Clique percolation in random networks

The notion of k-clique percolation in random graphs is introduced, where k is the size of the complete subgraphs whose large scale organizations are analytically and numerically investigated. For the Erdos-Rényi graph of N vertices we obtain that the percolation transition of k-cliques takes place when the probability of two vertices being connected by an edge reaches the threshold p(c) (k) = [(k - 1)N](-1/(k - 1)). At the transition point the scaling of the giant component with N is highly nontrivial and depends on k. We discuss why clique percolation is a novel and efficient approach to the identification of overlapping communities in large real networks.     

A self-organizing shortest path finding strategy on complex networks

The shortcomings of traditional methods to find the shortest path are revealed, and a strategy of finding the self-organizing shortest path based on thermal flux diffusion on complex networks is presented. In our method, the shortest paths between the source node and the other nodes are found to be self-organized by comparing node temperatures. The computation complexity of the method scales linearly with the number of edges on underlying networks. The effects of the method on several networks, including a regular network proposed by Ravasz and Barabási which is called the RB network, a real network, a random network proposed by Ravasz and Barabási which is called the ER network and a scale-free network, are also demonstrated. Analytic and simulation results show that the method has a higher accuracy and lower computational complexity than the conventional methods.     

Depletion-induced percolation in networks of nanorods

Above a certain density threshold, suspensions of rodlike colloidal particles form system-spanning networks. Using Monte Carlo simulations, we investigate how the depletion forces caused by spherical particles affect these networks in isotropic suspensions of rods. Although the depletion forces are strongly anisotropic and favor alignment of the rods, the percolation threshold of the rods decreases significantly. The relative size of the effect increases with the aspect ratio of the rods. The structural changes induced in the suspension by the depletant are characterized in detail and the system is compared to an ideal fluid of freely interpenetrable rods.     

二分网上的靴襻渗流

靴襻渗流最早应用于统计物理学中研究磁铁因非磁性杂质导致磁有序的降低并最终消失的现象. 随着复杂网络研究的深入, 许多学者展开网络上的靴襻渗流研究. 在自然界中, 许多系统自然呈现出二分结构, 二分网络是复杂网络中的一种重要的网络模式. 本文通过建立动力学方程和计算机仿真模拟的方法研究二分网上的靴襻渗流, 关注的参数是二分网中两类节点初始的活跃比例和活跃阈值, 分别用f₁, f₂和Ω₁, Ω₂表示, 得到二分网两类节点终态活跃比例随初始活跃比例的变化会发生相变等结论. 同时 验证了动力学方程与仿真模拟的一致性.     

Weighted percolation on directed networks

We present and numerically test an analysis of the percolation transition for general node removal strategies valid for locally treelike directed networks. On the basis of heuristic arguments we predict that, if the probability of removing node i is p(i), the network disintegrates if p(i) is such that the largest eigenvalue of the matrix with entries A(ij)(1-p(i)) is less than 1, where A is the adjacency matrix of the network. The knowledge or applicability of a Markov network model is not required by our theory, thus making it applicable to situations not covered by previous works.     

Connectivity and percolation in simulated grain-boundary networks

Random percolation theory is a common basis for modelling intergranular phenomena such as cracking, corrosion or diffusion. However, crystallographic constraints in real microstructures dictate that grain boundaries are not assembled at random. In this work a Monte Carlo method is used to construct physically realistic networks composed of high-angle grain boundaries that are susceptible to intergranular attack, as well as twin-variant boundaries that are damage resistant. When crystallographic constraints are enforced, the simulated networks exhibit triple-junction distributions that agree with experiment and reveal the non-random nature of grain-boundary connectivity. The percolation threshold has been determined for several constrained boundary networks and is substantially different from the classical result of percolation theory; compared with a randomly assembled network, about 50-75% more resistant boundaries are required to break up the network of susceptible boundaries. Triple-junction distributions are also shown to capture many details of the correlated percolation problem and to provide a simple means of ranking microstructures.     

Limited universality at the percolation threshold in 2 to 6 dimensions

The universality of the spanning fraction R(p) of percolation is confirmed by comparing bond percolation with site-bond percolation in four to six dimensions. However, different boundary conditions change the universality class, as shown also for site percolation in two dimensions.     

