Percolation Phase Transition Research Articles

Smeared phase transitions in percolation on real complex networks

Percolation on complex networks is used both as a model for dynamics on networks, such as network robustness or epidemic spreading, and as a benchmark for our models of networks, where our ability to predict percolation measures our ability to describe the networks themselves. In many applications, correctly identifying the phase transition of percolation on real-world networks is of critical importance. Unfortunately, this phase transition is obfuscated by the finite size of real systems, making it hard to distinguish finite size effects from the inaccuracy of a given approach that fails to capture important structural features. Here, we borrow the perspective of smeared phase transitions and argue that many observed discrepancies are due to the complex structure of real networks rather than to finite size effects only. In fact, several real networks often used as benchmarks feature a smeared phase transition where inhomogeneities in the topological distribution of the order parameter do not vanish in the thermodynamic limit. We find that these smeared transitions are sometimes better described as sequential phase transitions within correlated subsystems. Our results shed light not only on the nature of the percolation transition in complex systems, but also provide two important insights on the numerical and analytical tools we use to study them. First, we propose a measure of local susceptibility to better detect both clean and smeared phase transitions by looking at the topological variability of the order parameter. Second, we highlight a shortcoming in state-of-the-art analytical approaches such as message passing, which can detect smeared transitions but not characterize their nature.

The critical Ising model on amenable graphs of exponential growth

The purpose of this note is to point out that the proof of the recent result of Hutchcroft [Hut] concerning continuity of the phase transition in Bernoulli percolation is applicable to the setting of the Ising model with free boundary conditions. This observation, together with a recent result of Aizenman, Duminil-Copin, and Sidoravicius [ADS] implies that the Ising model on any amenable transitive graph with exponential growth undergoes a continuous phase transition.

The sharp phase transition for level set percolation of smooth planar Gaussian fields

We prove that the connectivity of the level sets of a wide class of smooth centred planar Gaussian fields exhibits a phase transition at the zero level that is analogous to the phase transition in Bernoulli percolation. In addition to symmetry, positivity and regularity conditions, we assume only that correlations decay polynomially with exponent larger than two - roughly equivalent to the integrability of the covariance kernel - whereas previously the phase transition was only known in the case of the Bargmann-Fock covariance kernel which decays super-exponentially. We also prove that the phase transition is sharp, demonstrating, without any further assumption on the decay of correlations, that in the sub-critical regime crossing probabilities decay faster than any polynomial. Key to our methods is the white-noise representation of a Gaussian field; we use this on the one hand to prove new quasi-independence results, inspired by the notion of influence from Boolean functions, and on the other hand to establish sharp thresholds via the OSSS inequality for i.i.d. random variables, following the recent approach of Duminil-Copin, Raoufi and Tassion

Jamming and percolation for deposition of k^{2}-mers on square lattices: A Monte Carlo simulation study.

Percolation and jamming of k×k square tiles (k^{2}-mers) deposited on square lattices have been studied by numerical simulations complemented with finite-size scaling theory and exact enumeration of configurations for small systems. The k^{2}-mers were irreversibly deposited into square lattices of sizes L×L with L/k ranging between 128 and 448 (64 and 224) for jamming (percolation) calculations. Jamming coverage θ_{j,k} was determined for a wide range of k values (2≤k≤100 with many intermediate k values to allow a fine scaling analysis). θ_{j,k} exhibits a decreasing behavior with increasing k, being θ_{j,k=∞}=0.5623(3) the limit value for large k^{2}-mer sizes. In addition, a finite-size scaling analysis of the jamming transition was carried out, and the corresponding spatial correlation length critical exponent ν_{j} was measured, being ν_{j}≈1. On the other hand, the obtained results for the percolation threshold θ_{c,k} showed that θ_{c,k} is an increasing function of k in the range 1≤k≤3. For k≥4, all jammed configurations are nonpercolating states and, consequently, the percolation phase transition disappears. An explanation for this phenomenon is offered in terms of the rapid increase with k of the number of surrounding occupied sites needed to reach percolation, which gets larger than the sufficient number of occupied sites to define jamming. In the case of k=2 and 3, the percolation thresholds are θ_{c,k=2}(∞)=0.60355(8) and θ_{c,k=3}=0.63110(9). Our results significantly improve the previously reported values of θ_{c,k=2}^{Naka}=0.601(7) and θ_{c,k=3}^{Naka}=0.621(6) [Nakamura, Phys. Rev. A 36, 2384 (1987)0556-279110.1103/PhysRevA.36.2384]. In parallel, a comparison with previous results for jamming on these systems is also done. Finally, a complete analysis of critical exponents and universality has been done, showing that the percolation phase transition involved in the system has the same universality class as the ordinary random percolation, regardless of the size k considered.

