ, g as well as the system size L with appropriate critical exponents. The values of the critical exponents are found to belong to the same universality class of percolation. Finally a phase dia- gram is developed in the ρ − g parameter space for percolating and non-percolating regions separated by a line of second order phase transition points. (doi: 10.5562/cca2313)

Generalized epidemic process and tricritical dynamic percolation.

The renowned general epidemic process describes the stochastic evolution of a population of individuals which are either susceptible, infected, or dead. A second order phase transition belonging to the universality class of dynamic isotropic percolation lies between the endemic and pandemic behavior of the process. We generalize the general epidemic process by introducing a fourth kind of individuals, viz., individuals which are weakened by the process but not yet infected. This weakening gives rise to a mechanism that introduces a global instability in the spreading of the process and therefore opens the possibility of a discontinuous transition in addition to the usual continuous percolation transition. The tricritical point separating the lines of first and second order transitions constitutes an independent universality class, namely, the universality class of tricritical dynamic isotropic percolation. Using renormalized field theory we work out a detailed scaling description of this universality class. We calculate the scaling exponents in an epsilon expansion below the upper critical dimension d(c) =5 for various observables describing tricritical percolation clusters and their spreading properties. In a remarkable contrast to the usual percolation transition, the exponents beta and beta(') governing the two order parameters, viz., the mean density and the percolation probability, turn out to be different at the tricritical point. In addition to the scaling exponents we calculate for all our static and dynamic observables logarithmic corrections to the mean-field scaling behavior at d(c) =5.