Evolution of Random Networks

Christensen, Kim; Donangelo, R.; Koiller, Belita; Sneppen, Kim

doi:10.1103/physrevlett.81.2380

Cited by 70 publications

(79 citation statements)

References 17 publications

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“…(The papers [194,195] as well as Refs. [196][197][198][199][200] are devoted to more complex models generically related to the biological evolution processes. )…”

Section: Non-scale-free Network With Preferential Linkingmentioning

confidence: 99%

Evolution of networks

Dorogovtsev¹,

Mendes²

2002

Advances in Physics

2,596

1,850

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We review the recent fast progress in statistical physics of evolving networks. Interest has focused mainly on the structural properties of random complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of such a kind came into existence recently. This opens a wide field for the study of their topology, evolution, and complex processes occurring in them. Such networks possess a rich set of scaling properties. A number of them are scale-free and show striking resilience against random breakdowns. In spite of large sizes of these networks, the distances between most their vertices are short -a feature known as the "smallworld" effect. We discuss how growing networks self-organize into scale-free structures and the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation in these networks. We present a number of models demonstrating the main features of evolving networks and discuss current approaches for their simulation and analytical study. Applications of the general results to particular networks in Nature are discussed. We demonstrate the generic connections of the network growth processes with the general problems of non-equilibrium physics, econophysics, evolutionary biology, etc. CONTENTS

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“…(The papers [194,195] as well as Refs. [196][197][198][199][200] are devoted to more complex models generically related to the biological evolution processes. )…”

Section: Non-scale-free Network With Preferential Linkingmentioning

confidence: 99%

Evolution of networks

Dorogovtsev¹,

Mendes²

2002

Advances in Physics

2,596

1,850

View full text Add to dashboard Cite

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“…While such an extension is natural, only few studies in that direction have been performed so far. Christensen et al [13] have studied the BS model on random networks [14]. Kulkarni et al [15] studied it on the small-world network introduced by Watts and Strogatz [16].…”

Section: Introductionmentioning

confidence: 99%

Extremal dynamics on complex networks: Analytic solutions

2005

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The Bak-Sneppen model displaying punctuated equilibria in biological evolution is studied on random complex networks. By using the rate equation and the random walk approaches, we obtain the analytic solution of the fitness threshold xc to be 1/( k f + 1), where k f = k 2 / k (= k ) in the quenched (annealed) updating case, where k n is the n-th moment of the degree distribution. Thus, the threshold is zero (finite) for the degree exponent γ < 3 (γ > 3) for the quenched case in the thermodynamic limit. The theoretical value xc fits well to the numerical simulation data in the annealed case only. Avalanche size, defined as the duration of successive mutations below the threshold, exhibits a critical behavior as its distribution follows a power law, Pa(s) ∼ s −3/2 .

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“…Since many things are known about the chaotic dynamics of low-dimensional non linear systems, a great progress has been achieved in the understanding of the dynamical behavior of chaotic systems coupled together in a simple, geometrical regular array (coupled chaotic maps [3]), or in a completely random way [4,5].…”

Section: Introductionmentioning

confidence: 99%

Economic small-world behavior in weighted networks

2003

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The small-world phenomenon has been already the subject of a huge variety of papers, showing its appeareance in a variety of systems. However, some big holes still remain to be filled, as the commonly adopted mathematical formulation suffers from a variety of limitations, that make it unsuitable to provide a general tool of analysis for real networks, and not just for mathematical (topological) abstractions. In this paper we show where the major problems arise, and how there is therefore the need for a new reformulation of the small-world concept. Together with an analysis of the variables involved, we then propose a new theory of small-world networks based on two leading concepts: efficiency and cost. Efficiency measures how well information propagates over the network, and cost measures how expensive it is to build a network. The combination of these factors leads us to introduce the concept of economic small worlds, that formalizes the idea of networks that are "cheap" to build, and nevertheless efficient in propagating information, both at global and local scale. This new concept is shown to overcome all the limitations proper of the so-far commonly adopted formulation, and to provide an adequate tool to quantitatively analyze the behaviour of complex networks in the real world. Various complex systems are analyzed, ranging from the realm of neural networks, to social sciences, to communication and transportation networks. In each case, economic small worlds are found. Moreover, using the economic small-world framework, the construction principles of these networks can be quantitatively analyzed and compared, giving good insights on how efficiency and economy principles combine up to shape all these systems.

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Evolution of Random Networks

Cited by 70 publications

References 17 publications

Evolution of networks

Evolution of networks

Extremal dynamics on complex networks: Analytic solutions

Economic small-world behavior in weighted networks

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