Percolation transitions are not always sharpened by making networks interdependent

We study a model for coupled networks introduced recently by Buldyrev et al., [Nature (London) 464, 1025 (2010)], where each node has to be connected to others via two types of links to be viable. Removing a critical fraction of nodes leads to a percolation transition that has been claimed to be more abrupt than that for uncoupled networks. Indeed, it was found to be discontinuous in all cases studied. Using an efficient new algorithm we verify that the transition is discontinuous for coupled Erd?s-Rényi networks, but find it to be continuous for fully interdependent diluted lattices. In 2 and 3 dimensions, the order parameter exponent β is larger than in ordinary percolation, showing that the transition is less sharp, i.e., further from discontinuity, than for isolated networks. Possible consequences for spatially embedded networks are discussed.     

Modeling wireless sensor networks using random graph theory

A critical issue in wireless sensor networks (WSNs) is represented by limited availability of energy within network nodes. Therefore, making good use of energy is necessary in modeling sensor networks. In this paper we proposed a new model of WSNs on a two-dimensional plane using site percolation model, a kind of random graph in which edges are formed only between neighbouring nodes. Then we investigated WSNs connectivity and energy consumption at percolation threshold when a so-called phase transition phenomena happen. Furthermore, we proposed an algorithm to improve the model; as a result the lifetime of networks is prolonged. We analyzed the energy consumption with Markov process and applied these results to simulation.     

Bulk transport properties and exponent inequalities for random resistor and flow networks

The properties of random resistor and flow networks are studied as a function of the density,p, of bonds which permit transport. It is shown that percolation is sufficient for bulk transport, in the sense that the conductivity and flow capacity are bounded away from zero wheneverp exceeds an appropriately defined percolation threshold. Relations between the transport coefficients and quantities in ordinary percolation are also derived. Assuming critical scaling, these relations imply upper and lower bounds on the conductivity and flow exponents in terms of percolation exponents. The conductivity exponent upper bound so derived saturates in mean field theory.Research supported by the NSF under Grant No. DMR-8314625Research supported by the DOE under Grant No. DE-AC02-83ER13044     

含随机裂纹网络孔隙材料渗透率的逾渗模型研究

采用逾渗理论对含随机裂纹网络的孔隙材料渗透性进行研究. 开裂孔隙材料渗透率的影响因素包括裂纹网络的几何特征、孔隙材料本体渗透率以及裂纹开度, 本文使用连续区逾渗理论模型建立了渗透率的标度律. 对于裂纹网络的几何特征, 本文基于连续区逾渗理论并考虑裂纹网络的分形特征提出了有限区域内二维随机裂纹网络的连通度定义; 对随机裂纹网络的几何分析表明, 随机裂纹局部团簇效应会降低裂纹网络的整体连通性, 随机裂纹网络的标度指数并非经典逾渗理论给出的固定值, 而是随着网络的分形维数的减小而增大. 本文在网络连通度和主裂纹团的曲折度的基础上, 提出了开裂孔隙材料渗透率标度律的解析表达, K=K₀(K_m,b)(ρ-ρ_c)^μ, 分别考虑了裂纹网络的几何逾渗特征 (ρ-ρ_c)^μ、孔隙材料渗透率K_m 以及裂纹开度比b; 对有限区域含有随机裂纹网络的孔隙材料渗透过程的有限元模拟表明, K₀ 在裂纹逾渗阈值附近与b呈指数关系, 但当裂纹的局部渗透率与K_m比值高于10⁶ 后, 开度比b对渗透率不再有影响.     

Maximal spanning trees,asset graphs and random matrix denoising in the analysis of dynamics of financial networks

We study the time dependence of maximal spanning trees and asset graphs based on correlation matrices of stock returns. In these networks the nodes represent companies and links are related to the correlation coefficients between them. Special emphasis is given to the comparison between ordinary and denoised correlation matrices. The analysis of single- and multi-step survival ratios of the corresponding networks reveals that the ordinary correlation matrices are more stable in time than the denoised ones. Our study also shows that some information about the cluster structure of the companies is lost in the denoising procedure. Cluster structure that makes sense from an economic point of view exists, and can easily be observed in networks based on denoised correlation matrices. However, this structure is somewhat clearer in the networks based on ordinary correlation matrices. Some technical aspects, such as the random matrix denoising procedure, are also presented.     

