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The current work establishes a pool boiling CHF prediction method based on percolation theory. For the first time, we observe the experimental bubble footprint’s power-law distributions with almost the same exponent in wire heaters’ water pool boiling crisis, which is borne out strongly that boiling crisis is a typical continuum percolative scale-free behavior, and its characteristics seems not to be influenced by the critical heat flux value. The proposed one-dimensional Monte Carlo(MC) method successfully simulates the phase transition of interactive near-wall bubbles. This research enriches and extends applications of continuum percolation theory in boiling phenomena, and could be an instruction for the followed critical heat flux enhancement studies.

For $d \geq 3$ we obtain an approximation of the zero-average Gaussian free field on the discrete $d$-dimensional torus of large side length $N$ by the Gaussian free field on $\mathbb Z^d$, valid in boxes of roughly side length $N - N^\delta$ with $\delta \in (\frac12,1)$. As an implication, the level sets of the zero-average Gaussian free field on the torus can be approximated by the level sets of the Gaussian free field on $\mathbb Z^d$. This leads to a series of applications related to level-set percolation. We show that level sets of the zero-average Gaussian free field on the torus for levels $h > h_\star$ (where $h_\star$ denotes the critical value for level-set percolation of the Gaussian free field on $\mathbb Z^d$) with high probability contain no connected component of volume comparable to the total volume of the torus. Moreover, level sets with $h < h_\star$ with high probability contain a connected component of (extrinsic) diameter comparable to the torus diameter $N$. We also show that level sets of the zero-average Gaussian free field on the torus for levels $h$ above a second critical parameter $h_{\star\star}(\geq h_\star)$, again defined via the Gaussian free field on $\mathbb Z^d$, with high probability only contain connected components negligible in their size when compared to the size of the torus. Similar results have been obtained by A. Teixeira and D. Windisch in [Comm. Pure Appl. Math., 64(12):1599-1646, 2011] and J. \v{C}ern\'y and A. Teixeira in [Ann. Appl. Probab., 26(5):2883-2914, 2016] for the vacant set of simple random walk on a large discrete torus with the help of random interlacements on $\mathbb Z^d$, introduced by A.-S. Sznitman in [Ann. of Math. (2), 171(3):2039-2087, 2010]., Comment: 23 pages, to appear in Stochastic Processes and their Applications

The emergence of system-spanning flow path is the prerequisite of fluid flow through porous composites. In statistic physics, the percolation threshold is often used to describe the formation of long-range connectivity in the system and previous studies on the percolation of random systems showed that the percolation of porous network is strongly affected by the geometrical shape of pores. In this work, the continuum percolation theory is applied to study the percolation properties of 2D/3D porous media composed of homogeneous solid matrix and overlapping pores of concave-shaped geometries (i.e., shape parameter m) from the “cross” limit to the square/octahedron, respectively. By using Monte Carlo simulation, a series of 2D/3D porous structures comprising mono-sized overlapping pores of regular geometries are generated. By combining these simplified structures with finite-size scaling technique, the corresponding percolation thresholds ϕc are derived and the relation between the characteristic of concave-shaped pores and the percolation threshold is quantified by a numerical approximation. The results reveal that for both 2D/3D porous structures, ϕc presents a monotonically increasing trend with the increasing m (i.e., the shape of pores evolves continuously from the “cross” limit to the square/octahedron). Moreover, a general percolation-based effective-medium approximation is further adopted by us to theoretically explore the solute diffusion in the saturated porous systems considering their percolation behavior. From the study, it is observed that both the pore shape m and threshold ϕc have significant effect on the effective diffusivity of these porous structures, especially when the solid matrix is regarded as an insulating component. The results may provide some guidance for the development of percolation theory and the design of composites.

In string percolation model, the study of colliding systems at high energies is based on a continuum percolation theory in two dimensions where the number of strings distributed in the surface of interest is strongly determined by the size and the energy of the colliding particles. It is also expected that the surface where the disks are lying be finite, defining a system without periodic boundary conditions. In this work, we report modifications to the fraction of the area covered by disks in continuum percolating systems due to a finite number of disks and bounded by different geometries: circle, ellipse, triangle, square and pentagon, which correspond to the first Fourier modes of the shape fluctuation of the initial state after the particle collision. We find that the deviation of the fraction of area covered by disks from its corresponding value in the thermodynamic limit satisfies a universal behavior, where the free parameters depend on the density profile, number of disks and the shape of the boundary. Consequently, it is also found that the color suppression factor of the string percolation model is modified by a damping function related to the small-bounded effects. Corrections to the temperature and the speed of sound defined in string systems are also shown for small and elliptically bounded systems., 19 pages, 12 figures

