The co-authorship network of scientists represents a prototype of complex evolving networks. By mapping the electronic database containing all relevant journals in mathematics and neuro-science for an eight-year period (1991-98), we infer the dynamic and the structural mechanisms that govern the evolution and topology of this complex system. First, empirical measurements allow us to uncover the topological measures that characterize the network at a given moment, as well as the time evolution of these quantities. The results indicate that the network is scale-free, and that the network evolution is governed by preferential attachment, affecting both internal and external links. However, in contrast with most model predictions the average degree increases in time, and the node separation decreases. Second, we propose a simple model that captures the network's time evolution. Third, numerical simulations are used to uncover the behavior of quantities that could not be predicted analytically.Comment: 14 pages, 15 figure

A key ingredient of current models proposed to capture the topological evolution of complex networks is the hypothesis that highly connected nodes increase their connectivity faster than their less connected peers, a phenomenon called preferential attachment. Measurements on four networks, namely the science citation network, Internet, actor collaboration and science coauthorship network indicate that the rate at which nodes acquire links depends on the node's degree, offering direct quantitative support for the presence of preferential attachment. We find that for the first two systems the attachment rate depends linearly on the node degree, while for the latter two the dependence follows a sublinear power law.PACS numbers: 89.65.-s, 89.75.-k, 05.10.-a Modeling the highly interconnected nature of various social, biological and communication systems as complex networks or graphs has attracted much attention in the last few years. [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. As for a long time these networks were modeled as completely random [15,16], the recent interest is motivated by the increasing evidence that real network display short length-scale clustering [1,2] and obey unexpected scaling laws [3,4], interpreted as signatures of deviation from randomness. Current approaches, using the tools of statistical physics [6,8,9] search for universalities both in the topology of these webs and in the dynamics governing their evolution. These efforts resulted in a class of models that view networks as evolving dynamical systems, rather than static graphs. Most evolving network models [4,6,9] are based on two ingredients [4]: growth and preferential attachment. The growth hypothesis suggests that networks continuously expand through the addition of new nodes and links between the nodes, while the preferential attachment hypothesis states that the rate Π(k) with which a node with k links acquires new links is a monotonically increasing function of k. While most versions of such evolving network models assume that Π(k) is linear in k [4,6,9], recently several authors proposed that Π(k) could follow a power-law [8,10]. Consequently, the time evolution of the degree k i of node i can be obtained from the first order differential equationwhere m is a constant and Π(k) has the formwith α > 0 an unknown scaling exponent. For α = 1 these models reduce to the scale-free model [4], for which the degree distribution P (k), giving the probability that a node has k links, follows P (k) ∝ k −γ with γ = 3. As Krapvisky, Redner and Leyvraz have shown [8], for α < 1 the degree distribution follows a stretched exponential, while for α > 1 a gelation-like phenomenon is expected, where a single site links to nearly all other nodes. On the other hand the hypothesis (2) Here we offer the first direct attempt to answer these questions in quantitative terms by proposing a numerical method that allows us to extract the functional form of Π(k) directly from dynamical data on real evolving networks. Our measurements indicate that Π(k...

We report on a series of measurements aimed to characterize the development and the dynamics of the rhythmic applause in concert halls. Our results demonstrate that while this process shares many characteristics of other systems that are known to synchronize, it also has features that are unexpected and unaccounted for in many other systems. In particular, we find that the mechanism lying at the heart of the synchronization process is the period doubling of the clapping rhythm. The characteristic interplay between synchronized and unsynchronized regimes during the applause is the result of a frustration in the systems. All results are understandable in the framework of the Kuramoto model.Comment: Revtex, 5 pages, 1 figur

Poisson Voronoi diagrams are useful for modeling and describing various natural patterns and for generating random lattices. Although this particular space tessellation is intensively studied by mathematicians, in two-and three dimensional spaces there is no exact result known for the size-distribution of Voronoi cells. Motivated by the simple form of the distribution function in the one-dimensional case, a simple and compact analytical formula is proposed for approximating the Voronoi cell's size distribution function in the practically important two-and three dimensional cases as well. Denoting the dimensionality of the space by d (d = 1, 2, 3) the f (y) = Const * y (3d−1)/2 exp(−(3d + 1)y/2) compact form is suggested for the normalized cell-size distribution function. By using large-scale computer simulations the validity of the proposed distribution function is studied and critically discussed.