Sharpness for inhomogeneous percolation on quasi-transitive graphs

In this note we study the phase transition for percolation on quasi-transitive graphs with quasi-transitive inhomogeneous edge-retention probabilities. A quasi-transitive graph is an infinite graph with finitely many different “types” of edges and vertices. We prove that the transition is sharp almost everywhere, i.e., that in the subcritical regime the expected cluster size is finite, and that in the subcritical regime the probability of the one-arm event decays exponentially. Our proof extends the proof of sharpness of the phase transition for homogeneous percolation on vertex-transitive graphs by Duminil-Copin and Tassion (2016) and the result generalizes previous results of Antunović and Veselić (2008) and Menshikov (1986).

Inverse percolation by removing straight rigid rods from square lattices in the presence of impurities

Numerical simulations and finite-size scaling analysis have been carried out to study the problem of inverse percolation by removing straight rigid rods from square lattices contaminated with non-conducting impurities. The presence of impurities provides a more realistic approach to the deposited monolayer, which usually presents inhomogeneities due to the irregular arrangement of surface and bulk atoms, the presence of various chemical species, etc. The process starts with an initial configuration, where all lattice sites are occupied by an impurity (with a concentration ) or a conducting particle (with a concentration ). Then, the system is diluted by randomly removing linear k-mers (linear clusters of k consecutive conducting particles) from the lattice. The impurities remain fixed in its position and cannot be removed. The central idea of this paper is based on finding the maximum concentration of conducting sites (minimum concentration of empty sites) for which the connectivity disappears. This particular value of the concentration is called inverse percolation threshold, and determines a well-defined geometrical phase transition in the system. On the other hand, the inverse jamming coverage is the coverage of the limit state, in which no more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbour sites of appropriate size. The dependence of percolation and jamming thresholds on the concentration of defects was investigated for different values of k, ranging from 1 to 120. The obtained results show that the behaviour of the system is significantly affected by the presence of impurities. In addition, the nature of the jamming and percolation transitions was studied. In the first case, the corresponding spatial correlation length critical exponent was measured, being . This value coincides with previous calculations of this exponent for the standard random sequential adsorption of linear k-mers on square lattices. Critical exponents were also calculated for the percolation phase transition, showing that the universality class corresponding to ordinary percolation is preserved regardless the values of k and considered.

Localized attack on networks with clustering

Network systems with clustering have been given much attention due to their wide occurrence in the real world. One focus of these studies has been on robustness of single clustered networks and interdependent clustered networks under random attack (RA) or hub-targeted attack. However, infrastructure networks could suffer from a damage that is localized, i.e. a group of neighboring nodes attacked or fail, a topic that was not studied earlier on clustered networks. In this paper, we analytically and via simulations study the robustness under localized attack (LA) of single Erdős–Rényi clustered network and interdependent clustered network. For generating networks with clustering we use two models: (i) double Poisson distribution (DPD) and (ii) fixed degree distribution (FDD). For the LA case, the DPD model shows a second order phase transition behavior for a single clustered network, while for dependent networks, the system undergoes a change of percolation phase transition from a first order (abrupt transition) to a second order (continuous) transition when the coupling strength q decreases below a critical value qc. Our results imply that single networks become significantly more vulnerable with increasing clustering coefficient c with respect to LA. This is in contrast to RA where the robustness is almost independent of c. We obtain similar results when testing different real networks. For LA on dependent networks, we also observe that the system becomes more vulnerable as c increases. This is again in contrast to RA, where for, q < qc, the system robustness is almost unaffected by increasing clustering. We also solved analytically the case of LA on random regular networks which are clustered and interdependent and find that as m (the number of clustered networks that each network depends on) or c increases, the system becomes significantly more vulnerable. We also analyzed via simulations the case of generating clustering in networks for the model of keeping a FDD, and find that the influence of clustering on the robustness of two partially interdependent networks under LA is smaller than for DPD, which is very different from these cases under RA.

