Scale-free networks by super-linear preferential attachment rule

A network growth model with geographic limitation of accessible information about the status of existing nodes is investigated. In this model, the probability Π(k) of an existing node of degree k is found to be super-linear with Π(k)∼k^α and α>1 when there are links from new nodes. The numerical results show that the constructed networks have typical power-law degree distributions P(k)∼k^−γ and the exponent γ depends on the constraint level. An analysis of local structural features shows the robust emergence of scale-free network structure in spite of the super-linear preferential attachment rule. This local structural feature is directly associated with the geographical connection constraints which are widely observed in many real networks.     

Degree correlation in scale-free graphs

We obtain closed form expressions for the expected conditional degree distribution and the joint degree distribution of the linear preferential attachment model for network growth in the steady state. We consider the multiple-destination preferential attachment growth model, where incoming nodes at each timestep attach to β existing nodes, selected by degree-proportional probabilities. By the conditional degree distribution p(?|k), we mean the degree distribution of nodes that are connected to a node of degree k. By the joint degree distribution p(k,?), we mean the proportion of links that connect nodes of degrees k and ?. In addition to this growth model, we consider the shifted-linear preferential growth model and solve for the same quantities, as well as a closed form expression for its steady-state degree distribution.     

Random pseudofractal networks with competition

In this paper, we present a simple rule which assigns fitness to each edge to generate random pseudofractal networks (RPNs). This RPN model is both scale-free and small-world. We obtain the theoretical results that the power-law exponent is γ=2+1/(1+α) for the tunable parameter α>-1, and that the degree distribution is of an exponential form for others. Analytical results also show that an RPN has a large clustering coefficient and can process hierarchical structure as C(k)∼k^-1 that is in accordance with many real networks. And we prove that the mean distance L(N) scales slower logarithmically with network size N. In particular, we explain the effect of nodes with degree 2 on the clustering coefficient. These results agree with numerical simulations very well.     

A Simple Asymmetric Evolving Random Network

We introduce a new oriented evolving graph model inspired by biological networks. A node is added at each time step and is connected to the rest of the graph by random oriented edges emerging from older nodes. This leads to a statistical asymmetry between incoming and outgoing edges. We show that the model exhibits a percolation transition and discuss its universality. Below the threshold, the distribution of component sizes decreases algebraically with a continuously varying exponent depending on the average connectivity. We prove that the transition is of infinite order by deriving the exact asymptotic formula for the size of the giant component close to the threshold. We also present a thorough analysis of aging properties. We compute local-in-time profiles for the components of finite size and for the giant component, showing in particular that the giant component is always dense among the oldest nodes but invades only an exponentially small fraction of the young nodes close to the threshold.     

Multiplicity distributions in the generalized Feynman gas model

Multiplicity distributions Ψ_n^(k) in the generalized Feynman gas model of order k (defined by saying that all integrated correlation functions f_n except f₁,…,f_k are zero) are derived and expressed in terms of Poisson distributions with different ”average multiplicities”, which are related to the integrated correlation functions. The relations between Ψ_n^(k) and Ψ_n^(j) for arbitrary positive integers k,j are found. An intuitive picture to gain a better feeling for these relations is developed.On the basis of our formulae we show that the experimentally observed multiplicity distributions (between 50 GeV/c and 303 GeV/c incoming momentum) can be well reproduced by those of a Feynman gas model of order two. Other applications of our formulae are suggested.     

Ranking spreaders by decomposing complex networks

Ranking the nodes? ability of spreading in networks is crucial for designing efficient strategies to hinder spreading in the case of diseases or accelerate spreading in the case of information dissemination. In the well-known k-shell method, nodes are ranked only according to the links between the remaining nodes (residual links) while the links connecting to the removed nodes (exhausted links) are entirely ignored. In this Letter, we propose a mixed degree decomposition (MDD) procedure in which both the residual degree and the exhausted degree are considered. By simulating the epidemic spreading process on real networks, we show that the MDD method can outperform the k-shell and degree methods in ranking spreaders.     

