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.     

Percolation in self-similar networks

We provide a simple proof that graphs in a general class of self-similar networks have zero percolation threshold. The considered self-similar networks include random scale-free graphs with given expected node degrees and zero clustering, scale-free graphs with finite clustering and metric structure, growing scale-free networks, and many real networks. The proof and the derivation of the giant component size do not require the assumption that networks are treelike. Our results rely only on the observation that self-similar networks possess a hierarchy of nested subgraphs whose average degree grows with their depth in the hierarchy. We conjecture that this property is pivotal for percolation in networks.     

Shift of percolation thresholds for epidemic spread between static and dynamic small-world networks

The study compares the epidemic spread on static and dynamic small-world networks. They are constructed as a 2-dimensional Newman and Watts model (500 × 500 square lattice with additional shortcuts), where the dynamics involves rewiring shortcuts in every time step of the epidemic spread. We assume susceptible-infectious-removed (SIR) model of the disease. We study the behaviour of the epidemic over the range of shortcut probability per underlying bond ϕ = 0–0.5. We calculate percolation thresholds for the epidemic outbreak, for which numerical results are checked against an approximate analytical model. We find a significant lowering of percolation thresholds on the dynamic network in the parameter range given. The result shows the behaviour of the epidemic on dynamic network is that of a static small world with the number of shortcuts increased by 20.7±1.4 %, while the overall qualitative behaviour stays the same. We derive corrections to the analytical model which account for the effect. For both dynamic and static small worlds we observe suppression of the average epidemic size dependence on network size in comparison with the finite-size scaling known for regular lattice. We also study the effect of dynamics for several rewiring rates relative to infectious period of the disease.     

Percolation in living neural networks

We study living neural networks by measuring the neurons' response to a global electrical stimulation. Neural connectivity is lowered by reducing the synaptic strength, chemically blocking neurotransmitter receptors. We use a graph-theoretic approach to show that the connectivity undergoes a percolation transition. This occurs as the giant component disintegrates, characterized by a power law with an exponent beta approximately or = 0.65. Beta is independent of the balance between excitatory and inhibitory neurons and indicates that the degree distribution is Gaussian rather than scale free.     

Synchronization in complex clustered networks

Synchronization in complex networks has been an active area of research in recent years. While much effort has been devoted to networks with the small-world and scale-free topology, structurally they are often assumed to have a single, densely connected component. Recently it has also become apparent that many networks in social, biological, and technological systems are clustered, as characterized by a number (or a hierarchy) of sparsely linked clusters, each with dense and complex internal connections. Synchronization is fundamental to the dynamics and functions of complex clustered networks, but this problem has just begun to be addressed. This paper reviews some progress in this direction by focusing on the interplay between the clustered topology and network synchronizability. In particular, there are two parameters characterizing a clustered network: the intra-cluster and the inter-cluster link density. Our goal is to clarify the roles of these parameters in shaping network synchronizability. By using theoretical analysis and direct numerical simulations of oscillator networks, it is demonstrated that clustered networks with random inter-cluster links are more synchronizable, and synchronization can be optimized when inter-cluster and intra-cluster links match. The latter result has one counterintuitive implication: more links, if placed improperly, can actually lead to destruction of synchronization, even though such links tend to decrease the average network distance. It is hoped that this review will help attract attention to the fundamental problem of clustered structures/synchronization in network science.      

Abnormal synchronization in complex clustered networks

Recent research has revealed that complex networks with a smaller average distance and more homogeneous degree distribution are more synchronizable. We find, however, that synchronization in complex, clustered networks tends to obey a different set of rules. In particular, the synchronizability of such a network is determined by the interplay between intercluster and intracluster links. The network is most synchronizable when the numbers of the two types of links are approximately equal. In the presence of a mismatch, increasing the number of intracluster links, while making the network distance smaller, can counterintuitively suppress or even destroy the synchronization. We provide theory and numerical evidence to establish this phenomenon.     

Bursting synchronization in clustered neuronal networks

Neuronal networks in the brain exhibit the modular (clustered) property, i.e., they are composed of certain subnetworks with differential internal and external connectivity. We investigate bursting synchronization in a clustered neuronal network. A transition to mutual-phase synchronization takes place on the bursting time scale of coupled neurons, while on the spiking time scale, they behave asynchronously. This synchronization transition can be induced by the variations of inter- and intra- coupling strengths, as well as the probability of random links between different subnetworks. Considering that some pathological conditions are related with the synchronization of bursting neurons in the brain, we analyze the control of bursting synchronization by using a time-periodic external signal in the clustered neuronal network. Simulation results show a frequency locking tongue in the driving parameter plane, where bursting synchronization is maintained, even in the presence of external driving. Hence, effective synchronization suppression can be realized with the driving parameters outside the frequency locking region.     