We report on a series of measurements aimed to characterize the development and the dynamics of the rhythmic applause in concert halls. Our results demonstrate that while this process shares many characteristics of other systems that are known to synchronize, it also has features that are unexpected and unaccounted for in many other systems. In particular, we find that the mechanism lying at the heart of the synchronization process is the period doubling of the clapping rhythm. The characteristic interplay between synchronized and unsynchronized regimes during the applause is the result of a frustration in the systems. All results are understandable in the framework of the Kuramoto model.

Fracture in quasi-statically driven systems is studied by means of a discrete spring-block model. Developed from close comparison with desiccation experiments, it describes crack formation induced by friction on a substrate. The model produces cellular, hierarchical patterns of cracks, characterized by a mean fragment size linear in the layer thickness, in agreement with experiments. The selection of a stationary fragment size is explained by exploiting the correlations prior to cracking. A scaling behavior associated with the thickness and substrate coupling, derived and confirmed by simulations, suggests why patterns have similar morphology despite their disparity in scales.PACS numbers: 05.65.+b, 62.60.Mk, 46.50.+a Nature is full of fascinating patterns [1]. A ubiquitous yet relatively less explored class of patterns is that produced by the fracture of solids [2], as often seen in cracked structures, battered roads and dried out fields. Although there have been early observations and characterizations [1,3,4], only rather recently has it been studied systematically [5][6][7][8][9]. The mathematical problem of a network of interacting cracks [11] is a formidable one, in view of the difficulties facing a lone crack propagating in a homogeneous medium [10]. One way to make progress is to turn to a statistical description. This view has been pursued extensively in investigations of the fracture of disordered media [12], borrowing concepts such as percolation and universality from studies of phase transitions. For crack patterns, an immediate observation is the geometrical similarities of patterns over a wide range of scales, from microns [5] to kilometers [3]. This suggests some universal mechanism is at work and microscopic details may be unimportant. Hence, analogous to phase transitions, a mesoscopic, coarse-grained description may be sufficient and more useful than a microscopic one, provided essential features are captured.Crack patterns often arise from slow physical (e.g., embrittlement, contraction) or chemical (e.g., oxidation) variations, or their combination, in the material properties of an overlayer on top of an inert substrate. The overlayer may fail in different ways [14], such as decohesion, buckling, spalling and in-plane cracking. In this paper, we address the pattern-selection aspects of quasi-static, in-plane cracking by identifying the dominant mechanism and control parameters that determine the ensuing patterns. By understanding a simplified, coarse-grained model, we hope to achieve the same for the phenomena in general.A Bundle-Block Model -In a mesoscopic approach, the grains in the overlayer are represented by an array of blocks. Each pair of neighboring blocks is connected by a bundle of H bonds (coil springs) each of which has spring constant k ≡ 1 and relaxed length l. Initially, the blocks are randomly displaced by r = (x, y), where | r| ≪ l, about their mean positions on a triangular lattice. For slow cracking on a frictional substrate, the motion of the grains is overdamped, the ...

Human mobility is investigated using a continuum approach that allows to calculate the probability to observe a trip to any arbitrary region, and the fluxes between any two regions. The considered description offers a general and unified framework, in which previously proposed mobility models like the gravity model, the intervening opportunities model, and the recently introduced radiation model are naturally resulting as special cases. A new form of radiation model is derived and its validity is investigated using observational data offered by commuting trips obtained from the United States census data set, and the mobility fluxes extracted from mobile phone data collected in a western European country. The new modeling paradigm offered by this description suggests that the complex topological features observed in large mobility and transportation networks may be the result of a simple stochastic process taking place on an inhomogeneous landscape.

The continuum percolation problem of permeable and isotropically oriented sticks ͑with the form of capped cylinders͒ is reconsidered by Monte Carlo simulations in three dimensions. Errors in earlier studies are revealed and results in agreement with the excluded volume rule are presented. Finite-size effects are discussed.