Understanding the effect of percolation behavior of complex geometrical pores on the tortuosity and thermal conductivity of porous composites is very crucial to the design and optimization of porous composites. In this work, we adopt the continuum percolation theory to accurately determine the nonlinear thermal conductivity and tortuosity of porous composites composed of homogeneous solid matrix and three-dimensional pores of geometrical morphologies from the isotropic sphere to anisotropic polyhedra. Through extensive Monte Carlo simulations and the finite-size scaling analysis, the percolation threshold of spherical and polyhedral pores is obtained. Two continuum percolation-based models are respectively presented to derive the tortuosity and thermal conductivity of porous composites over the whole porosities range, including near the percolation threshold. Comparison with extensive experimental, numerical and theoretical results confirms that the present models are capable of accurately determining the percolation threshold and tortuosity of complex geometrical porous networks and the effective thermal conductivity of porous composites as conductor-superconductor and insulator-conductor media. Furthermore, we use the proposed models to probe the influences of pore shape and porosity on the tortuosity and thermal conductivity of porous composites. The results elucidate the intrinsic interplay of component, structure, and thermal conductivity of porous composites, which can provide sound guidance for porous composite design and evaluation.

To address some physical properties of percolating systems it can be useful to know the degree distributions in finite clusters along with their size distribution. Here we show that to achieve this aim for classical bond percolation one can use the $q \to 1$ limit of suitably modified q-state Potts model. We consider a version of such model with the additional complex variables and show that its partition function gives generating function for the size and degree distribution in this limit. We derive this distribution analytically for bond percolation on Bethe lattice and complete graph. The possibility to expand the applications of present method to other clusters' characteristics and to models of correlated percolation is discussed., 13 pages

Synchronization and percolation are two essential phenomena in complex dynamical networks. They have been studied widely, but previously treated as unrelated. In this paper, the relationship between synchronization and percolation are revealed for regular networks. Firstly, we discovered a bridge between synchronization and percolation by using the eigenvalues of the Laplacian matrix to describe the synchronizability and using the eigenvalues of the adjacency matrix to describe the percolation threshold. Then, we proposed a method to find the relationship for regular networks based on the topology of networks. Particularly, if the degree distribution of the network is subject to delta function, we show that only the eigenvalues of the adjacency matrix need to be calculated. Finally, several examples are provided to demonstrate how to apply our proposed method to discover the relationship between synchronization and percolation for regular networks.

We study how fractal features of an infinitely ramified network affect its percolation properties. The fractal attributes are characterized by the Hausdorff ( D H ), topological Hausdorff ( D t H ), and spectral ( d s ) dimensions. Monte Carlo simulations of site percolation were performed on pre-fractal standard Sierpinski carpets with different fractal attributes. Our findings suggest that within the universality class of random percolation the values of critical percolation exponents are determined by the set of dimension numbers ( D H , D t H , d s ), rather than solely by the spatial dimension (d). We also argue that the relevant dimension number for the percolation threshold is the topological Hausdorff dimension D t H , whereas the hyperscaling relations between critical exponents are governed by the Hausdorff dimension D H . The effect of the network connectivity on the site percolation threshold is revealed.

It has been shown that a hot and dense deconfined nuclear matter state produced in ultra-relativistic heavy-ion collisions, can be quantitatively described by the String Percolation phenomenological model. The model address the phase transition in terms of the two-dimensional continuum percolation theory over strings, which are schematic representations of the fundamental interactions among the partons of the colliding nuclei in the initial state. In this work, we present an extension of the critical string density results including the eccentricity dependence on the initial state geometry focus on small string number with different density profile, small deviations from the different profile densities are found. The percolation threshold shows consistency with the thermodynamic limit for the uniform density profile with a large number of strings in the case of circular boundary system. A significant dependence on the eccentricity for a small number of strings compared to high occupancy systems is exhibited, the implications may become relevant in hadron-hadron or hadron-nucleus collision systems., Comment: 17 pages, 8 figures

We consider supercritical percolation on Z d ( d ≥ 3 ) induced by random walk loop soup. Two vertices are in the same cluster if they are connected through a sequence of intersecting loops. We obtain quenched parabolic Harnack inequalities, Gaussian heat kernel bounds, the invariance principle and the local central limit theorem for the simple random walks on the unique infinite cluster. We also show that the diameter of finite clusters have exponential tails like in Bernoulli bond percolation. Our results hold for all d ≥ 3 and all supercritical intensities despite polynomial decay of correlations.