Topological percolation on hyperbolic simplicial complexes

Simplicial complexes are increasingly used to understand the topology of complex systems as different as brain networks and social interactions. It is therefore of special interest to extend the study of percolation to simplicial complexes. Here we propose a topological theory of percolation for discrete hyperbolic simplicial complexes. Specifically we consider hyperbolic manifolds in dimension $d=2$ and $d=3$ formed by simplicial complexes, and we investigate their percolation properties in the presence of topological damage, i.e., when nodes, links, triangles or tetrahedra are randomly removed. We show that in $d=2$ simplicial complexes there are four topological percolation problems and in $d=3$, there are six. We demonstrate the presence of two percolation phase transitions characteristic of hyperbolic spaces for the different variants of topological percolation. While most of the known results on percolation in hyperbolic manifolds are in $d=2$, here we uncover the rich critical behavior of $d=3$ hyperbolic manifolds, and show that triangle percolation displays a Berezinskii-Kosterlitz-Thouless (BKT) transition. Finally we provide evidence that topological percolation can display a critical behavior that is unexpected if only node and link percolation are considered.

Directed percolation phase transition at the onset of STI in an inhomogeneous coupled map lattice

A model for inhomogeneously coupled logistic maps is considered to find some critical exponents in the transition from inhomogeneous steady state to spatiotemporal chaos through spatiotemporal intermittency. The laminar state in the model is described by inhomogeneous steady state with spatial period two. We obtain a complete set of static exponents which match with the corresponding directed percolation (DP) values in (1+1) dimension. We also find four nonuniversal spreading exponents in which three exponents are in agreement with DP values. The model in which absorbing state is inhomogeneous steady state, contributes a new example in evidence of Pomeau's [18] conjecture that the onset of STI in a deterministic system belongs to DP universality class.

Evaluation of ENSO simulations in CMIP5 models: A new perspective based on percolation phase transition in complex networks

In this study, the performance of CMIP5 models in simulating the El Niño-Southern Oscillation (ENSO) is evaluated by using a new metric based on percolation theory. The surface air temperatures (SATs) over the tropical Pacific Ocean are constructed as a SAT network, and the nodes within the network are linked if they are highly connected (e.g., high correlations). It has been confirmed from reanalysis datasets that the SAT network undergoes an abrupt percolation phase transition when the influences of the sea surface temperature anomalies (SSTAs) below are strong enough. However, from simulations of the CMIP5 models, most models are found incapable of capturing the observed phase transition at a proper critical point Pc. For the 15 considered models, four even miss the phase transition, indicating that the simulated SAT network is too stable to be significantly changed by the SSTA below. Only four models can be considered cautiously with some skills in simulating the observed phase transition of the SAT network. By comparing the simulated SSTA patterns with the node vulnerabilities, which is the chance of each node being isolated during a ENSO event, we find that the improperly simulated sea-air interactions are responsible for the missing of the observed percolation phase transition. Accordingly, a careful study of the sea-air couplers, as well as the atmospheric components of the CMIP5 models is suggested. Since the percolation phase transition of the SAT network is a useful phenomenon to indicate whether the ENSO impacts can be transferred remotely, it deserves more attention for future model development.

Hot Dense Matter: Deconfinement and Clustering of Color Sources in Nuclear Collisions

Within the first few microseconds from after the Big Bang, the hot dense matter was in the form of the Quark Gluon Plasm (QGP) consisting of free quarks and gluons. By colliding heavy nuclei at RHIC and LHC at a velocity close to the speed of light, we were able to create the primordial matter and observe the matter after expansion and cooling. In this report we present the thermodynamics and transport coefficients obtained in the framework of clustering of color sources in both hadron-hadron and nucleus-nucleus collisions at RHIC and LHC energies. Multiparticle production at high energies can be described in terms of color strings stretched between the projectile and target. At high string density single strings overlap and form color sources. This addition belongs to the non-perturbative domain of Quantum Chromo Dynamics (QGP) and manifests its most fundamental features. The Schwinger QED 2 mechanism produces color neutral q q ¯ pairs when color source strings break. Subsequent hardonization produces the observed hadrons. With growing energy and atomic number of the colliding nuclei the density of strings grows and more color sources form clusters in the transverse plane. At a certain critical density a macroscopic cluster appears, which marks the percolation phase transition. This is the Color String Percolation Model (CSPM). The critical density is identified as the deconfinement transition and happens at the hadronization temperature. The stochastic thermalization in p p and A-A is a consequence of the quantum tunneling through the event horizon introduced by the confining color fields, the Hawking-Unruh effect. The percolation approach within CSPM is successfully used to describe the crossover phase transition in the soft collision region. The same phenomenology when applied to both hadron-hadron and nucleus-nucleus collisions emphasizes the importance of color string density, creating a macroscopic cluster which identifies the connectivity required for a finite droplet of the QGP.

