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In some systems, the connecting probability (and thus the percolation process) between two sites depends on the geometric distance between them. To understand such process, we propose gravitationally correlated percolation models for link-adding networks on the two-dimensional lattice $G$ with two strategies $S_{\rm max}$ and $S_{\rm min}$, to add a link $l_{i,j}$ to connect site $i$ and site $j$ with mass $m_i$ and $m_j$, respectively; $m_i$ and $m_j$ are sizes of the clusters which contain site $i$ and site $j$, respectively. The probability to add the link $l_{i,j}$ is related to the generalized gravity $g_{ij} \equiv m_i m_j/r_{ij}^d$, where $r_{ij}$ is the geometric distance between $i$ and $j$, and $d$ is an adjustable decaying exponent. In the beginning of the simulation, all sites of $G$ are occupied and there is no link. In the simulation process, two inter-cluster links $l_{i,j}$ and $l_{k,n}$ are randomly chosen and the generalized gravities $g_{ij}$ and $g_{kn}$ are computed. In the strategy $S_{\rm max}$, the link with larger generalized gravity is added. In the strategy $S_{\rm min}$, the link with smaller generalized gravity is added, which include percolation on the Erd\H os-R\'enyi random graph and the Achlioptas process of explosive percolation as the limiting cases, $d \to \infty$ and $d \to 0$, respectively. Adjustable strategies facilitate or inhibit the network percolation in a generic view. We calculate percolation thresholds $T_c$ and critical exponents $\beta$ by numerical simulations. We also obtain various finite-size scaling functions for the node fractions in percolating clusters or arrival of saturation length with different intervening strategies., Comment: 26 pages, 8 figures, Chinese Journal of Physics, accepted for publication

The distribution of the geometric distances of connected neurons is a practical factor underlying neural networks in the brain. It can affect the brain\'s dynamic properties at the ground level. Karbowski derived a power-law decay distribution that has not yet been verified by experiment. In this work, we check its validity using simulations with a phenomenological model. Based on the in vitro two-dimensional development of neural networks in culture vessels by Ito, we match the synapse number saturation time to obtain suitable parameters for the development process, then determine the distribution of distances between connected neurons under such conditions. Our simulations obtain a clear exponential distribution instead of a power-law one, which indicates that Karbowski's conclusion is invalid, at least for the case of in vitro neural network development in two-dimensional culture vessels.

Flight delay happens every day in airports all over the world. However, systemic investigation in large scales remains a challenge. We collect primary data of domestic departure records from Bureau of Transportation Statistics of United States, and do empirical statistics with them in form of complementary cumulative distributions functions (CCDFs) and transmission function of the delays. Fourteen main airlines are characterized by two types of CCDFs: shifted power-law and exponentially truncated shifted power-law. By setting up two phenomenological models based on mean-field approximation in temporal regime, we convert effect from other delay factors into a propagation one. Three parameters meaningful in measuring airlines emerge as universal metrics. Moreover, method used here could become a novel approach to revealing practical meanings hidden in temporal big data in wide fields.

In the coevolving voter model, each voter has one of two diametrically opposite opinions, and a voter encountering a neighbor with the opposite opinion may either adopt it or rewire the connection to another randomly chosen voter sharing the same opinion. As we smoothly change the relative frequency of rewiring compared to that of adoption, there occurs a phase transition between an active phase and a frozen phase. By performing extensive Monte Carlo calculations, we show that the phase transition is characterized by critical exponents {\beta}=0.54(1) and {\nu} =1.5(1), which differ from the existing mean-field-type prediction. We furthermore extend the model by introducing a contrarian type that tries to have neighbors with the opposite opinion, and show that the critical behavior still belongs to the same universality class irrespective of such contrarians' fraction., Comment: 11 pages, 5 figures

