Self-organized scale-free networks

Park, Kwangho; Lai, Ying–Cheng; Ye, Nong

doi:10.1103/physreve.72.026131

Cited by 67 publications

(58 citation statements)

References 22 publications

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“…This is the role of the factors of (1 − Π). Such terms are not normally found in the master equations for network rewiring [3,5,6,7,8,9,11]. It is crucial that we do this otherwise we will not have the correct behaviour at the boundaries k = 0 and k = E.…”

Section: The Modelmentioning

confidence: 99%

“…The classic example of Watts and Strogatz [1] is of this type and such models are often studied in their own right [2,3,4,5,6,7]. Network rewiring is also related to to some multi-Urn models [8,9,10,11] which include what are termed Backgammon or Balls-in-Boxes models [12] used for glasses [13,14], simplicial gravity [15] and wealth distributions [16].…”

Section: Introductionmentioning

confidence: 99%

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Exact solution for the time evolution of network rewiring models

Evans

Plato

2007

Phys. Rev. E

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We consider the rewiring of a bipartite graph using a mixture of random and preferential attachment. The full mean field equations for the degree distribution and its generating function are given. The exact solution of these equations for all finite parameter values at any time is found in terms of standard functions. It is demonstrated that these solutions are an excellent fit to numerical simulations of the model. We discuss the relationship between our model and several others in the literature including examples of Urn, Backgammon, and Balls-in-Boxes models, the Watts and Strogatz rewiring problem and some models of zero range processes. Our model is also equivalent to those used in various applications including cultural transmission, family name and gene frequencies, glasses, and wealth distributions. Finally some Voter models and an example of a Minority game also show features described by our model.

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Section: The Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exact solution for the time evolution of network rewiring models

Evans

Plato

2007

Phys. Rev. E

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“…In addition, mechanisms have been proposed that can produce stable topological metrics in networks of constant size (9)(10)(11)(12). Notably, methods based on percolation theory have significantly contributed to our understanding of how robust the static network structures generated by these assembly mechanisms are to fragmentation under random and targeted attack (8,(13)(14)(15).…”

mentioning

confidence: 99%

Asymmetric disassembly and robustness in declining networks

Saavedra

Reed‐Tsochas²,

Uzzi

2008

Proc. Natl. Acad. Sci. U.S.A.

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Mechanisms that enable declining networks to avert structural collapse and performance degradation are not well understood. This knowledge gap reflects a shortage of data on declining networks and an emphasis on models of network growth. Analyzing >700,000 transactions between firms in the New York garment industry over 19 years, we tracked this network's decline and measured how its topology and global performance evolved. We find that favoring asymmetric (disassortative) links is key to preserving the topology and functionality of the declining network. Based on our findings, we tested a model of network decline that combines an asymmetric disassembly process for contraction with a preferential attachment process for regrowth. Our simulation results indicate that the model can explain robustness under decline even if the total population of nodes contracts by more than an order of magnitude, in line with our observations for the empirical network. These findings suggest that disassembly mechanisms are not simply assembly mechanisms in reverse and that our model is relevant to understanding the process of decline and collapse in a broad range of biological, technological, and financial networks.complex networks ͉ contraction ͉ socioeconomic systems R esearch on the dynamics and robustness of complex networks (1-3) has emphasized the study of global network growth, identifying assembly mechanisms such as preferential attachment (4, 5), vertex fitness (6), vertex duplication (7), and fractal network growth (8), which, at the macroscopic level, generate stable topological characteristics despite large fluctuations in the microscopic network parameters. In addition, mechanisms have been proposed that can produce stable topological metrics in networks of constant size (9-12). Notably, methods based on percolation theory have significantly contributed to our understanding of how robust the static network structures generated by these assembly mechanisms are to fragmentation under random and targeted attack (8,(13)(14)(15).Less is known about the dynamics and robustness of networks under sustained decline. In declining networks, new nodes and links may be added over time but the net process is a progressive loss of nodes and links. Hence, although assembly mechanisms may come into play, the emphasis must be on which disassembly mechanisms help preserve the topological characteristics and performance of the network. For example, Alzheimer's research examines how degradation in mental performance can be related to the progressive loss and disconnection of neurons with age (16). In an ecological context, analogous questions arise regarding the vulnerability of food webs to habitat loss and fragmentation (17, 18). In social and economic systems, network decline has raised questions about the preservation of social capital (19), resource allocation in developing economies (20), the prevention of financial collapse (21), effective coordination among suppliers (22), the restructuring of political systems (23), and lock-in in...