Structural resilience of spatial networks with inter-links behaving as an external field

Many real systems such as, roads, shipping routes, and infrastructure systems can be modeled based on spatially embedded networks. The inter-links between two distant spatial networks, such as those formed by transcontinental airline flights, play a crucial role in optimizing communication and transportation over such long distances. Still, little is known about how inter-links affect the structural resilience of such systems. Here, we develop a framework to study the structural resilience of interlinked spatially embedded networks based on percolation theory. We find that the inter-links can be regarded as an external field near the percolation phase transition, analogous to a magnetic field in a ferromagnetic–paramagnetic spin system. By defining the analogous critical exponents δ and γ, we find that their values for various inter-links structures follow Widom’s scaling relations. Furthermore, we study the optimal robustness of our model and compare it with the analysis of real-world networks. The framework presented here not only facilitates the understanding of phase transitions with external fields in complex networks but also provides insight into optimizing real-world infrastructure networks.

Effect of a Cocurrent Gas Flow on the Velocity and Concentration Limits of Combustion Wave Propagation in Granulated Ti + C + xAl2O3 Mixtures

In this paper, we studied the effects of a cocurrent gas flow on the velocity and combustion limits of a granulated Ti + C mixture diluted with inert corundum granules. In the whole range of dilutions, an increase in the combustion rate was observed for the flow of active and inert gases. This effect is several times greater than that calculated based on the filtration combustion theory. Concentration combustion limits (75 wt %), the incomplete combustion of Ti + C granules, and the ratio of the combustion rates of the undiluted mixture and the mixture at the propagation limit (from 2 to 3) are well predicted by the percolation theory. The same combustion limit of concentration in the flow of inert and active gases and without flow indicates a percolation-phase transition as the reason for the cessation of combustion.

The Sign Clusters of the Massless Gaussian Free Field Percolate on $${\mathbb{Z}^{d}, d \geqslant 3}$$ Z d , d ⩾ 3 (and more)

We investigate the percolation phase transition for level sets of the Gaussian free field on $\mathbb{Z}^d$, with $d\geqslant 3$, and prove that the corresponding critical parameter $h_*(d)$ is strictly positive for all $d\geqslant3$, thus settling an open question from arXiv:1202.5172. In particular, this implies that the sign clusters of the Gaussian free field percolate on $\mathbb{Z}^d$, for all $d\geqslant 3$. Among other things, our construction of an infinite cluster above small, but positive level $h$ involves random interlacements at level $u>0$, a random subset of $\mathbb{Z}^d$ with desirable percolative properties, introduced in arXiv:0704.2560 in a rather different context, a certain Dynkin-type isomorphism theorem relating random interlacements to the Gaussian free field, see arXiv:1111.4818, and a recent coupling from arXiv:1402.0298 of these two objects, lifted to a continuous metric graph structure over $\mathbb{Z}^d$.

Resilience of networks with community structure behaves as if under an external field

Although detecting and characterizing community structure is key in the study of networked systems, we still do not understand how community structure affects systemic resilience and stability. We use percolation theory to develop a framework for studying the resilience of networks with a community structure. We find both analytically and numerically that interlinks (the connections among communities) affect the percolation phase transition in a way similar to an external field in a ferromagnetic- paramagnetic spin system. We also study universality class by defining the analogous critical exponents δ and γ, and we find that their values in various models and in real-world coauthor networks follow the fundamental scaling relations found in physical phase transitions. The methodology and results presented here facilitate the study of network resilience and also provide a way to understand phase transitions under external fields.

Molecular jenga: the percolation phase transition (collapse) in virus capsids

Virus capsids are polymeric protein shells that protect the viral cargo. About half of known virus families have icosahedral capsids that self-assemble from tens to thousands of subunits. Capsid disassembly is critical to the lifecycles of many viruses yet is poorly understood. Here, we apply a graph and percolation theory to examine the effect of removing capsid subunits on capsid stability and fragmentation. Based on the structure of the icosahedral capsid of hepatitis B virus (HBV), we constructed a graph of rhombic subunits arranged with icosahedral symmetry. Though our approach neglects dependence on energetics, time, and molecular detail, it quantitatively predicts a percolation phase transition consistent with recent in vitro studies of HBV capsid dissociation. While the stability of the capsid graph followed a gradual quadratic decay, the rhombic tiling abruptly fragmented when we removed more than 25% of the 120 subunits, near the percolation threshold observed experimentally. This threshold may also affect results of capsid assembly, which also experimentally produces a preponderance of 90 mer intermediates, as the intermediate steps in these reactions are reversible and can thus resemble dissociation. Application of percolation theory to understanding capsid association and dissociation may prove a general approach to relating virus biology to the underlying biophysics of the virus particle.

Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^{d}$

Nous considérons le modèle Booléen $Z$ sur $\mathbb{R}^{d}$ avec des grains compacts aléatoires de diamètres bornés, c’est-à-dire $Z:=\bigcup_{i\in\mathbb{N}}(Z_{i}+X_{i})$ où $\{X_{1},X_{2},\dots\}$ est un processus de Poisson d’intensité $t$ et $(Z_{1},Z_{2},\dots)$ est une suite i.i.d. de grains compacts (non nécessairement des boules) de diamètres p.s. bornés par une constante. Nous montrons une décroissance exponentielle dans le régime sous-critique, ce qui veut dire que le volume et le rayon du cluster d’un grain typique dans $Z$ a une queue exponentielle. Pour cela, nous adaptons des résultats de (A new proof of the sharpness of the phase transition for Bernoulli percolation on $\mathbb{Z}^{d}$ (2015) Preprint) et appliquons une nouvelle construction du cluster d’un grain typique avec des arguments issus des processus de branchement. Dans la seconde partie du papier, nous obtenons de nouvelles bornes inférieures pour le modèle booléen avec grains déterministes. Certaines des ces bornes sont rigoureuses, alors que d’autres sont obtenues par simulation. Les bornes obtenues par simulation sont fournies avec des intervalles de confiance et sont beaucoup plus précises que celles obtenues rigoureusement. Elles améliorent les résultats connus (J. Chem. Phys. 137 (2012) 074106) en dimension 6 et plus.

Standard and inverse bond percolation of straight rigid rods on square lattices.

Numerical simulations and finite-size scaling analysis have been carried out to study standard and inverse bond percolation of straight rigid rods on square lattices. In the case of standard percolation, the lattice is initially empty. Then, linear bond k-mers (sets of k linear nearest-neighbor bonds) are randomly and sequentially deposited on the lattice. Jamming coverage p_{j,k} and percolation threshold p_{c,k} are determined for a wide range of k (1≤k≤120). p_{j,k} and p_{c,k} exhibit a decreasing behavior with increasing k, p_{j,k→∞}=0.7476(1) and p_{c,k→∞}=0.0033(9) being the limit values for large k-mer sizes. p_{j,k} is always greater than p_{c,k}, and consequently, the percolation phase transition occurs for all values of k. In the case of inverse percolation, the process starts with an initial configuration where all lattice bonds are occupied and, given that periodic boundary conditions are used, the opposite sides of the lattice are connected by nearest-neighbor occupied bonds. Then, the system is diluted by randomly removing linear bond k-mers from the lattice. The central idea here is based on finding the maximum concentration of occupied bonds (minimum concentration of empty bonds) for which connectivity disappears. This particular value of concentration is called the inverse percolation threshold p_{c,k}^{i}, and determines a geometrical phase transition in the system. On the other hand, the inverse jamming coverage p_{j,k}^{i} is the coverage of the limit state, in which no more objects can be removed from the lattice due to the absence of linear clusters of nearest-neighbor bonds of appropriate size. It is easy to understand that p_{j,k}^{i}=1-p_{j,k}. The obtained results for p_{c,k}^{i} show that the inverse percolation threshold is a decreasing function of k in the range 1≤k≤18. For k>18, all jammed configurations are percolating states, and consequently, there is no nonpercolating phase. In other words, the lattice remains connected even when the highest allowed concentration of removed bonds p_{j,k}^{i} is reached. In terms of network attacks, this striking behavior indicates that random attacks on single nodes (k=1) are much more effective than correlated attacks on groups of close nodes (large k's). Finally, the accurate determination of critical exponents reveals that standard and inverse bond percolation models on square lattices belong to the same universality class as the random percolation, regardless of the size k considered.

Percolation Thresholds and Phase Transitions in Binary Composites

The metal–insulator and metal–superconductor phase transitions related to the percolation thresholds in two-component composites are considered. The behavior of effective conductivity σ e in the vicinity of both thresholds is described in terms of the similarity hypothesis. A one-to-one correspondence between the equations derived for σ e in both critical regions is found for randomly heterogeneous systems.

Sharpness of the phase transition for continuum percolation in $$\mathbb {R}^2$$ R 2

We study the phase transition of random radii Poisson Boolean percolation: Around each point of a planar Poisson point process, we draw a disc of random radius, independently for each point. The behavior of this process is well understood when the radii are uniformly bounded from above. In this article, we investigate this process for unbounded (and possibly heavy tailed) radii distributions. Under mild assumptions on the radius distribution, we show that both the vacant and occupied sets undergo a phase transition at the same critical parameter $$\lambda _c$$ . Moreover, The techniques we develop in this article can be applied to other models such as the Poisson Voronoi and confetti percolation.