Analyzing percolation rules of real networks has some great realistic significance. In this paper, we develop a relatively simple model based on generating function method to study percolation properties of real networks. We construct our model for both site and bond percolation, compare its estimates with those of the message passing algorithm and simulation results on computer-generated networks as well as practical networks, and discuss causes of the inaccuracy. The conclusions show that the accuracy of our model could be accepted though it is lower than that of the message passing algorithm and the discrepancies between the estimates of our model and the simulation values mainly come from the disagreement of those real networks with the model hypotheses.

We analyze a mathematical model of a cognitive radio network introduced in Yemini et al. (2016). Our analysis reveals several surprising features of the model. We explain some of these features using ideas from percolation theory and stochastic geometry.

The vertices of thekth power of a directed path withnvertices are exposed one by one to a selector in some random order. At any time the selector can see the graph induced by the vertices that have already appeared. The selector's aim is to choose online the maximal vertex (i.e. the vertex with no outgoing edges). We give upper and lower bounds for the asymptotic behaviour ofpn,kn1/(k+1), wherepn,kis the probability of success under the optimal algorithm. In order to derive the upper bound, we consider a model in which the selector obtains some extra information about the edges that have already appeared. We give the exact asymptotics of the probability of success under the optimal algorithm in this case. In order to derive the lower bound, we analyse a site percolation process on a sequence of thekth powers of a directed path withnvertices.

In this paper, we study inhomogeneous site percolation on an irregular Bethe lattice, for considering that percolation often occurs on irregular grids or lattices with variable site neighbours in real-world problems. The explicit expression for cluster-size distribution of this percolation is derived based on probability theory. Moreover, the exact formulas for critical occupation probability, mean cluster size, and percolation probability are obtained using generating function method and generalised recursive approach. In addition, sensitivity analysis and numerical simulation are given to deepen and illustrate the results.

The effectiveness of secondary recovery methods in reservoir development studies depends on the knowledge about how fluid-carrying regions (i.e. good-quality rock types) are connected between injection and production wells. To estimate reservoir performance uncertainty, comprehensive simulations on many reservoir model realisations are necessary, which is very CPU consuming and time demanding. Alternatively, we can use much simpler and physically based methods such as percolation approach. Classic percolation assumes connectivity between opposite 2-D faces of a 3-D system; whereas, hydrocarbon production is achieved through active wells that are one-dimensional lines (e.g. vertical, horizontal or deviated wells). The main contribution of this study is to analyse the percolation properties of 3-D continuum percolation models with more realistic well representations during secondary recovery. In particular, the connection of randomly distributed sands (i.e. good-quality rock types) between two lines (representing two wells) located at two corners of the system are modelled by Monte Carlo simulations. Subsequently, the connectivity and conductivity of such a line-to-line well representation is compared with that of face-to-face well representations in the previously published results. The critical percolation properties of those systems as well as the universality concept are also investigated. As there are many rooms for connections in 3-D models, we found that the principal percolation properties will not be altered significantly when the problem with a face-to-face connection is transformed to a line-to-line connection model.

Stochastic simulations are used to study the linking properties of solutions of circular polymers in slit confinement. Specifically, we consider dispersions of semiflexible rings at various densities ϕ and slit height, H. The competing length scales in the system have significant effects on the interchain entanglement. We observe that the linking probability is largest for a specific slit height that is about independent of solution density. However, when ϕ is large, links with given number of components can significantly depart from the overall linking trend with H. In this case, binary links are found to be least probable when the overall incidence of links is maximum. We show that this intriguing dichotomy and other properties, including upper bounds on links abundance, can be quantitatively captured with an approximate model based on continuum percolation theory. Our results suggest that slit-like confinement could be used in applicative contexts to control independently both the average abundance and...

Connectivity and automation are expected to enhance safety and efficiency in transportation systems. Connectivity will provide information to drivers/autonomous vehicles to enhance decision-making reliability at the operational and tactical levels. Consequently, drivers are more likely to execute safe and efficient maneuvers and autonomous vehicles will have a more accurate perception of the traffic condition and an “error-free” execution of the driving maneuvers. At the operational level, ensuring string stability is a key consideration since unstable traffic flow results in shockwave propagation and possibly crashes. While several studies have examined the effects of information availability on string stability in a connected environment, most of the approaches are focused on automated driving (e.g., Cooperative Adaptive Cruise Control systems) and do not consider a mixed environment with regular, connected, and autonomous vehicles. To ensure connectivity in such a mixed environment, the correlation between communication range and connected vehicles density should be considered. To capture the effects of this correlation, this study uses the Continuum Percolation theory to determine the effects of the vehicular density and communication range on the connectivity level in telecommunications network and consequently, on the string stability of traffic flow. Based on the Continuum Percolation theory, there is a critical density of connected vehicles above which the entire system is connected. This critical density also depends on the communication range. This study presents an analytical derivation of this critical density. Moreover, this study evaluates the string stability under different communication ranges and market penetration rates of connected and autonomous vehicles. Results revealed that as communication range increases, the system becomes more stable and at high communication ranges, the system performs similar to the system with full connectivity. Moreover, results indicated the existence of an optimal communication range to maximize stability and ensure information delivery.

