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111
Evolutionary games on graphs
, 2007
"... Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to ..."
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Cited by 143 (0 self)
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Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in nonequilibrium statistical physics. This review gives a tutorialtype overview of the field for physicists. The first four sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fifth section surveys the topological complications implied by nonmeanfieldtype social network structures in general. The next three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner’s Dilemma, the Rock–Scissors–Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.
Stochastic neural field theory and the systemsize expansion
 SIAM J. Appl. Math
, 2009
"... Abstract. We analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons. The state of the network is specified by the fraction of active or spiking neurons in each population, and tra ..."
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Cited by 29 (5 self)
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Abstract. We analyze a master equation formulation of stochastic neurodynamics for a network of synaptically coupled homogeneous neuronal populations each consisting of N identical neurons. The state of the network is specified by the fraction of active or spiking neurons in each population, and transition rates are chosen so that in the thermodynamic or deterministic limit (N →∞)we recover standard activitybased or voltagebased rate models. We derive the lowest order corrections to these rate equations for large but finite N using two different approximation schemes, one based on the Van Kampen systemsize expansion and the other based on path integral methods. Both methods yield the same series expansion of the moment equations, which at O(1/N) can be truncated to form a closed system of equations for the first and secondorder moments. Taking a continuum limit of the moment equations while keeping the system size N fixed generates a system of integrodifferential equations for the mean and covariance of the corresponding stochastic neural field model. We also show how the path integral approach can be used to study large deviation or rare event statistics underlying escape from the basin of attraction of a stable fixed point of the meanfield dynamics; such an analysis is not possible using the systemsize expansion since the latter cannot accurately determine exponentially small transitions. Key words. neural field theory, master equations, stochastic processes, systemsize expansion, path integrals
Evolutionary game theory: temporal and spatial effects beyond replicator dynamics
, 2009
"... Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the socalled replicator equation, that describes mathematically the idea that those indiv ..."
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Cited by 23 (1 self)
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Evolutionary game dynamics is one of the most fruitful frameworks for studying evolution in different disciplines, from Biology to Economics. Within this context, the approach of choice for many researchers is the socalled replicator equation, that describes mathematically the idea that those individuals performing better have more offspring and thus their frequency in the population grows. While very many interesting results have been obtained with this equation in the three decades elapsed since it was first proposed, it is important to realize the limits of its applicability. One particularly relevant issue in this respect is that of nonmean field effects, that may arise from temporal fluctuations or from spatial correlations, both neglected in the replicator equation. This review discusses these temporal and spatial effects focusing on the nontrivial modifications they induce when compared to the outcome of replicator dynamics. Alongside this question, the hypothesis of linearity and its relation to the choice of the rule for strategy update is also analyzed. The discussion is presented in terms of the emergence of cooperation, as one of the current key problems in Biology and in other disciplines.
Asynchronism induces second order phase transitions in elementary cellular automata
 Journal of Cellular Automata
"... Cellular automata are widely used to model natural or artificial systems. Classically they are run with perfect synchrony, i.e., the local rule is applied to each cell at each time step. A possible modification of the updating scheme consists in applying the rule with a fixed probability, called the ..."
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Cited by 22 (8 self)
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Cellular automata are widely used to model natural or artificial systems. Classically they are run with perfect synchrony, i.e., the local rule is applied to each cell at each time step. A possible modification of the updating scheme consists in applying the rule with a fixed probability, called the synchrony rate. For some particular rules, varying the synchrony rate continuously produces a qualitative change in the behaviour of the cellular automaton. We investigate the nature of this change of behaviour using MonteCarlo simulations. We show that this phenomenon is a secondorder phase transition, which we characterise more specifically as belonging to the directed percolation or to the parity conservation universality classes studied in statistical physics.
Wijland: Thermodynamic formalism for systems with Markov dynamics
 J. Stat. Phys
, 2008
"... The thermodynamic formalism allows one to access the chaotic properties of equilibrium and outofequilibrium systems, by deriving those from a dynamical partition function. The definition that has been given for this partition function within the framework of discrete time Markov chains was not sui ..."
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Cited by 20 (3 self)
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The thermodynamic formalism allows one to access the chaotic properties of equilibrium and outofequilibrium systems, by deriving those from a dynamical partition function. The definition that has been given for this partition function within the framework of discrete time Markov chains was not suitable for continuous time Markov dynamics. Here we propose another interpretation of the definition that allows us to apply the thermodynamic formalism to continuous time. We also generalize the formalism –a dynamical Gibbs ensemble construction– to a whole family of observables and their associated large deviation functions. This allows us to make the connection between the thermodynamic formalism and the observable involved in the muchstudied fluctuation theorem. We illustrate our approach on various physical systems: random walks, exclusion processes, an Ising model and the contact process. In the latter cases, we identify a signature of the occurrence of dynamical phase transitions. We show that this signature can already be unraveled using the simplest dynamical ensemble one could define, based on the number of configuration changes a system has undergone over an asymptotically large time window. 1 1
Evolutionary Game Theory: Theoretical Concepts and Applications to Microbial Communities
, 2010
"... Ecological systems are complex assemblies of large numbers of individuals, interacting competitively under multifaceted environmental conditions. Recent studies using microbial laboratory communities have revealed some of the selforganization principles underneath the complexity of these systems. A ..."
