Quantum computing of delocalization in small-world networks

We study a quantum small-world network with disorder and show that the system exhibits a delocalization transition. A quantum algorithm is built up which simulates the evolution operator of the model in a polynomial number of gates for exponential number of vertices in the network. The total computational gain is shown to depend on the parameters of the network and a larger than quadratic speed-up can be reached. We also investigate the robustness of the algorithm in presence of imperfections.Comment: 4 pages, 5 figures, research done at http://www.quantware.ups-tlse.fr

Statistical Analysis of Airport Network of China

Through the study of airport network of China (ANC), composed of 128 airports (nodes) and 1165 flights (edges), we show the topological structure of ANC conveys two characteristics of small worlds, a short average path length (2.067) and a high degree of clustering (0.733). The cumulative degree distributions of both directed and undirected ANC obey two-regime power laws with different exponents, i.e., the so-called Double Pareto Law. In-degrees and out-degrees of each airport have positive correlations, whereas the undirected degrees of adjacent airports have significant linear anticorrelations. It is demonstrated both weekly and daily cumulative distributions of flight weights (frequencies) of ANC have power-law tails. Besides, the weight of any given flight is proportional to the degrees of both airports at the two ends of that flight. It is also shown the diameter of each sub-cluster (consisting of an airport and all those airports to which it is linked) is inversely proportional to its density of connectivity. Efficiency of ANC and of its sub-clusters are measured through a simple definition. In terms of that, the efficiency of ANC's sub-clusters increases as the density of connectivity does. ANC is found to have an efficiency of 0.484.Comment: 6 Pages, 5 figure

Preferencial growth: exact solution of the time dependent distributions

We consider a preferential growth model where particles are added one by one to the system consisting of clusters of particles. A new particle can either form a new cluster (with probability q) or join an already existing cluster with a probability proportional to the size thereof. We calculate exactly the probability \Pm_i(k,t) that the size of the i-th cluster at time t is k. We analyze the asymptotics, the scaling properties of the size distribution and of the mean size as well as the relation of our system to recent network models.Comment: 8 pages, 4 figure

Exact results and scaling properties of small-world networks

We study the distribution function for minimal paths in small-world networks. Using properties of this distribution function, we derive analytic results which greatly simplify the numerical calculation of the average minimal distance, $\bar{\ell}$, and its variance, $\sigma^2$. We also discuss the scaling properties of the distribution function. Finally, we study the limit of large system sizes and obtain some analytic results.Comment: RevTeX, 4 pages, 5 figures included. Minor corrections and addition

Efficiency of informational transfer in regular and complex networks

We analyze the process of informational exchange through complex networks by measuring network efficiencies. Aiming to study non-clustered systems, we propose a modification of this measure on the local level. We apply this method to an extension of the class of small-worlds that includes {\it declustered} networks, and show that they are locally quite efficient, although their clustering coefficient is practically zero. Unweighted systems with small-world and scale-free topologies are shown to be both globally and locally efficient. Our method is also applied to characterize weighted networks. In particular we examine the properties of underground transportation systems of Madrid and Barcelona and reinterpret the results obtained for the Boston subway network.Comment: 10 pages and 9 figure

Citation Networks in High Energy Physics

The citation network constituted by the SPIRES data base is investigated empirically. The probability that a given paper in the SPIRES data base has $k$ citations is well described by simple power laws, $P(k) \propto k^{-\alpha}$, with $\alpha \approx 1.2$ for $k$ less than 50 citations and $\alpha \approx 2.3$ for 50 or more citations. Two models are presented that both represent the data well, one which generates power laws and one which generates a stretched exponential. It is not possible to discriminate between these models on the present empirical basis. A consideration of citation distribution by subfield shows that the citation patterns of high energy physics form a remarkably homogeneous network. Further, we utilize the knowledge of the citation distributions to demonstrate the extreme improbability that the citation records of selected individuals and institutions have been obtained by a random draw on the resulting distribution.Comment: 9 pages, 6 figures, 2 table

Search in weighted complex networks

We study trade-offs presented by local search algorithms in complex networks which are heterogeneous in edge weights and node degree. We show that search based on a network measure, local betweenness centrality (LBC), utilizes the heterogeneity of both node degrees and edge weights to perform the best in scale-free weighted networks. The search based on LBC is universal and performs well in a large class of complex networks.Comment: 14 pages, 5 figures, 4 tables, minor changes, added a referenc

Diffusion Processes on Small-World Networks with Distance-Dependent Random-Links

We considered diffusion-driven processes on small-world networks with distance-dependent random links. The study of diffusion on such networks is motivated by transport on randomly folded polymer chains, synchronization problems in task-completion networks, and gradient driven transport on networks. Changing the parameters of the distance-dependence, we found a rich phase diagram, with different transient and recurrent phases in the context of random walks on networks. We performed the calculations in two limiting cases: in the annealed case, where the rearrangement of the random links is fast, and in the quenched case, where the link rearrangement is slow compared to the motion of the random walker or the surface. It has been well-established that in a large class of interacting systems, adding an arbitrarily small density of, possibly long-range, quenched random links to a regular lattice interaction topology, will give rise to mean-field (or annealed) like behavior. In some cases, however, mean-field scaling breaks down, such as in diffusion or in the Edwards-Wilkinson process in "low-dimensional" small-world networks. This break-down can be understood by treating the random links perturbatively, where the mean-field (or annealed) prediction appears as the lowest-order term of a naive perturbation expansion. The asymptotic analytic results are also confirmed numerically by employing exact numerical diagonalization of the network Laplacian. Further, we construct a finite-size scaling framework for the relevant observables, capturing the cross-over behaviors in finite networks. This work provides a detailed account of the self-consistent-perturbative and renormalization approaches briefly introduced in two earlier short reports.Comment: 36 pages, 27 figures. Minor revisions in response to the referee's comments. Furthermore, some typos were fixed and new references were adde

Multifractal analysis of complex networks

Complex networks have recently attracted much attention in diverse areas of science and technology. Many networks such as the WWW and biological networks are known to display spatial heterogeneity which can be characterized by their fractal dimensions. Multifractal analysis is a useful way to systematically describe the spatial heterogeneity of both theoretical and experimental fractal patterns. In this paper, we introduce a new box covering algorithm for multifractal analysis of complex networks. This algorithm is used to calculate the generalized fractal dimensions $D_{q}$ of some theoretical networks, namely scale-free networks, small world networks and random networks, and one kind of real networks, namely protein-protein interaction networks of different species. Our numerical results indicate the existence of multifractality in scale-free networks and protein-protein interaction networks, while the multifractal behavior is not clear-cut for small world networks and random networks. The possible variation of $D_{q}$ due to changes in the parameters of the theoretical network models is also discussed.Comment: 18 pages, 7 figures, 4 table

Efficient Behavior of Small-World Networks

We introduce the concept of efficiency of a network, measuring how efficiently it exchanges information. By using this simple measure small-world networks are seen as systems that are both globally and locally efficient. This allows to give a clear physical meaning to the concept of small-world, and also to perform a precise quantitative a nalysis of both weighted and unweighted networks. We study neural networks and man-made communication and transportation systems and we show that the underlying general principle of their construction is in fact a small-world principle of high efficiency.Comment: 1 figure, 2 tables. Revised version. Accepted for publication in Phys. Rev. Let