The application of the backbone effect in the transport throughput capacity of materials is considered.A novel parallel implementation of the backbone algorithm is proposed for percolation lattices.The theoretical algorithm complexity is estimated for data-flow implementation.The results are analyzed regarding the classical graph-based approaches and the execution efficiency. The paper proposes a new data-flow based approach for the identification of backbones in infinite clusters on 2-D percolation site lattices of dimension L × L. The infinite cluster is identified first, then a multi step algorithm is applied for the reduction of the infinite cluster to its backbone. Algorithm performances are evaluated theoretically and experimentally. The algorithm is local and can therefore be efficiently implemented on data-flow parallel platforms in ź(L) time if applied on percolation lattices near the critical percolation probability or in ź(L2) in the worst case. The proposed methodology could resolve the problem of stack overflow at large systems that can appear with classical graph based algorithms, and has potential for a higher execution speed-up on parallel architectures.

The aim of this article is to study the forest composed by point-to-line geodesics in the last-passage percolation model with exponential weights. We will show that the location of the root can be described in terms of the maxima of a random walk, whose distribution will depend on the geometry of the substrate (line). For flat substrates, we will get power law behaviour of the height function, study its scaling limit, and describe it in terms of variational problems involving the Airy process.

We review results of two previous papers on the asymptotic behavior of finite connection probabilities in three or more dimensions for Bernoulli percolation and the Fortuin–Kasteleyn random-cluster model. In the introduction, we prove a multidimensional renewal theorem that is needed for these results and previous results on Ornstein–Zernike behavior; the proof is significantly simpler than that originally derived by Doney (1966) and those of other subsequent works on this subject.

We consider a directed variant of the negative-weight percolation model in a two-dimensional, periodic, square lattice. The problem exhibits edge weights which are taken from a distribution that allows for both positive and negative values. Additionally, in this model variant all edges are directed. For a given realization of the disorder, a minimally weighted loop/path configuration is determined by performing a non-trivial transformation of the original lattice into a minimum weight perfect matching problem. For this problem, fast polynomial-time algorithms are available, thus we could study large systems with high accuracy. Depending on the fraction of negatively and positively weighted edges in the lattice, a continuous phase transition can be identified, whose characterizing critical exponents we have estimated by a finite-size scaling analyses of the numerically obtained data. We observe a strong change of the universality class with respect to standard directed percolation, as well as with respect to undirected negative-weight percolation. Furthermore, the relation to directed polymers in random media is illustrated., Comment: 8 pages, 8 figures

Permeability k and formation factor F are important petrophysical properties that have broad applications to reservoir engineering. Although well-log analysis has advanced reservoir characterizations, direct measurement of k is still not available. Accordingly, resistivity logs and more particularly formation factor have been frequently used to estimate k. In this study, we present a theoretical framework to relate k to F. We use power-law scaling from continuum percolation theory to develop a theoretic relationship between k and F, which is similar in form to a published power-law equation with empirical coefficients. Our result gives a physical interpretation to this empirical relationship. We also compare power-law scaling laws from percolation theory with data derived from experimental measurements and numerical simulations in the literature. Using the Curve Fitting toolbox of MATLAB, the parameters of the power-law equations are determined, and the significance of the non-universal power laws are discussed.

Graphene-based polymers exhibit a conductive microstructure formed by aggregates in a matrix which drastically enhances their transmitting properties. We develop a new numerical framework for predicting the electrical conductivity based on continuum percolation theory in a two dimensional stochastically-generated medium. We analyze the role of the flake shape and its aspect ratio and consequently predict the onset of percolation based on the particle density and the domain scale. Simultaneously, we have performed experiments and have achieved very high electrical conductivity for such composites compared to other film fabrication techniques, which have verified the results of computing the homogenized electrical conductivity. As well, the proximity to and a comparison with other analytical models and other experimental techniques are presented. The numerical model can predict the composite transmitting conductivity in a larger range of particle geometry. Such quantification is exceedingly useful for effective utilization and optimization of graphene filler densities and their spatial distribution during manufacturing.