Percolation on bipartite scale-free networks

Recent studies introduced biased (degree-dependent) edge percolation as a model for failures in real-life systems. In this work, such process is applied to networks consisting of two types of nodes with edges running only between nodes of unlike type. Such bipartite graphs appear in many social networks, for instance in affiliation networks and in sexual-contact networks in which both types of nodes show the scale-free characteristic for the degree distribution. During the depreciation process, an edge between nodes with degrees k and q is retained with a probability proportional to (kq)^−α, where α is positive so that links between hubs are more prone to failure. The removal process is studied analytically by introducing a generating functions theory. We deduce exact self-consistent equations describing the system at a macroscopic level and discuss the percolation transition. Critical exponents are obtained by exploiting the Fortuin-Kasteleyn construction which provides a link between our model and a limit of the Potts model.     

Percolation theory and fragmentation measures in social networks

We study the statistical properties of a recently proposed social networks measure of fragmentation F after removal of a fraction q of nodes or links from the network. The measure F   is defined as the ratio of the number of pairs of nodes that are not connected in the fragmented network to the total number of pairs in the original fully connected network. We compare this measure with the one traditionally used in percolation theory, _P∞$P_{\infty }$, the fraction of nodes in the largest cluster relative to the total number of nodes. Using both analytical and numerical methods, we study Erd?s–Rényi (ER) and scale-free (SF) networks under various node removal strategies. We find that for a network obtained after removal of a fraction q   of nodes above criticality, _P∞≈(1-F)^1/2$P_{\infty }\approx (1-F{)}^{1/2}$. For fixed _P∞$P_{\infty }$ and close to criticality, we show that 1-F$1-F$ better reflects the actual fragmentation. For a given _P∞$P_{\infty }$, 1-F$1-F$ has a broad distribution and thus one can improve significantly the fragmentation of the network. We also study and compare the fragmentation measure F   and the percolation measure _P∞$P_{\infty }$ for a real national social network of workplaces linked by the households of the employees and find similar results.     

Percolation thresholds and percolation conductivities of octagonal and dodecagonal quasicrystalline lattices

The octagonal and dodecagonal quaislattices were generated by means of the grid method. Monte Carlo simulation and cluster counting procedure were used for numerical determination of the site and bond percolation thresholds. Two types of connectivity called ferromagnetic and chemical were studied. The estimated site percolation thresholds are 0.5435… and 0.585… for octagonal lattice and 0.617… and 0.628… for dodecagonal lattice respectively. The obtained spanning fraction curves (for site percolation) seem to approach the 50% value at the percolation threshold. The site percolation conductivity for these lattices was studied by means of a transfer-matrix approach. The critical behavior was found to be the same as for the periodic lattices.     

Percolation on networks with dependence links

As a classical model of statistical physics, the percolation theory provides a powerful approach to analyze the network structure and dynamics. Recently, to model the relations among interacting agents beyond the connection of the networked system, the concept of dependence link is proposed to represent the dependence relationship of agents. These studies suggest that the percolation properties of these networks differ greatly from those of the ordinary networks. In particular,unlike the well known continuous transition on the ordinary networks, the percolation transitions on these networks are discontinuous. Moreover, these networks are more fragile for a broader degree distribution, which is opposite to the famous results for the ordinary networks. In this article, we give a summary of the theoretical approaches to study the percolation process on networks with inter- and inner-dependence links, and review the recent advances in this field, focusing on the topology and robustness of such networks.     

Optimization of synchronization in complex clustered networks

There has been mounting evidence that many types of biological or technological networks possess a clustered structure. As many system functions depend on synchronization, it is important to investigate the synchronizability of complex clustered networks. Here we focus on one fundamental question: Under what condition can the network synchronizability be optimized? In particular, since the two basic parameters characterizing a complex clustered network are the probabilities of intercluster and intracluster connections, we investigate, in the corresponding two-dimensional parameter plane, regions where the network can be best synchronized. Our study yields a quite surprising finding: a complex clustered network is most synchronizable when the two probabilities match each other approximately. Mismatch, for instance caused by an overwhelming increase in the number of intracluster links, can counterintuitively suppress or even destroy synchronization, even though such an increase tends to reduce the average network distance. This phenomenon provides possible principles for optimal synchronization on complex clustered networks. We provide extensive numerical evidence and an analytic theory to establish the generality of this phenomenon.     