Core percolation and onset of complexity in boolean networks

The determination and classification of fixed points of large Boolean networks is addressed in terms of a constraint-satisfaction problem. We develop a general simplification scheme that, removing all those variables and functions belonging to trivial logical cascades, returns the computational core of the network. The transition line from an easy to a complex regulatory phase is described as a function of the parameters of the model, identifying thereby both theoretically and algorithmically the relevant regulatory variables.     

From invasion percolation to flow in rock fracture networks

The main purpose of this work is to simulate two-phase flow in the form of immiscible displacement through anisotropic, three-dimensional (3D) discrete fracture networks (DFN). The considered DFNs are artificially generated, based on a general distribution function or are conditioned on measured data from deep geological investigations. We introduce several modifications to the invasion percolation (MIP) to incorporate fracture inclinations, intersection lines, as well as the hydraulic path length inside the fractures. Additionally a trapping algorithm is implemented that forbids any advance of the invading fluid into a region, where the defending fluid is completely encircled by the invader and has no escape route. We study invasion, saturation, and flow through artificial fracture networks, with varying anisotropy and size and finally compare our findings to well studied, conditioned fracture networks.     

Height-height correlations for surface growth on percolation networks

The height-height correlations of the surface growth for equilibrium and nonequilibrium restricted solid-on-solid (RSOS) model were investigated on randomly diluted lattices, i.e., on infinite percolation networks. It was found that the correlation function calculated over the chemical distances reflected the dynamics better than that calculated over the geometrical distances. For the equilibrium growth on a critical percolation network, the correlation function for the evolution time t?1 yielded a power-law behavior with the power ζ^′, associated with the roughness exponent ζ via the relation ζ=ζ^′d_f/d_l, with d_f and d_l being, respectively, the fractal dimension and the chemical dimension of the substrate. For the nonequilibrium growth, on the other hand, the correlation functions did not yield power-law behaviors for the concentration of diluted sites x less than or equal to the critical concentration x_c.     

Growth, percolation, and correlations in disordered fiber networks

This paper studies growth, percolation, and correlations in disordered fiber networks. We start by introducing a 2D continuum deposition model with effective fiber-fiber interactions represented by a parameterp which controls the degree of clustering. Forp=1 the deposited network is uniformly random, while forp=0 only a single connected cluster can grow. Forp=0 we first derive the growth law for the average size of the cluster as well as a formula for its mass density profile. Forp>0 we carry out extensive simulations on fibers, and also needles and disks, to study the dependence of the percolation threshold onp. We also derive a mean-field theory for the threshold nearp=0 andp=1 and find good qualitative agreement with the simulations. The fiber networks produced by the model display nontrivial density correlations forp<1. We study these by deriving an approximate expression for the pair distribution function of the model that reduces to the exactly known case of a uniformly random network. We also show that the two-point mass density correlation function of the model has a nontrivial form, and discuss our results in view of recent experimental data on mss density correlations in paper sheets.     

Accessibility in complex networks

This Letter describes a method for the quantification of the diversity of non-linear dynamics in complex networks as a consequence of self-avoiding random walks. The methodology is analyzed in the context of theoretical models and illustrated with respect to the characterization of the accessibility in urban streets.     

Synchronization in complex networks

Synchronization processes in populations of locally interacting elements are the focus of intense research in physical, biological, chemical, technological and social systems. The many efforts devoted to understanding synchronization phenomena in natural systems now take advantage of the recent theory of complex networks. In this review, we report the advances in the comprehension of synchronization phenomena when oscillating elements are constrained to interact in a complex network topology. We also take an overview of the new emergent features coming out from the interplay between the structure and the function of the underlying patterns of connections. Extensive numerical work as well as analytical approaches to the problem are presented. Finally, we review several applications of synchronization in complex networks to different disciplines: biological systems and neuroscience, engineering and computer science, and economy and social sciences.     

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.     

On the numerical study of percolation and epidemic critical properties in networks

The static properties of the fundamental model for epidemics of diseases allowing immunity (susceptible-infected-removed model) are known to be derivable by an exact mapping to bond percolation. Yet when performing numerical simulations of these dynamics in a network a number of subtleties must be taken into account in order to correctly estimate the transition point and the associated critical properties. We expose these subtleties and identify the different quantities which play the role of criticality detector in the two dynamics.