The connectivity between two points in porous media is important for evaluating hydrocarbon recovery in underground reservoirs or toxic migration in waste disposal. For example, the connectivity between a producer and an injector in a hydrocarbon reservoir impact the fluid dispersion throughout the system. The conventional approach, flow simulation, is computationally very expensive and time consuming. Alternative method employs percolation theory. Classical percolation approach investigates the connectivity between two lines (representing the wells) in 2D cross sectional models whereas we look for the connectivity between two points (representing the wells) in 2D aerial models. In this study, site percolation is used to determine the fraction of permeable regions connected between two cells at various occupancy probabilities and system sizes. The master curves of mean connectivity and its uncertainty are then generated by finite size scaling. The results help to predict well-to-well connectivity without need to any further simulation.

We consider a random graph created by the long-range percolation on the nth stage finite subset of a fractal lattice called the pre-Sierpinski gasket. The long-range percolation is a stochastic model in which any pair of two points is connected by a random bond independently. On the random graph obtained as above, we consider a discrete-time random walk. We show that the mixing time of the random walk is of order \(2^{(s-d)n}\) if \(d

During routine operation, electrically percolating nanocomposites are subjected to high voltages, leading to spatially heterogeneous current distribution. The heterogeneity implies localized self-heating that may (self-consistently) reroute the percolation pathways and even irreversibly damage the material. In the absence of experiments that can spatially resolve the current distribution and a nonlinear percolation model suitable to interpret them, one relies on empirical rules and safety factors to engineer these materials. In this paper, we use ultrahigh resolution thermo-reflectance imaging, coupled with a new imaging processing technique, to map the spatial distribution ΔT(x, y; I) and histogram f(ΔT) of temperature rise due to self-heating in two types of 2D networks (percolating and copercolating). Remarkably, we find that the self-heating can be described by a simple two-parameter Weibull distribution, even under voltages high enough to reconfigure the percolation pathways. Given the generality of the phenomenological argument supporting the distribution, other percolating networks are likely to show similar stress distribution in response to sufficiently large stimuli. Furthermore, the spatial evolution of the self-heating of network was investigated by analyzing the spatial distribution and spatial correlation, respectively. An estimation of degree of hotspot clustering reveals a mechanism analogous to crystallization physics. The results should encourage nonlinear generalization of percolation models necessary for predictive engineering of nanocomposite materials.

In this paper we are concerned with contact processes on open clusters of oriented bond percolation in Zd, where the disease spreads along the direction of open edges. We show that the two critical values in the quenched and annealed cases are equal with probability one and are asymptotically equal to (dp)−1 as the dimension d grows to infinity, where p is the probability that the edge is ‘open’.

Percolation theory is used to model intrinsic and relative permeabilities as well as tortuosity in anisotropic carbon paper gas diffusion layers (GDL) and compared with existing results from lattice-Boltzmann (LB) simulations and experimental measurements. Although single- and two-phase characteristics of the carbon paper GDL are mainly affected by medium geometrical and topological properties, e.g., pore-size distribution, connectivity, and pore geometry, analyzing capillary pressure curves implies that the pore-size distribution of the carbon paper GDL is very narrow. This suggests that its effect on tortuosity and wetting- and nonwetting-phase relative permeabilities is trivial. However, integrated effects of pore geometry, surface area, connectivity, and tortuosity on intrinsic permeability might be substantial. Universal power laws from percolation theory predict the tortuosity-porosity and relative permeability–saturation curves accurately, indicating both characteristics not affected by the pore-size distribution. The permeability–porosity relationship, however, conforms to nonuniversality.

We study the percolation transition in evolving scale-free networks. A new node is added at each step and is connected to a random number of old nodes according to the preferential attachment mechanism. We give the critical value of the emergence of the giant component and prove that the transition is of infinite order. We also obtain asymptotic expressions for the cluster size distribution in the subcritical, critical, and supercritical regimes.

We studied a model of semi-directed percolation on finite strips of the square and triangular lattices. Using the transfer-matrix method, combined with phenomenological renormalization group approach, we obtain good numerical estimates for critical probabilities and correlation lengths critical exponents. Our results confirm the conjecture that semi-directed percolation belongs to the universality class of the usual fully-directed percolation model.