Percolation thresholds in the homogenization of spheroidal particles oriented in two directions

We consider percolation thresholds which arise in the homogenization of composite mediums based on three different types of component particles. An extension of the standard Bruggeman homogenization formalism is implemented in order to take account of the sizes, shapes, and orientations of the component particles. The relationships between the geometric attributes of the component particles and the constitutive parameters of the homogenized composite mediums are investigated. In particular, percolation thresholds arising in the homogenization of conducting component particles oriented in two directions and nonconducting component particles are explored via representative numerical examples. Anisotropies in these percolation thresholds are highlighted.     

Photoionization thresholds and structures of third group metals clustered with ammonia

A photoionization study of the Me(NH₃) clusters formed in the reaction of photoablated third group metal vapor with gaseous ammonia is reported. The photoionization spectra exhibit some features due to vibrational excitation of ionic clusters and to transitions to neutral Rydberg states leading to autoionization. DFT quantum chemical calculations are performed on the Me(NH₃). The cluster geometries are fully optimized imposing the C_3v symmetry. The calculated values of the IPs are in agreement with those experimentally determined. Received: 16 February 1998 / Revised and Accepted: 7 May 1998     

Percolation on infinitely ramified fractal 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 ($D_{tH}$), and spectral ($d_s$) dimensions. Monte Carlo simulations of site percolation were performed on pre-fractal standard Sierpiński 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 ($D_H$, $D_{tH}$, $d_s$), 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 $D_{tH}$, 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.     

Percolation on shopping and cashback electronic commerce networks

Many realistic networks live in the form of multiple networks, including interacting networks and interdependent networks. Here we study percolation properties of a special kind of interacting networks, namely Shopping and Cashback Electronic Commerce Networks (SCECNs). We investigate two actual SCECNs to extract their structural properties, and develop a mathematical framework based on generating functions for analyzing directed interacting networks. Then we derive the necessary and sufficient condition for the absence of the system-wide giant in- and out- component, and propose arithmetic to calculate the corresponding structural measures in the sub-critical and supercritical regimes. We apply our mathematical framework and arithmetic to those two actual SCECNs to observe its accuracy, and give some explanations on the discrepancies. We show those structural measures based on our mathematical framework and arithmetic are useful to appraise the status of SCECNs. We also find that the supercritical regime of the whole network is maintained mainly by hyperlinks between different kinds of websites, while those hyperlinks between the same kinds of websites can only enlarge the sizes of in-components and out-components.     

Immunization and epidemic dynamics in complex networks

We study the behavior of epidemic spreading in networks, and, in particular, scale free networks. We use the Susceptible-Infected-Removed (SIR) epidemiological model. We give simulation results for the dynamics of epidemic spreading. By mapping the model into a static bond-percolation model we derive analytical results for the total number of infected individuals. We study this model with various immunization strategies, including random, targeted and acquaintance immunization.Received: 3 November 2003, Published online: 14 May 2004PACS: 02.50.Cw Probability theory - 02.10.Ox Combinatorics; graph theory - 89.20.Hh World Wide Web, Internet - 64.60.-i General studies of phase transitions     

簇间连接方式不同的簇网络的同步过程研究

本文对簇间连接方式不同的三类簇网络的同步能力和同步过程进行研究. 构成簇网络的两个子网均为BA无标度网络, 当簇间连接方式是双向耦合时, 称其为TWD网络模型, 当簇间连接是大子网驱动小子网时, 称其为BDS网络模型, 当簇间连接是小子网驱动大子网时, 称其为SDB网络模型. 研究表明, 当小子网和大子网节点数目的比值大于某一临界值时, TWD网络模型的同步能力大于BDS网络模型的同步能力, 当该比值小于某一临界值时, TWD网络模型的同步能力小于BDS网络模型的同步能力, SDB网络模型的同步能力是三种网络结构中最差的. 对于簇间连接具有方向性的单向驱动网络, 簇网络的整体同步能力与被驱动子网的节点数和簇间连接数有关, 与驱动网络自身节点数无关. 增加簇间连接数在开始时会降低各子网的同步速度, 但最终各子网到达完全同步的时间减少, 网络的整体同步能力增强. 文中以Kuramoto相振子作为网络节点, 研究了不同情况下三种簇网络的同步过程, 证明了所得结论的正确性. 关键词： 簇网络 有向连接 同步能力 Kuramoto振子