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Power laws, Pareto distributions and Zipf’s law
 Contemporary Physics
, 2005
"... When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf’s law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, econ ..."
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Cited by 176 (0 self)
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When the probability of measuring a particular value of some quantity varies inversely as a power of that value, the quantity is said to follow a power law, also known variously as Zipf’s law or the Pareto distribution. Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. For instance, the distributions of the sizes of cities, earthquakes, solar flares, moon craters, wars and people’s personal fortunes all appear to follow power laws. The origin of powerlaw behaviour has been a topic of debate in the scientific community for more than a century. Here we review some of the empirical evidence for the existence of powerlaw forms and the theories proposed to explain them. I.
Models of the small world
 J. Stat. Phys
, 2000
"... It is believed that almost any pair of people in the world can be connected to one another by a short chain of intermediate acquaintances, of typical length about six. This phenomenon, colloquially referred to as the ``six degrees of separation,' ' has been the subject of considerable recent interes ..."
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Cited by 86 (2 self)
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It is believed that almost any pair of people in the world can be connected to one another by a short chain of intermediate acquaintances, of typical length about six. This phenomenon, colloquially referred to as the ``six degrees of separation,' ' has been the subject of considerable recent interest within the physics community. This paper provides a short review of the topic. KEY WORDS: social networks. Small world; networks; disordered systems; graph theory;
Nonequilibrium critical phenomena and phase transitions into absorbing states
 ADVANCES IN PHYSICS
, 2000
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Optimization with extremal dynamics
 Physical Review Letters
"... We explore a new generalpurpose heuristic for finding highquality solutions to hard optimization problems. The method, called extremal optimization, is inspired by selforganized criticality, a concept introduced to describe emergent complexity in physical systems. Extremal optimization successive ..."
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Cited by 31 (2 self)
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We explore a new generalpurpose heuristic for finding highquality solutions to hard optimization problems. The method, called extremal optimization, is inspired by selforganized criticality, a concept introduced to describe emergent complexity in physical systems. Extremal optimization successively replaces extremely undesirable variables of a single suboptimal solution with new, random ones. Large fluctuations ensue, that efficiently explore many local optima. With only one adjustable parameter, the heuristic’s performance has proven competitive with more elaborate methods, especially near phase transitions which are believed to coincide with the hardest instances. We use extremal optimization to elucidate the phase transition in the 3coloring problem, and we provide independent confirmation of previously reported extrapolations for the groundstate energy of±J spin glasses in d = 3 and 4. PACS number(s): 02.60.Pn, 05.65.+b, 75.10.Nr, 64.60.Cn. Many natural systems have, without any centralized organizing facility, developed into complex structures that optimize their use of resources in sophisticated ways [1]. Biological evolution has formed efficient
Computability and Evolutionary Complexity: Markets as Complex Adaptive Systems
 CAS). Economic Journal 115 (504) (2005), F159–F192. Available online at SSRN: http://ssrn.com/abstract=745578
"... Few will argue that the epiphenomena of biological systems and socioeconomic systems are anything but complex. The purpose of this Feature is to examine critically and contribute to the burgeoning multidisciplinary literature on markets as complex adaptive systems (CAS). The new sciences of compl ..."
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Cited by 26 (9 self)
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Few will argue that the epiphenomena of biological systems and socioeconomic systems are anything but complex. The purpose of this Feature is to examine critically and contribute to the burgeoning multidisciplinary literature on markets as complex adaptive systems (CAS). The new sciences of complexity, the principles of selforganisation and emergence along with the methods of evolutionary computation and artificially intelligent agent models have been developed in a multidisciplinary fashion. The cognoscenti here consider that complex systems whether natural or artificial, physical, biological or socioeconomic can be characterised by a unifying set of principles. Further, it is held that these principles mark a paradigm shift from earlier ways of viewing such phenomenon. The articles in this Feature aim to provide detailed insights and examples of both the challenges and the prospects for economics that are offered by the new methods of the complexity sciences. The applicability or not of the optimisation framework of conventional economics depends on the domain of the problem and in particular the modern theories behind noncomputability are outlined to explain why adaptive or emergent methods of computation and agentbased
Nature's way of optimizing
, 2000
"... We propose a generalpurpose method for finding highquality solutions to hard optimization problems, inspired by selforganizing processes often found in nature. The method, called Extremal Optimization, successively eliminates extremely undesirable components of suboptimal solutions. Drawing upon ..."