In this paper we consider the Ising model on the triangular percolation lattice and analyze its geometrical interfaces and spin clusters. The (site) percolation lattice is tuned by the occupancy parameter p which is the probability that a site is magnetic. Some statistical observables are studied in terms of temperature (T) and p. We find two separate (second order) transition lines, namely magnetic and percolation transition lines. The finite size analysis shows that the magnetic transition line is a critical one with varying exponents, having its root in the fact that the line is composed of individual critical points, or that a cross-over occurs between two (UV and IR) fixed points. For the percolation transition line however the exponents seem to be identical. Schramm–Loewner evolution (SLE) is employed to address the problem of conformal invariance at the points on the magnetic transition line. We find that at p ≃ 0.9 the model is described by κ ≃ 4 whose corresponding central charge is maximum with respect to the others.

The overlapping sphere (OS) percolation model gives a two-phase microstructure (matrix plus inclusions) that is useful for testing composite material ideas and other applications, as well as serving as a paradigm of overlapping object percolation and phase transitions. Real materials often have more than two phases, so it is of interest to extend the applicability of the OS model. A flexible variant of the OS model can be constructed by randomly assigning the spheres different phase labels, according to a uniform probability distribution, as they are inserted one by one into the matrix. The resulting three or more phase models can have different amounts of percolating and non-percolating phases, depending on the volume fraction of each phase and the total OS volume fraction. A three-dimensional digital image approach is used to approximately map out the percolation phase diagram of such models, explicitly up to four phases (one matrix plus three spherical inclusion phases) and implicitly for N > 4 phases. For the three phase model, it was found that a single OS sub-phase has a percolation threshold that ranges from about a volume fraction of 0.16, when the matrix volume fraction is about 0.01, to about 0.30, at a matrix volume fraction of about 0.7. The approximate analytical dependence of this sub-phase percolation threshold on the defining model parameters serves to guide the building of the percolation phase diagram for the N -phase model, and is used to determine the maximum value of N ( N = 6 ) at which all N phases can be simultaneously percolated.

In this paper, we investigate connectivity and transmission delay of secondary users in large-scale wireless cognitive radio (CR) ad hoc networks from a percolation-based perspective. Using the random connection model, we study the unreliability of wireless secondary links in CR ad hoc networks, which has not been well studied in the existing literature. By introducing two auxiliary random graphs and using continuum percolation theory, we study the impacts of key system parameters on connectivity and transmission delay in a CR network. We first characterize three behavioral regions of connectivity for a secondary network, i.e., disconnectivity, long-term connectivity and instantaneous connectivity regions. We show that the unreliability of secondary links does not affect the disconnectivity region, but affects the long-term connectivity and instantaneous connectivity regions. Using the ergodic theorem, we then study the scaling behavior of transmission delay with respect to the distance between two randomly chosen secondary users in a connected secondary network for two cases. Specifically, when propagation delay is negligible, we show that transmission delay scales linearly and sub-linearly with distance in the long-term connectivity and instantaneous connectivity regions, respectively. When propagation delay is considered, we show that transmission delay scales linearly with respect to distance in both the long-term connectivity and instantaneous connectivity regions.

We introduce a guided network growth model, which we call the degree product rule process, that uses solely local information when adding new edges. For small numbers of candidate edges our process gives rise to a second order phase transition, but becomes first order in the limit of global choice. We provide the set of critical exponents required to characterize the nature of this percolation transition. Such a process permits interventions which can delay the onset of percolation while tempering the explosiveness caused by cluster product rule processes., 6 pages, 5 figures, 1 table

We consider the Poisson Boolean percolation model in R2, where the radius of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in Rd, for any d≥2, finite moment of order d is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set., Electronic Communications in Probability, 23, ISSN:1083-589X

Given an initially uncovered field, and as more and more directional sensors (sensors with sector shape sensing area) are continuously added to the sensor network, the size of partial covered areas increases. At some point, the situation abruptly changes from small fragmented covered areas to a single large covered area. This abrupt change is called the sensing-coverage phase transition (SCPT). Likewise, given an originally disconnected sensor network, as more and more sensors are added, the number of connected components changes such that the sensor network suddenly becomes connected. This sudden change is called the network connectivity phase transition (NCPT). Such phase transitions occur in a certain density which is called critical density and finding it is a central topic of Percolation Theory. In this paper, we introduce fixed-orientation directional sensor networks (FIODSNs) and analytically compute critical density of nodes for both SCPT and NCPT in FIODSNs, for all field-of-view angles between 0 and π using continuum percolation. In FIODSNs which are the most common type of directional sensor networks, sensor nodes are deployed based on Poisson Point Process, and orientation of them is distributed between 0 and 2 π , independently and uniformly. Due to percolation theory, the critical density is the infimum density for densities above it SCPT and NCPT almost surely occur. Therefore, the results could be used to prepare coverage and connectivity in FIODSNs. Moreover, to solve the SCPT and NCPT problems together, we propose a model for percolation in directional sensor networks which could be used in other related researches.

