Emergence of Scaling in Random Networks

Abstract

Systems as diverse as genetic networks or the world wide web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature is found to be a consequence of the two generic mechanisms that networks expand continuously by the addition of new vertices, and new vertices attach preferentially to already well connected sites. A model based on these two ingredients reproduces the observed stationary scalefree distributions, indicating that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems. * alb@nd.edu 1

Complex network theory, streamflow, and hydrometric monitoring system design

“…From Fig. 4, it is immediately clear that the streamflow network is not a regular network because, by definition, each node in a regular network has the same number of links, i.e., P (k) = δ k , where δ k is the Kronecker delta function located at a single value of k. Furthermore, the streamflow network degree distribution is not consistent with the expected degree distribution for a scale-free network because scale-free networks have an asymmetric degree distribution which asymptotes to P (k) ∝ k −γ at sufficiently large values of k, where γ ranges from 2.1 to 4 for a wide array of observed networks (Barabási and Albert, 1999). The stream- .…”

“…Examples of the latter include the need to monitor a particular river at a particular location to constrain the design of a bridge or highway, set instream flow requirements for a river with special ecological significance, monitor high-flow conditions for a downstream inhabited flood plain, estimate water availability for a particular water supply utility, provide key input information to an environmental assessment process around a proposed natural resource development project, and so forth. That said, there is a long history of using quantitative analysis of environmental data to provide information that might enable improved sampling system design, including correlation, cluster, principal component, information theoretic (entropic), geostatistical, and other types of analysis (e.g., Bras and Rodríguez-Iturbe, 1976;Caselton and Husain, 1980;Flatman and Yfantis, 1984;Burn and Goulter, 1991;Yang and Burn, 1994;Norberg and Rosén, 2006;Fleming, 2007;Pires et al, 2008;Mishra and Coulibaly, 2010;Archfield and Kiang, 2011;Neuman et al, 2012;Putthividhya and Tanaka, 2012;Mishra and Coulibaly, 2014). A review specifically of streamflow monitoring system design applications of such methods is provided by Mishra and Coulibaly (2009), and for a recent example of continued innovation in this field, see Hannaford et al (2013).…”

Semi-automatic generation of a corpus of Wikipedia articles on science and technology

“…As stated, we did not include 225,380 category pages. Due to its nature, Wikipedia can be seen as a scale-free network (Barabási;Albert, 1999), where the distribution of node out-degrees (i.e. number of links) emerges automatically from a stochastic growth model in which new nodes are added continuously and attach themselves preferentially to existing nodes, with probability proportional to the degree of the target node--so richly connected nodes get richer (Strogatz, 2001), becoming hubs to hundreds or even thousands of pages (like annex pages).…”

Internet as a Growing and Dynamic Network: An Economic View

“…By exploring several large databases describing the topology of large networks, AB found that, for most large networks, the degree distribution deviates from the Poisson law and that, in most of cases, it follows a power-law for large K. Since power-laws are independent of the unit of measure, these networks are called "scale-free" [17] [18]. This topological characteristic is determined by two mechanisms that interact inside the network: growth and preferential attachment.…”