Conducting to non-conducting transition in dual transmission lines using a ternary model with long-range correlated disorder

In this work we study the behavior of the allowed and forbidden frequencies in disordered classical dual transmission lines when the values of capacitances {Cj} are distributed according to a ternary model with long-range correlated disorder. We introduce the disorder from a random sequence with a power spectrum S(k)∝k−(2α−1), where α?0.5 is the correlation exponent. From this sequence we generate an asymmetric ternary map using two map parameters b₁ and b₂, which adjust the occupancy probability of each possible value of the capacitances Cj={CA,CB,CC,}. If the sequence of capacitance values is totally at random α=0.5 (white noise), the electrical transmission line is in the non-conducting state for every frequency ω. When we introduce long-range correlations in the distribution of capacitances, the electrical transmission lines can change their conducting properties and we can find a transition from the non-conducting to conducting state for a fixed system size. This implies the existence of critical values of the map parameters for each correlation exponent α. By performing finite-size scaling we obtain the asymptotic value of the map parameters in the thermodynamic limit for any α. With these data we obtain a phase diagram for the symmetric ternary model, which separates the non-conducting state from the conducting one. This is the fundamental result of this Letter. In addition, introducing one or more impurities in random places of the long-range correlated distribution of capacitances, we observe a dramatic change in the conducting properties of the electrical transmission lines, in such a way that the system jumps from conducting to non-conducting states. We think that this behavior can be considered as a possible mechanism to secure communication.     

A model for cascading failures in scale-free networks with a breakdown probability

Considering that not all overload nodes will be removed from networks due to some effective measures to protect them, we propose a new cascading model with a breakdown probability. Adopting the initial load of a node j to be L_j=[k_j(∑_m∈Γjk_m)]^α with k_j and Γ_j being the degree of the node j and the set of its neighboring nodes, respectively, where α is a tunable parameter, we investigate the relationship between some parameters and universal robustness characteristics against cascading failures on scale-free networks. According to a new measure originated from a phase transition from the normal state to collapse, the numerical simulations show that Barabási-Albert (BA) networks reach the strongest robustness level against cascading failures when the tunable parameter α=0.5, while not relating to the breakdown probability. We furthermore explore the effect of the average degree 〈k〉 for network robustness, thus obtaining a positive correlation between 〈k〉 and network robustness. We then analyze the effect of the breakdown probability on the network robustness and confirm by theoretical predictions this universal robustness characteristic observed in simulations. Our work may have practical implications for controlling various cascading-failure-induced disasters in the real world.     

Connectivity of growing networks with link constraints

We propose a growing network model with link constraint, in which new nodes are continuously introduced into the system and immediately connected to preexisting nodes, and any arbitrary node cannot receive new links when it reaches a maximum number of links k_m. The connectivity of the network model is then investigated by means of the rate equation approach. For the connection kernel A(k)=k^γ, the degree distribution n_k takes a power law if γ≥1 and decays stretched exponentially if 0≤γ< 1. We also consider a network system with the connection kernel A(k)=k^α(k_m-k)^β. It is found that n_k approaches a power law in the α> 1 case and has a stretched exponential decay in the 0≤α< 1 case, while it can take a power law with exponential truncation in the special α=β=1 case. Moreover, n_k may have a U-type structure if α> β.     

Degree and connectivity of the Internet’s scale-free topology

This paper theoretically and empirically studies the degree and connectivity of the Internet's scale-free topology at an autonomous system (AS) level. The basic features of scale-free networks influence the normalization constant of degree distribution p(k). It develops a new mathematic model for describing the power-law relationships of Internet topology. From this model we theoretically obtain formulas to calculate the average degree, the ratios of the k_min-degree (minimum degree) nodes and the k_max-degree (maximum degree) nodes, and the fraction of the degrees (or links) in the hands of the richer (top best-connected) nodes. It finds that the average degree is larger for a smaller power-law exponent λ and a larger minimum or maximum degree. The ratio of the k_min-degree nodes is larger for larger λ and smaller k_min or k_max. The ratio of the k_max-degree ones is larger for smaller λ and k_max or larger k_min. The richer nodes hold most of the total degrees of Internet AS-level topology. In addition, it is revealed that the increased rate of the average degree or the ratio of the k_min-degree nodes has power-law decay with the increase of k_min. The ratio of the k_max-degree nodes has a power-law decay with the increase of k_max, and the fraction of the degrees in the hands of the richer 27% nodes is about 73% (the '73/27 rule'). Finally, empirically calculations are made, based on the empirical data extracted from the Border Gateway Protocol, of the average degree, ratio and fraction using this method and other methods, and find that this method is rigorous and effective for Internet AS-level topology.     