In this paper, a domain in a cube is called a coverage hole if it is not covered by the largest component of the random geometric graph in this cube. We obtain asymptotic properties of the size of the largest coverage hole in the cube. In addition, we give an exponentially decaying tail bound for the probability that a line with length s do not intersect with the coverage of the infinite component of continuum percolation. These results have applications in communication networks and especially in wireless ad-hoc sensor networks.

Extension of ways of connection to 13 compared to 4 possible ways in prior arts.Uni-direction connection is considered instead of bi-direction connection.Novel small structures are proposed to help derive desired theorems.New conditions for percolation is established.New conditions for full connection is established. In this work, we discuss when two users are able to exchange multimedia content and when that exchange is always possible. Two thresholds are of interest, a threshold for percolation and a threshold for full connectivity. To derive these thresholds, when discussing the bond percolation, we consider not only four directions, upper, right, left, and lower, but also more directions like upper-right, lower-right, upper-left, lower-left. In addition, small structures are proposed to help obtain the necessary conditions for percolation and full connectivity. We find that if the probability of four directions being open is greater than 0.3118 or the transmission radius of every node is longer than 1.52 d / k , we can find a crossing path in our network and the percolation ensues. Furthermore, if that is greater than 0.3799 or the transmission radius of every node is longer than 1.579 d / k , a fully-connected network graph exists for sure.

This paper investigates the permeability of microcracked porous solids containing 2D random crack networks. The past works on permeability of crack networks are firstly reviewed. The geometry analysis is performed on numerical samples of crack networks with different crack length distributions, crack densities, domain size ratios and clustering degrees. The parameters from continuum percolation theory are used to characterize the geometry of random networks including the percolation threshold, the scaling exponents for percolation probability and correlation length of crack clusters, and the fractal dimension of spanning clusters. The crack density is used as the basic percolation variable, and a new connectivity factor is proposed for the cluster spanning in finite domain. Then the effective permeability of porous matrix containing 2D random crack networks is analyzed on numerical samples via finite element method. A scaling law for effective permeability is established near the percolation threshold taking into account the matrix permeability, crack opening aperture, crack connectivity and tortuosity. The results from geometry analysis and permeability analysis show that: (1) The new connectivity factor is proved pertinent to network percolation, related to both crack density and crack clustering degree; (2) the percolation parameters of uncorrelated crack networks are rather near to the universal values from the continuum percolation theory, but their values change with the clustering degree of crack networks; (3) the numerical results confirm the scaling law proposed for effective permeability, and the permeability is found to scale with the crack opening through a power law.

Statistical simulation of distributed sets of objects integrated into a large network and forming an operating environment is considered. The simulation is carried out on square matrices with random filling using percolation theory. Statistical features of the clusters of objects that arise in this case are analyzed. Along with well-known stochastic percolation, the concept of a controlled percolation of the operating environment is introduced, which is implemented in two phases: in the first phase, a stochastic basis with a relatively low concentration of objects is created, and in the second phase, additional objects are introduced in the intercluster intervals in order to obtain the shortest percolation path while minimizing the total cost of such two-phase operation. This model is used to consider vehicle navigation by visual landmarks randomly distributed in the operating environment.

We investigate a percolation process where an additional parameter q is used to interpolate between the classical Erdös–Rényi (ER) network model and the smallest cluster (SC) model. This model becomes the ER network at q = 1, which is characterized by a robust second order phase transition. When q = 0, this model recovers to the SC model which exhibits a first order phase transition. To study how the percolation phase transition changes from second order to first order with the decrease of the value of q from 1 to 0, the numerical simulations study the final vanishing moment of the each existing cluster except the N-cluster in the percolation process. For the continuous phase transition, it is shown that the tail of the graph of the final vanishing moment has the characteristic of the convexity. While for the discontinuous phase transition, the graph of the final vanishing moment possesses the characteristic of the concavity. Just before the critical point, it is found that the ratio between the maximum of the sequential vanishing clusters sizes and the network size N can be used to decide the phase transition type. We show that when the ratio is larger than or equal to zero in the thermodynamic limit, the percolation phase transition is first or second order respectively. For our model, the numerical simulations indicate that there exists a tricritical point qc which is estimated to be between 0.2 < qc < 0.25 separating the two phase transition types.