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“…Previous research showed that the degree exponents from a scale-free network such as the World Wide Web were typically g Ն2 ; however, those from scale-free spatial networks of biological origin have been found to be P(k)ϰ k Ϫ1 (Hermann et al, 2003). Park et al (2005) suggested that two classes of model exhibited the scale-free property with and without growth of the network. According to these authors, the scale-free property of the network might arise through a self-organising mechanism via the natural evolution of links in the network, even without any growth mechanism.…”

Section: ϫGmentioning

confidence: 99%

Scale-free Property of the Passing Behaviour in a Team Sport

Yamamoto

2009

Int. J. Sport Health Sci.

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This study examined the application of network theory to skilled passing behaviour in a team sport by considering small-world and scale-free network models. Using data obtained from a 2006 soccer game in Japan between Japan and Ghana, we counted the number of passes by each player within 5-minute intervals. The structural properties of the passing behaviour, which included a characteristic path length and clustering coefficient, and the degree of distribution were analysed. This showed that the structural property of the passing behaviour represented neither a complete graph nor a random graph; rather, it reflected a small-world or scale-free network. In addition, the probabilities of outgoing and incoming passes reflected links that followed a power-law distribution. Passing behaviour in a soccer match appeared to be similar to behaviour in social networks with smaller vertices in terms of the scale-free property and a self-organising mechanism.Key word: network theory, soccer, power-law distribution Our study represents another approach to exploring the dynamics of team sports. We examined the application of network theory to skilled passing behaviour in a team sport to investigate the merits of the small-world (Watts & Strogatz, 1998) and scale-free (Barabási & Albert, 1999) network models. Brief review of network theoryNetwork theory is rooted in Euler's graph theory (Biggs et al., 1976), which was originally designed to solve the problem of the seven bridges of Königsberg, Prussia (now Kaliningrad, Russia) mathematically. The question was whether it was possible to follow a route that crossed each bridge exactly once. Euler (1736) formulated the problem by graphing Königsberg. First, he eliminated all of the features except the landmasses and the bridges connecting them; next, he replaced each landmass with a dot, called a vertex or node, and each bridge with a line, called an edge or link. The geographic relationships between the landmasses and the distances of the bridges were not considered in this model and only the connectivities of vertices deemed important.Subsequently, Euler's graph theory (Biggs et al., 1976) has been expanded to include various types of graph, including star, tree, and wheel graphs (See Figure 1). Studies by Erdös and Rényi (1959, 1960, 1961, cited by Karoński and Ruciński (1997), elaborated on these graph theories by introducing the graph generation approach. This approach uses a regular graph and a random graph; a graph is regular when all vertices have equal edges and a random graph is reconstructed from a regular graph by random linkages between vertices with a particular probability p. Random graphs can be used to examine the phase transition and percolation of graph properties. In this context, the terms graph and network are synonymous.The small-world phenomenon refers to the relatively short path between two vertices that occurs in most networks despite their frequently large size. During the 1960s, social psychologist Stanley Milgram examined this phenomenon with a sophis...

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Self-organized scale-free networks

Cited by 67 publications

References 22 publications

Exact solution for the time evolution of network rewiring models

Exact solution for the time evolution of network rewiring models

Asymmetric disassembly and robustness in declining networks

Scale-free Property of the Passing Behaviour in a Team Sport

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