Motivated by the importance of geometric information in real systems, a new model for long-range correlated percolation in link-adding networks is proposed with the connecting probability decaying with a power-law of the distance on the two-dimensional(2D) plane. By overlapping it with Achlioptas process, it serves as a gravity model which can be tuned to facilitate or inhibit the network percolation in a generic view, cover a broad range of thresholds. Moreover, it yields a set of new scaling relations. In the present work, we develop an approach to determine critical points for them by simulating the temporal evolutions of type-I, type-II and type-III links(chosen from both inter-cluster links, an intra-cluster link compared with an inter-cluster one, and both intra-cluster ones, respectively) and corresponding average lengths. Numerical results have revealed objective competition between fractions, average lengths of three types of links, verified the balance happened at critical points. The variation of decay exponents $a$ or transmission radius $R$ always shifts the temporal pace of the evolution, while the steady average lengths and the fractions of links always keep unchanged just as the values in Achlioptas process. Strategy with maximum gravity can keep steady average length, while that with minimum one can surpass it. Without the confinement of transmission range, $\bar{l} \to \infty$ in thermodynamic limit, while $\bar{l}$ does not when with it. However, both mechanisms support critical points. In two-dimensional free space, the relevance of correlated percolation in link-adding process is verified by validation of new scaling relations with various exponent $a$, which violates the scaling law of Weinrib's.

In this paper, critical global connectivity of mobile ad hoc communication networks (MAHCN) is investigated. We model the two-dimensional plane on which nodes move randomly with a triangular lattice. Demanding the best communication of the network, we account the global connectivity $\eta$ as a function of occupancy $\sigma$ of sites in the lattice by mobile nodes. Critical phenomena of the connectivity for different transmission ranges $r$ are revealed by numerical simulations, and these results fit well to the analysis based on the assumption of homogeneous mixing . Scaling behavior of the connectivity is found as $\eta \sim f(R^{\beta}\sigma)$, where $R=(r-r_{0})/r_{0}$, $r_{0}$ is the length unit of the triangular lattice and $\beta$ is the scaling index in the universal function $f(x)$. The model serves as a sort of site percolation on dynamic complex networks relative to geometric distance. Moreover, near each critical $\sigma_c(r)$ corresponding to certain transmission range $r$, there exists a cut-off degree $k_c$ below which the clustering coefficient of such self-organized networks keeps a constant while the averaged nearest neighbor degree exhibits a unique linear variation with the degree k, which may be useful to the designation of real MAHCN., Comment: 6 pages, 6 figures

Competitive exclusion, a key principle of ecology, can be generalized to understand many other complex systems. Individuals under surviving pressure tend to be different from others, and correlations among them change correspondingly to the updating of their states. We show with numerical simulation that these aptitudes can contribute to group formation or speciation in social fields. Moreover, they can lead to power-law topological correlations of complex networks. By coupling updating states of nodes with variation of connections in a network, structural properties with power-laws and functions like multifractality, spontaneous ranking and evolutionary branching of node states can emerge out simultaneously from the present self-organized model of coevolutionary process., Comment: 5 figures and 5 pages

Motivated by a recently introduced network growth mechanism that rely on the ranking of node prestige measures [S. Fortunato \emph{et al}., Phys. Rev. Lett. \textbf{96}, 218701 (2006)], a rank-based model for weighted network evolution is studied. The evolution rule of the network is based on the ranking of node strength, which couples the topological growth and the weight dynamics. Both analytical solutions and numerical simulations show that the generated networks possess scale-free distributions of degree, strength, and weight in the whole region of the growth dynamics parameter ($\alpha>0$). We also characterize the clustering and correlation properties of this class of networks. It is showed that at $\alpha=1$ a structural phase transition occurs, and for $\alpha>1$ the generated network simultaneously exhibits hierarchical organization and disassortative degree correlation, which is consistent with a wide range of biological networks., Comment: 4 pages, 5 figures

We propose a model of network growth that generalizes the deactivation model previously suggested for complex networks. Several topological features of this generalized model, such as the degree distribution and clustering coefficient, have been investigated analytically and by simulations. A scaling behavior of clustering coefficient $C \sim 1/M$ is theoretically obtained, where $M$ refers to the number of active nodes in the network. We discuss the relationship between the recently observed numerical behavior of clustering coefficient in the coauthor and paper citation networks and our theoretical result. It shows that both of them are induced by deactivation mechanism. By introducing a perturbation, the generated network undergoes a transition from large- to small-world, meanwhile the scaling behavior of $C$ is conserved. It indicates that $C \sim 1/M$ is a universal scaling behavior induced by deactivation mechanism., Comment: 8 pages, 9 figures. To be published in Physical Review E