Complex networks appear in almost every aspect of science and technology. Previous work in network theory has focused primarily on analyzing single networks that do not interact with other networks, despite the fact that many real-world networks interact with and depend on each other. Very recently an analytical framework for studying the percolation properties of interacting networks has been introduced. Here we review the analytical framework and the results for percolation laws for a Network Of Networks (NONs) formed by n interdependent random networks. The percolation properties of a network of networks differ greatly from those of single isolated networks. In particular, because the constituent networks of a NON are connected by node dependencies, a NON is subject to cascading failure. When there is strong interdependent coupling between networks, the percolation transition is discontinuous (first-order) phase transition, unlike the wellknown continuous second-order transition in single isolated networks. Moreover, although networks with broader degree distributions, e.g., scale-free networks, are more robust when analyzed as single networks, they become more vulnerable in a NON. We also review the effect of space embedding on network vulnerability. It is shown that for spatially embedded networks any finite fraction of dependency nodes will lead to abrupt transition. 2014 Elsevier Ltd. All rights reserved.

A mathematical model has been proposed for a neural network whose morphological structure varies dynamically depending on activity. This is the property of the so-called structural plasticity typical of developed neural systems of a brain. It has been shown that the spontaneous generation and propagation of a signal in such networks correspond to a percolation transition and the appearance of the connectivity component covering the entire system. Furthermore, adaptive change in the geometric structure of a network results in the clustering of cells and in the reduction of the effective percolation threshold, which corresponds to experimental neurobiological observations.

We study site percolation on Angel & Schramm's uniform infinite planar triangulation. We compute several critical and near-critical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes (subcritical, critical and supercritical). We prove in particular that the scaling limit of the boundary of large critical percolation clusters is the random stable looptree of index 3/2, which was introduced in a companion paper. We also give a conjecture linking looptrees of any index in (1,2) with scaling limits of cluster boundaries in random triangulations decorated with O(N) models., 30 pages, 12 figures. Published version (Probab. Theory Related Fields)

Hyperbolic lattices interpolate between finite-dimensional lattices and Bethe lattices and are interesting in their own right with ordinary percolation exhibiting not one, but two, phase transitions. We study four constraint percolation models---$k$-core percolation (for $k=1,2,3$) and force-balance percolation---on several tessellations of the hyperbolic plane. By comparing these four different models, our numerical data suggests that all of the $k$-core models, even for $k=3$, exhibit behavior similar to ordinary percolation, while the force-balance percolation transition is discontinuous. We also provide a proof, for some hyperbolic lattices, of the existence of a critical probability that is less than unity for the force-balance model, so that we can place our interpretation of the numerical data for this model on a more rigorous footing. Finally, we discuss improved numerical methods for determining the two critical probabilities on the hyperbolic lattice for the $k$-core percolation models., 10 pages, 15 figures

Percolation theory characterizing the robustness of a network has applications ranging from biology, to epidemic spreading, and complex infrastructures. Percolation theory, however, only concern the typical response of a infinite network to random damage of its nodes while in real finite networks, fluctuations are observable. Consequently for finite networks there is an urgent need to evaluate the risk of collapse in response to rare configurations of the initial damage. Here we build a large deviation theory of percolation characterizing the response of a sparse network to rare events. This general theory includes the second order phase transition observed typically for random configurations of the initial damage but reveals also discontinuous transitions corresponding to rare configurations of the initial damage for which the size of the giant component is suppressed., (11 pages, 4 figures)

We consider first passage percolation on the configuration model with n vertices, and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a uniform X2 logX-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path namely the hopcount. Writing Ln for the weight of the optimal path, we show that Ln- (log n)/αn converges to a limiting random variable, for some sequence αn. Furthermore, the hopcount satisfies a central limit theorem (CLT) with asymptotic mean and variance of order log n. The sequence αn and the norming constants for the CLT are expressible in terms of the parameters of an associated continuous-time branching process that describes the growth of neighborhoods around a uniformly chosen vertex in the random graph. The limit of Ln-(log n)/ αn equals the sum of the logarithm of the product of two independent martingale limits, and a Gumbel random variable. So far, for sparse random graph models, such results have only been shown for the special case where the edge weights have an exponential distribution, wherein the Markov property of this distribution plays a crucial role in the technical analysis of the problem. The proofs in the paper rely on a refined coupling between shortest path trees and continuous-time branching processes, and on a Poisson point process limit for the potential closing edges of shortest-weight paths between the source and destination.