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Cited by 18 (1 self)
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Ecological systems are complex assemblies of large numbers of individuals, interacting competitively under multifaceted environmental conditions. Recent studies using microbial laboratory communities have revealed some of the selforganization principles underneath the complexity of these systems. A major role of the inherent stochasticity of its dynamics and the spatial segregation of different interacting species into distinct patterns has thereby been established. It ensures viability of microbial colonies by allowing for species diversity, cooperative behavior and other kinds of “social ” behavior. A synthesis of evolutionary game theory, nonlinear dynamics, and the theory of stochastic processes provides the mathematical tools and conceptual framework for a deeper understanding of these ecological systems. We give an introduction into the modern formulation of these theories and illustrate their effectiveness focussing on selected examples of microbial
BOUNDARY PRESERVING SEMIANALYTIC NUMERICAL ALGORITHMS FOR STOCHASTIC DIFFERENTIAL EQUATIONS ∗
, 1525
"... Abstract. Construction of splittingstep methods and properties of related nonnegativity and boundary preserving semianalytic numerical algorithms for solving stochastic differential equations (SDEs) of Itô type are discussed. As the crucial assumption, we oppose conditions such that one can decompo ..."
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Cited by 10 (1 self)
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Abstract. Construction of splittingstep methods and properties of related nonnegativity and boundary preserving semianalytic numerical algorithms for solving stochastic differential equations (SDEs) of Itô type are discussed. As the crucial assumption, we oppose conditions such that one can decompose the original system of SDEs into subsystems for which one knows either the exact solution or its conditional transition probability. We present convergence proofs for a newly designed splittingstep algorithm and simulation studies for numerous wellknown numerical examples ranging from stochastic dynamics occurring in asset pricing in mathematical finance (Cox–Ingersoll–Ross (CIR) and constant elasticity of variance (CEV) models) to measurevalued diffusion and superBrownian motion (stochastic PDEs (SPDEs)) as met in biology and physics.
S.: Boundary Conditions and Phase Transitions in Neural Networks. Theoretical Results
"... Abstract. This paper gives new simulation results on the asymptotic behaviour of theoretical neural networks on Z and Z2 following an extended Hopfield law. It specifically focuses on the influence of fixed boundary conditions on such networks. First, we will generalise the theoretical results alr ..."
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Cited by 10 (8 self)
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Abstract. This paper gives new simulation results on the asymptotic behaviour of theoretical neural networks on Z and Z2 following an extended Hopfield law. It specifically focuses on the influence of fixed boundary conditions on such networks. First, we will generalise the theoretical results already obtained for attractive networks in one dimension to more complicated neural networks. Then, we will focus on twodimensional neural networks. Theoretical results have already been found for the nearest neighbours Ising model in 2D with translationinvariant local isotropic interactions. We will detail what happens for this kind of interaction in neural networks and we will also focus on more complicated interactions, i.e., interactions that are not local, neither isotropic, nor translationinvariant. For all these kinds of interactions, we will show that fixed boundary conditions have significant impacts on the asymptotic behaviour of such networks. These impacts result in the emergence of phase transitions whose geometric shape will be numerically characterised.
Scale and scaling in ecological and economic systems
 Environmental and Resource Economics
, 2003
"... Abstract. We review various aspects of the notion of scale applied to natural systems, in particular complex adaptive systems. We argue that scaling issues are not only crucial from the standpoint of basic science, but also in many applied issues, and discuss tools for detecting and dealing with mul ..."
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Cited by 8 (0 self)
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Abstract. We review various aspects of the notion of scale applied to natural systems, in particular complex adaptive systems. We argue that scaling issues are not only crucial from the standpoint of basic science, but also in many applied issues, and discuss tools for detecting and dealing with multiple scales, both spatial and temporal. We also suggest that the techniques of statistical mechanics, which have been successful in describing many emergent patterns in physical systems, can also prove useful in the study of complex adaptive systems.
Robustness of the critical behaviour in a discrete stochastic reactiondiffusion medium
 in Proceedings of IWNC 2009
, 2010
"... Abstract. We study the steady states of a reactiondiffusion medium modelled by a stochastic 2D cellular automaton. We consider the GreenbergHastings model where noise and topological irregularities of the grid are taken into account. The decrease of the probability of excitation changes qualitati ..."
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Cited by 6 (2 self)
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Abstract. We study the steady states of a reactiondiffusion medium modelled by a stochastic 2D cellular automaton. We consider the GreenbergHastings model where noise and topological irregularities of the grid are taken into account. The decrease of the probability of excitation changes qualitatively the behaviour of the system from an “active” to an “extinct ” steady state. Simulations show that this change occurs near a critical threshold; it is identified as a nonequilibrium phase transition which belongs to the directed percolation universality class. We test the robustness of the phenomenon by introducing persistent defects in the topology: directed percolation behaviour is conserved. Using experimental and analytical tools, we suggest that the critical threshold varies as the inverse of the average number of neighbours per cell.