In this paper, by extending the concept of Kuramoto oscillator to the left-invariant flow on general Lie group, we investigate the generalized phase synchronization on networks. The analyses and simulations of some typical dynamical systems on Watts-Strogatz networks are given, including the $n$-dimensional torus, the identity component of 3-dimensional general linear group, the special unitary group, and the special orthogonal group. In all cases, the greater disorder of networks will predict better synchronizability, and the small-world effect ensures the global synchronization for sufficiently large coupling strength. The collective synchronized behaviors of many dynamical systems, such as the integrable systems, the two-state quantum systems and the top systems, can be described by the present phase synchronization frame. In addition, it is intuitive that the low-dimensional systems are more easily to synchronize, however, to our surprise, we found that the high-dimensional systems display obviously synchronized behaviors in regular networks, while these phenomena can not be observed in low-dimensional systems., Comment: 5 pages, and 3 figures

We propose a simple rule that generates scale-free networks with very large clustering coefficient and very small average distance. These networks are called simplex triangulation networks(STNs) as they can be considered as a kind of network representation of simplex triangulation. We obtain the analytic results of power-law exponent $\gamma =2+\frac{1}{d-1}$ for $d$-dimensional STNs, and clustering coefficient $C$. We prove that the increasing tendency of average distance of STNs is a little slower than the logarithm of the number of nodes in STNs. In addition, the STNs possess hierarchical structure as $C(k)\sim k^{-1}$ when $k\gg d$ that in accord with the observations of many real-life networks., Comment: 6 pages, no figures

A dynamical model of small-world network, with directed links which describe various correlations in social and natural phenomena, is presented. Random responses of every site to the imput message are introduced to simulate real systems. The interplay of these ingredients results in collective dynamical evolution of a spin-like variable S(t) of the whole network. In the present model, global average spreading length \langel L >_s and average spreading time _s are found to scale as p^-\alpha ln N with different exponents. Meanwhile, S behaves in a duple scaling form for N>>N^*: S ~ f(p^-\beta q^\gamma t'_sc), where p and q are rewiring and external parameters, \alpha, \beta, \gamma and f(t'_sc) are scaling exponents and universal functions, respectively. Possible applications of the model are discussed., Comment: 4 pages, 6 Figures

We investigate the localization behavior of electrons in a random lattice which is constructed from a quasi-one-dimensional chain with large coordinate number $Z$ and rewired bonds, resembling the small-world network proposed recently but with site-energy disorder and quantum links instead of classical ones. The random rewiring of bonds in the chain with large $Z$ enhances both the topological disorder and the effective dimensionality. From the competition between disorder and dimensionality enhancement a transition from localization to delocalization is found by using the level statistics method combined with the finite-size scaling analysis. The critical value of the rewiring rate for this transition is determined numerically. We obtain a universal critical integrated distribution of level spacing $s$ in the form $I_{p_{c}}(s)\propto \exp (-A_{c}s^{\alpha})$, with $A_{c}\simeq 1.50$ and $\alpha \simeq 1.0$. This reveals the possible existence of metal-insulator transition in materials with chains as the backbones., Comment: 12 pages, 4 figures

We prove that the hierarchical fractal model recently proposed for describing the stretched polymers [A. N. Samukhin et al, Phys. Rev. Lett.{\bf 78}, 326(1997)] is equivalent to a one-dimensional chain with hierarchical aperiodic structure. By use of the transfer matrix technique we calculate the electronic transmission and the dc conductance. We find that there exist sharp-edged transmission subbands and gaps, but the transmission subbands are rich in substructures which show self-similarity. The temperature dependence of conductance $g (T)$ is sensitive to the variation of the Fermi level and to the structure parameters $m$ and $n$ of the original fractal structure. The relationship of the obtained results to the experimental data and other theoretical works is discussed., Comment: 5 pages, 7 figures