The emergence of scale-free networks with a seceding mechanism

In order to explore further the underlying mechanism of scale-free networks, we study stochastic secession as a mechanism for the creation of complex networks. In this evolution the network growth incorporates the addition of new nodes, the addition of new links between existing nodes, the deleting and rewiring of some existing links, and the stochastic secession of nodes. To random growing networks with preferential attachment, the model yields scale-free behavior for the degree distribution. Furthermore, we obtain an analytical expression of the power-law degree distribution with scaling exponent γ ranging from 1.1 to 9. The analytical expressions are in good agreement with the numerical simulation results.     

Influence of self-affine and mound roughness on the surface impedance and skin depth of conductive materials

In this work, we explore the influence of self-affine and mound surface roughness on the surface impedance and skin depth. For self-affine roughness, the surface impedance and skin depth increases with decreasing the roughness exponent H (for k_Fξ?1 with k_F the Fermi wave-vector), and/or increasing roughness ratio w/ξ, where w is the rms roughness amplitude and ξ the in-plane roughness correlation length. For mound roughness, the surface impedance and skin depth decrease monotonically with increasing average mound separation λ when λ>ζ with ζ the correlation length (assuming k_Fζ?1), while for λ<ζ they are both decreased with increasing λ.     

On-shell T-matrices in multiple scattering

The transition operator T for the scattering of a particle from N potentials V_j(x) can be expanded into a series featuring the transition operators t_j associated with the individual potentials. For V_j(x) both absolutely and square integrable in x, we show, using an analytic continuation argument, that if T is on-shell, i.e. in 〈k|T(k₀²±i0)|k′〉, |k| = |k′| = k₀, then each t_j is also on-shell.     

The Combined Effect of Connectivity and Dependency Links on Percolation of Networks

Percolation theory is extensively studied in statistical physics and mathematics with applications in diverse fields. However, the research is focused on systems with only one type of links, connectivity links. We review a recently developed mathematical framework for analyzing percolation properties of realistic scenarios of networks having links of two types, connectivity and dependency links. This formalism was applied to study Erdős-Rényi (ER) networks that include also dependency links. For an ER network with average degree [`(k)]\bar{k} that is composed of dependency clusters of size s, the fraction of nodes that belong to the giant component, P <sub>∞</sub>, is given by P<sub>￥</sub>=p<sup>s-1</sup>[1-exp(-[`(k)]pP_￥) ]^sP_{\infty}=p^{s-1}[1-\exp{(-\bar{k}pP_{\infty})} ]^{s} where 1−p is the initial fraction of randomly removed nodes. Here, we apply the formalism to the study of random-regular (RR) networks and find a formula for the size of the giant component in the percolation process: P _∞=p ^s−1(1−r ^k ) ^s where r is the solution of r=p ^s (r ^k−1−1)(1−r ^k )+1, and k is the degree of the nodes. These general results coincide, for s=1, with the known equations for percolation in ER and RR networks respectively without dependency links. In contrast to s=1, where the percolation transition is second order, for s>1 it is of first order. Comparing the percolation behavior of ER and RR networks we find a remarkable difference regarding their resilience. We show, analytically and numerically, that in ER networks with low connectivity degree or large dependency clusters, removal of even a finite number (zero fraction) of the infinite network nodes will trigger a cascade of failures that fragments the whole network. Specifically, for any given s there exists a critical degree value, [`(k)]<sub>min</sub>\bar{k}_{\min}, such that an ER network with [`(k)] ￡ [`(k)]_min\bar{k}\leq \bar{k}_{\min} is unstable and collapse when removing even a single node. This result is in contrast to RR networks where such cascades and full fragmentation can be triggered only by removal of a finite fraction of nodes in the network.     