A self-consistent Ornstein-Zernike approximation for the site-diluted Ising model

We propose a theory for the site-diluted Ising model which is an extension to disordered systems of the self-consistent Ornstein-Zernike approximation of Hoye and Stell. By using the replica method in the context of liquid-state theory, we treat the concentration of impurities as an ordinary thermodynamic variable. This approach is not limited to the weak-disorder regime or to the vicinity of the percolation point. A preliminary analysis using series expansion shows that it can predict accurately the dependence of the critical temperature on dilution and can reproduce the nonuniversal behavior of the effective exponents. The theory also gives a reasonable estimate of the percolation threshold.     

Asymmetrically coupled directed percolation systems

We introduce a dynamical model of coupled directed percolation systems with two particle species. The two species A and B are coupled asymmetrically in that A particles branch B particles, whereas B particles prey on A particles. This model may describe epidemic spreading controlled by reactive immunization agents. We study nonequilibrium phase transitions with attention focused on the multicritical point where both species undergo the absorbing phase transition simultaneously. In one dimension, we find that the inhibitory coupling from B to A is irrelevant and the model belongs to the unidirectionally coupled directed percolation class. On the contrary, a mean-field analysis predicts that the inhibitory coupling is relevant and a new universality appears with a variable dynamic exponent. Numerical simulations on small-world networks confirm our predictions.     

Exact solution of a drop-push model for percolation

Motivated by a computer science algorithm known as "linear probing with hashing," we study a new type of percolation model whose basic features include a sequential "dropping" of particles on a substrate followed by their transport via a "pushing" mechanism. Our exact solution in one dimension shows that, unlike the ordinary random percolation model, the drop-push model has nontrivial spatial correlations generated by the dynamics itself. The critical exponents in the drop-push model are also different from those of the ordinary percolation. The relevance of our results to computer science is pointed out.     

Transport in dispersed ionic conductors: Effect of insulating particle size

Dispersed ionic conductors are random mixtures of a solid salt, e.g. AgI, LiI, with fine particles of an insulating second phase, like Al₂O₃ or SiO₂. These composites can show a dramatic increase in ionic conductivity compared to the pure homogeneous system. Generally, this observation is attributed to an increased conductivity along the internal interface between the conducting salt and the insulating material. In this work a three-component random resistor network (RRN) model for dispersed ionic conductors is reviewed. In the model, the ionic conductor is represented by normally conducting bonds, the insulating material by non-conducting bonds and the interface between the two phases by highly conducting bonds. A special feature of the model is the existence of two critical concentrations of the insulating phase, p′_c and p″_c , for interface percolation and bulk conduction, respectively, where critical transport properties corresponding to conductor/superconductor and conductor/insulator networks are predicted. The model describes satisfactorily the dependence on composition of the conductivity and activation energy of dispersed ionic conductors. Furthermore, the observed effect on the conductivity of the size of dispersed particles can be described qualitatively well by a generalized version of the RRN model, which in addition predicts a sensitive dependence of the critical thresholds on particle size. Non-universality features in the critical exponents for the conductivity are also discussed within a continuum percolation analog of the model.     