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Cited by 25 (5 self)
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We propose a generalpurpose method for finding highquality solutions to hard optimization problems, inspired by selforganizing processes often found in nature. The method, called Extremal Optimization, successively eliminates extremely undesirable components of suboptimal solutions. Drawing upon models used to simulate farfromequilibrium dynamics, it complements approximation methods inspired by equilibrium statistical physics, such as Simulated Annealing. With only one adjustable parameter, its performance proves competitive with, and often superior to, more elaborate stochastic optimization procedures. We demonstrate it here on two classic hard optimization problems: graph partitioning and the traveling salesman problem.
The Influence of PredatorPrey Population Dynamics on the Longterm Evolution of Food Web Structure
 J. theor. Biol
, 2001
"... We develop a set of equations to describe the population dynamics of many interacting species in food webs. Predatorprey interactions are nonlinear, and are based on ratiodependent functional responses. The equations account for competition for resources between members of the same species, and b ..."
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Cited by 24 (2 self)
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We develop a set of equations to describe the population dynamics of many interacting species in food webs. Predatorprey interactions are nonlinear, and are based on ratiodependent functional responses. The equations account for competition for resources between members of the same species, and between members of different species. Predators divide their total hunting/foraging effort between the available prey species according to an evolutionarily stable strategy (ESS). The ESS foraging behaviour does not correspond to the predictions of optimal foraging theory. We use the population dynamics equations in simulations of the Webworld model of evolving ecosystems. New species are added to an existing food web due to speciation events, whilst species become extinct due to coevolution and competition. We study the dynamics of speciesdiversity in Webworld on a macroevolutionary timescale. Coevolutionary interactions are strong enough to cause continuous overturn of species, in contrast to our previous Webworld simulations with simpler population dynamics. Although there are significant fluctuations in species diversity because of speciation and extinction, 1 very large scale extinction avalanches appear to be absent from the dynamics, and we find no evidence for selforganised criticality. 1
Scale Invariance in Biology: Coincidence Or Footprint of a Universal Mechanism?
, 2001
"... In this article, we present a selfcontained review of recent work on complex biological systems which exhibit no characteristic scale. This property can manifest itself with fractals (spatial scale invariance), flicker noise or 1}fnoise where f denotes the frequency of a signal (temporal scale i ..."
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Cited by 23 (1 self)
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In this article, we present a selfcontained review of recent work on complex biological systems which exhibit no characteristic scale. This property can manifest itself with fractals (spatial scale invariance), flicker noise or 1}fnoise where f denotes the frequency of a signal (temporal scale invariance) and power laws (scale invariance in the size and duration of events in the dynamics of the system). A hypothesis recently put forward to explain these scalefree phenomomena is criticality, a notion introduced by physicists while studying phase transitions in materials, where systems spontaneously arrange themselves in an unstable manner similar, for instance, to a row of dominoes. Here, we review in a critical manner work which investigates to what extent this idea can be generalized to biology. More precisely, we start with a brief introduction to the concepts of absence of characteristic scale (powerlaw distributions, fractals and 1}fnoise) and of critical phenomena. We then review typical mathematical models exhibiting such properties : edge of chaos, cellular automata and selforganized critical models. These notions are then brought together to see to what extent they can account for the scale invariance observed in ecology, evolution of species, type III epidemics and some aspects of the central nervous system. This article also discusses how the notion of scale invariance can give important insights into the workings of biological systems.
SelfOrganization in PeertoPeer Systems
 In Proceedings of the 2002 SIGOPS European Workshop, St. Emilion
, 2002
"... This paper addresses the problem of forming groups in peertopeer (P2P) systems and examines what dependability means in decentralized distributed systems. Much of the literature in this field assumes that the participants form a local picture of global state, yet little research has been done disc ..."
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Cited by 23 (2 self)
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This paper addresses the problem of forming groups in peertopeer (P2P) systems and examines what dependability means in decentralized distributed systems. Much of the literature in this field assumes that the participants form a local picture of global state, yet little research has been done discussing how this state remains stable as nodes enter and leave the system. We assume that nodes remain in the system long enough to benefit from retaining state, but not sufficiently long that the dynamic nature of the problem can be ignored. We look at the components that describe a system's dependability and argue that nextgeneration decentralized systems must explicitly delineate the information dispersal mechanisms (e.g., probe, eventdriven, broadcast), the capabilities assumed about constituent nodes (bandwidth, uptime, reentry distributions), and distribution of information demands (needles in a haystack vs. hay in a haystack [13]). We evaluate two systems based on these criteria: Chord [22] and a heterogeneousnode hierarchical grouping scheme [11]. The former gives a failed request rate under normal P2P conditions and a prototype of the latter a similar rate under more strenuous conditions with an order of magnitude more organizational messages. This analysis suggests several methods to greatly improve the prototype.