A multi-channel scattering theory for some time dependent Hamiltonians,charge transfer problem

Scattering theory for time dependent HamiltonianH(t)=?(1/2) Δ+ΣV _j (x?q _j (t)) is discussed. The existence, asymptotic orthogonality and the asymptotic completeness of the multi-channel wave operators are obtained under the conditions that the potentials are short range: |V _j (x)|≦C _j (1+|x|)^?2?ε, roughly spoken; and the trajectoriesq _j (t) are straight lines at remote past and far future, and |q _j (t)?q _k (t)| → ∞ ast → ± ∞ (j≠k).     

Gibbs Distributions for Random Partitions Generated by a Fragmentation Process

In this paper we study random partitions of {1,…,n} where every cluster of size j can be in any of w _j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n,k,w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences w _j , the time-reversed process is the discrete Marcus–Lushnikov coalescent process with affine collision rate K _i,j = a+b(i+j) for some real numbers a and b. Under further restrictions on a and b, the fragmentation process can be realized by conditioning a Galton–Watson tree with suitable offspring distribution to have n nodes, and cutting the edges of this tree by random sampling of edges without replacement, to partition the tree into a collection of subtrees. Suitable offspring distributions include the binomial, negative binomial and Poisson distributions. Research supported in part by N.S.F. Grant DMS-0405779.     

Experimente und Berechnungen zur Hyperfeinstruktur der 3d 9 5g-Konfiguration im Kupfer II-Spektrum

The 3d ⁹ 5g-configuration in the Cu II-spectrum is an example for extreme(j _d l _g )K-coupling. It is shown how under these circumstances experimental hyperfinestructure separations can be explained and calculated by means of the concept of intermediate coupling between two basic coupling schemes. The basic schemes used are built on the vectorK coupled either to the spins of the outer 5g-electron or to the nuclear spinI. The value of the experimental magnetic separation factorya _j (3d) can be understood as the consequence of the core polarization of the inners-electrons by the spin of the unfilled 3d-shell. — The measured effective spin-orbit-constantζ _5g is three times larger than the theoretical result. Reasons for this discrepancy will be discussed.     

Discontinuities in disordered systems

The entropyS _T (j) of a two-dimensional Ising spin glass with an independent distribution of the random couplingp(J)=x·δ(J+1)+(1-x)δ(J-j) is discontinuous for temperatureT=0 and rationalj>0 and continuous elsewhere. The integrated density of frequenciesk _M (ω ²) of an one-dimensional chain of coupled oscillators with an independent distribution of the random massesp(m)=x·δ(m-1)+(1-x)δ(m-M) has the same behaviour, whereω ² corresponds toj andM to 1/T. The discontinuity points for infiniteM are, for sufficiently large but finiteM, special, frequencies, wherek _M (ω ²) has a Lifshitz singularity.     

Discretized kinetic theory on scale-free networks

The network of interpersonal connections is one of the possible heterogeneous factors which affect the income distribution emerging from micro-to-macro economic models. In this paper we equip our model discussed in [1, 2] with a network structure. The model is based on a system of n differential equations of the kinetic discretized-Boltzmann kind. The network structure is incorporated in a probabilistic way, through the introduction of a link density P(α) and of correlation coefficients P(β|α), which give the conditioned probability that an individual with α links is connected to one with β links. We study the properties of the equations and give analytical results concerning the existence, normalization and positivity of the solutions. For a fixed network with P(α) = c/α^q, we investigate numerically the dependence of the detailed and marginal equilibrium distributions on the initial conditions and on the exponent q. Our results are compatible with those obtained from the Bouchaud-Mezard model and from agent-based simulations, and provide additional information about the dependence of the individual income on the level of connectivity.     

A generalized O(2,1) expansion in the space-space region

Following the approach of Jones, Low and Young, a generalized O(2, 1) expansion is developed for amplitudes that have a power bounded growth asymptotically. The expansion, set up in an O(1, 1) basis, holds in a new kinematical region, where all the incoming and outgoing clusters have space-like O(2, 1) momenta.