Percolation thresholds of a group of anisotropic three-dimensional fracture networks

Percolation thresholds (average number of connections per object) of two models of anisotropic three-dimensional (3D) fracture networks made of mono-disperse hexagons have been calculated numerically. The first model is when the fracture networks are comprised of two groups of fractures that are distributed in an anisotropic manner about two orthogonal mean directions, i.e., Z- and X-directions. We call this model bipolar anisotropic fracture network (BFN). The second model is when three groups of fractures are distributed about three orthogonal mean directions, that is Z-, X-, and Y-directions. In this model three families of fractures about three orthogonal mean directions are oriented in 3D space. We call this model tripolar anisotropic fracture network (TFN). The finite-size scaling method is used to predict the infinite percolation thresholds. The effect of anisotropicity on percolation thresholds in X-, Y-, and Z-directions is investigated. We have revealed that as the anisotropicity of networks increases, the percolation thresholds in X-, Y-, and Z-directions span the range of 2.3 to 2.0, where 2.3 and 2.0 are extremums of percolation thresholds for isotropic and non-isotropic orthogonal fracture networks, respectively.     

Network robustness and fragility: percolation on random graphs

Recent work on the Internet, social networks, and the power grid has addressed the resilience of these networks to either random or targeted deletion of network nodes or links. Such deletions include, for example, the failure of Internet routers or power transmission lines. Percolation models on random graphs provide a simple representation of this process but have typically been limited to graphs with Poisson degree distribution at their vertices. Such graphs are quite unlike real-world networks, which often possess power-law or other highly skewed degree distributions. In this paper we study percolation on graphs with completely general degree distribution, giving exact solutions for a variety of cases, including site percolation, bond percolation, and models in which occupation probabilities depend on vertex degree. We discuss the application of our theory to the understanding of network resilience.     

Percolation of the Loss of Tension in an Infinite Triangular Lattice

We introduce a new class of bootstrap percolation models where the local rules are of a geometric nature as opposed to simple counts of standard bootstrap percolation. Our geometric bootstrap percolation comes from rigidity theory and convex geometry. We outline two percolation models: a Poisson model and a lattice model. Our Poisson model describes how defects--holes is one of the possible interpretations of these defects--imposed on a tensed membrane result in a redistribution or loss of tension in this membrane; the lattice model is motivated by applications of Hooke spring networks to problems in material sciences. An analysis of the Poisson model is given by Menshikov et al. ⁽⁴⁾ In the discrete set-up we consider regular and generic triangular lattices on the plane where each bond is removed with probability 1–p. The problem of the existence of tension on such lattice is solved by reducing it to a bootstrap percolation model where the set of local rules follows from the geometry of stresses. We show that both regular and perturbed lattices cannot support tension for any p<1. Moreover, the complete relaxation of tension--as defined in Section 4--occurs in a finite time almost surely. Furthermore, we underline striking similarities in the properties of the Poisson and lattice models.     

Impurity diffusion in a harmonic potential and nonhomogeneous temperature

Classical blockmodel is known as the simplest among models of networks with community structure. The model can be also seen as an extremely simply example of interconnected networks. For this reason, it is surprising that the percolation transition in the classical blockmodel has not been examined so far, although the phenomenon has been studied in a variety of much more complicated models of interconnected and multiplex networks. In this paper we derive the self-consistent equation for the size the global percolation cluster in the classical blockmodel. We also find the condition for percolation threshold which characterizes the emergence of the giant component. We show that the discussed percolation phenomenon may cause unexpected problems in a simple optimization process of the multilevel network construction. Numerical simulations confirm the correctness of our theoretical derivations.     

An anti-attack model based on complex network theory in P2P networks

Complex network theory is a useful way to study many real systems. In this paper, an anti-attack model based on complex network theory is introduced. The mechanism of this model is based on a dynamic compensation process and a reverse percolation process in P2P networks. The main purpose of the paper is: (i) a dynamic compensation process can turn an attacked P2P network into a power-law (PL) network with exponential cutoff; (ii) a local healing process can restore the maximum degree of peers in an attacked P2P network to a normal level; (iii) a restoring process based on reverse percolation theory connects the fragmentary peers of an attacked P2P network together into a giant connected component. In this way, the model based on complex network theory can be effectively utilized for anti-attack and protection purposes in P2P networks.     

Amorphous freezing in two dimensions: From soft coils to rigid particles

The topic of the gel transition in two dimensions is revisited by considering data on the shear elasticity of Langmuir monolayers of different spherical objects. Amorphous freezing can be associated to structural percolation in a lattice able to resist shear stresses. The shear modulus and its dependence on the packing fraction are found to strongly depend on the details of the interaction potential and largely differ from expectations for entropic networks. This behaviour can be interpreted in terms of more elaborated percolation theories including central forces and bond-bending forces.     

Connectivity and percolation behaviour of grain boundary networks in three dimensions

Grain boundary networks are subject to crystallographic constraints at both triple junctions (first-order constraints) and quadruple nodes (second-order constraints). First-order constraints are known to influence the connectivity and percolation behaviour in two-dimensional grain boundary networks, and here we extend these considerations to fully three-dimensional microstructures. Defining a quadruple node distribution (QND) to quantify both the composition and isomerism of quadruple nodes, we explore how the QNDs for crystallographically consistent networks differ from that expected in a randomly assembled network. Configurational entropy is used to quantify the relative strength of each type of constraint (i.e., first- and second-order), with first-order triple junction constraints accounting for at least 75% of the non-random correlations in the network. As the dominant effects of constraint are captured by considering the triple junctions alone, a new analytical model is presented which allows the 3-D network connectivity to be estimated from data on 2-D microstructural sections. Finally, we show that the percolation thresholds for 3-D crystallographically consistent networks differ by as much as ±0.07 from those of standard percolation theory.     

Description of the degree of reinforcement of polymer/carbon nanotube nanocomposites in the framework of percolation models

A radically new percolation model for describing the extremal dependence of the degree of reinforcement of polymer/carbon nanotube nanocomposites on the nanofiller content has been proposed. It has been shown that, for this nanofiller, the percolation threshold almost coincides with the aggregation threshold on the concentration scale. From the structural point of view, the extremum of this dependence is caused by the change in the type of the reinforcing component (from interphase regions to the skeleton of carbon nanotubes). From the mathematical point of view, the behavior of the degree of reinforcement is described by the general percolation relationship with replacement of the critical exponents near the percolation threshold. Neither the functionalization of the nanofiller nor the preliminary ultrasound treatment qualitatively change the dependence under study.     

Explosive percolation transition is actually continuous

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

System of mobile agents to model social networks

We propose a model of mobile agents to construct social networks, based on a system of moving particles by keeping track of the collisions during their permanence in the system. We reproduce not only the degree distribution, clustering coefficient, and shortest path length of a large database of empirical friendship networks recently collected, but also some features related with their community structure. The model is completely characterized by the collision rate, and above a critical collision rate we find the emergence of a giant cluster in the universality class of two-dimensional percolation. Moreover, we propose possible schemes to reproduce other networks of particular social contacts, namely, sexual contacts.     

Low-concentration series in general dimension

We discuss recent work on the development and analysis of low-concentration series. For many models, the recent breakthrough in the extremely efficient no- free-end method of series generation facilitates the derivation of 15th-order series for multiple moments in general dimension. The 15th-order series have been obtained for lattice animals, percolation, and the Edwards-Anderson Ising spin glass. In the latter cases multiple moments have been found. From complete graph tables through to 13th order, general dimension 13th-order series have been derived for the resistive susceptibility, the moments of the logarithms of the distribution of currents in resistor networks, and the average transmission coefficient in the quantum percolation problem, 11th-order series have been found for several other systems, including the crossover from animals to percolation, the full resistance distribution, nonlinear resistive susceptibility and current distribution in dilute resistor networks, diffusion on percolation clusters, the dilute Ising model, dilute antiferromagnet in a field, and random field Ising model and self-avoiding walks on percolation clusters. Series for the dilute spin-1/2 quantum Heisenberg ferromagnet are in the process of development. Analysis of these series gives estimates for critical thresholds, amplitude ratios, and critical exponents for all dimensions. Where comparisons are possible, our series results are in good agreement with both-expansion results near the upper critical dimension and with exact results (when available) in low dimensions, and are competitive with other numerical approaches in intermediate realistic dimensions.