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Topic: Replicator equations

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In the News (Mon 22 Jul 19)

  Replicator Equations, Maximal Cliques, and Graph Isomorphism - Pelillo (ResearchIndex)
Replicator Equations, Maximal Cliques, and Graph Isomorphism (1998)
To solve the program we use the so-called "replicator" equations, a class of straightforward continuous- and...
Pelillo, "Replicator equations, maximal cliques, and graph isomorphism," Neural Computation, to appear.
citeseer.ist.psu.edu /pelillo98replicator.html   (1000 words)

 Energy Lyapunov function for generalized replicator equations   (Site not responding. Last check: 2007-10-26)
Replicator dynamics is an evolutionary strategy well established in different disciplines of biological sciences.
Besides this, replicator selection has been applied to problem solving in combinatorial optimization and to learning in neural networks and also in fluid mechanics, game and laser theory.
In this paper the author introduces the new class of generalized replicator equations with nonlinear response functions and constructs an energy Lyapunov function for this system.
csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/phycon/&toc=comp/proceedings/phycon/2003/7939/01/7939toc.xml&DOI=10.1109/PHYCON.2003.1236830   (167 words)

The third equations shows that the total replicator number is constant, and so replacing x2 with C-x1, in the equation for x1', we can obtain an expression for x1' purely in terms of x1.
For parabolic replicators with only active constituent decay, there is no co-existence, since the inactive constituent can replenish the active constituent as the active constituent decays, and so serve as a buffer store of active replicators, specific for the species concerned.
A Model of Prebiotic Replication: Survival of the Fittest versus Extinction of the Unfittest.
www.cogs.susx.ac.uk /users/ctf20/dphil_2005/Coexistence.htm   (7248 words)

 Inhomogeneous Replicator Equations   (Site not responding. Last check: 2007-10-26)
The inhomogeneous replicator equation is derived as the continuous time model for parallel first and second order autocatalytic replication of macromolecules in a flow reactor based on mass action kinetics.
It is shown that the total concentration of replicating material determines the relative importance of the first order and the second order mechanism.
A minimal prebiotic scenario with the potentiality to develop cooperation is derived from the inhomogeneous replicator equation.
www.tbi.univie.ac.at /papers/Abstracts/91-00-002abs.html   (149 words)

 Preprint: On the Linearity of Replicator Equations   (Site not responding. Last check: 2007-10-26)
We show that replicator equations follow naturally from the exponential affine structure of the simplex known from information geometry.
It is then natural to call replicator equations linear if their fitness function is affine.
For such linear replicator equations an explicit solution can be found.
www.bioinf.uni-leipzig.de /Publications/ABSTRACTS/03-022abs.html   (67 words)

 Immune Networks Modeled by Replicator Equations
We show that by transforming into relative concentrations, the B-cell network equations can be brought into a form that closely resembles the replicator equation.
The number of equilibria and their stability are then characterized using methods developed for the study of second-order replicator equations.
A particularly interesting result of our analysis is the fact that even though the immune network equations are not second-order, the number and stability of their equilibria can be obtained by a superposition of second-order replicator systems.
www.santafe.edu /research/publications/wpabstract/199307048   (234 words)

 Amazon.ca: Evolutionary Games and Population Dynamics: Books: Josef Hofbauer,Karl Sigmund   (Site not responding. Last check: 2007-10-26)
Replicator equations describe how successful strategies spread and thereby create new conditions that can alter the basis of their success, i.e., to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions that punctuate evolution.
Indeed, it is a wonderful way to learn differential equation theory, since one immediately is faced with meaningful problems to solve.
First, HS deal with Lotka-Volterra equations of the type prevalent in predator-prey models, which they extend to ecological models and several populations.
www.amazon.ca /Evolutionary-Games-Population-Dynamics-Hofbauer/dp/052162570X   (879 words)

 Math 539   (Site not responding. Last check: 2007-10-26)
This is a graduate course on ordinary differential equations and dynamical systems.
The main focus of the course will be on the geometric (or qualitative) theory of nonlinear differential equations, and applications of these ideas to problems in mathematical biology.
It is one of the most cited references in current research on differential equations and biological applications.
www.webpages.uidaho.edu /~krone/m539.html   (198 words)

 On the Asymptotics of an Infinite-Dimensional Stochastic Dynamical System
I'm a little paranoid about being scooped, so I won't say much more, only that there are certain connections with genetic algorithms, and that the finite-dimensional analog is related to the replicator equation of evolutionary biology.
Now, the replicator equation can be re-written as a linear system (or so Nihat tells me), which may be helpful...
Nihat Ay and Ionas Erb, "On the Linearity of Replicator Equations", SFI Working Paper 2003-10-053 [I think this is applicable to my problem, but I need to carefully re-read it to make sure.
cscs.umich.edu /~crshalizi/notebooks/infinite-stochastic-dyn-sys.html   (750 words)

 Bulletin of the American Mathematical Society
In this survey, we present an overview of the many brands of deterministic dynamical systems motivated by evolutionary game theory, including ordinary differential equations (and, in particular, the replicator equation), differential inclusions (the best response dynamics), difference equations (as, for instance, fictitious play) and reaction-diffusion systems.
A recurrent theme (the so-called `folk theorem of evolutionary game theory') is the close connection of the dynamical approach with the Nash equilibrium, but we show that a static, equilibrium-based viewpoint is, on principle, unable to always account for the long-term behaviour of players adjusting their behaviour to maximise their payoff.
H. Nikaido: Stability of equilibrium by the Brown-von Neumann differential equation, Econometrica 27 (1959), 654-671.
www.ams.org /bull/2003-40-04/S0273-0979-03-00988-1/home.html   (2799 words)

 Autocatalytic Networks with Translation
We consider the kinetics of an autocatalytic reaction network in which replication and catalytic action is separated by a translation step.
We find that the behavior of such a system is closely related to second order replicator equations, which describe the kinetic of autocatalytic reaction networks in which the replicators act also as catalysts.
The influence of mutations on replication translation networks is also very similar to the behavior of selection-mutation equations.
ideas.repec.org /p/wop/safiwp/95-07-063.html   (272 words)

 Generalized URN models of evolutionary processes, Michel Benaïm, Sebastian J. Schreiber, Pierre Tarrès
Previously, it has been shown that on the event of population growth, the limiting distribution of genotypes is a connected internally chain recurrent set for the mean limit ODE.
Second, the population distribution almost surely does not converge to sets where growth is not expected and almost surely does not converge to “nondegenerate” unstable equilibria or periodic orbits of the mean limit ODE.
Applications to stochastic analogs of the replicator equations and fertility-selection equations of population genetics are given.
projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.aoap/1089736292   (435 words)

 Replicator equation - Wikipedia, the free encyclopedia
The replicator equation is a differential equation that defines the dynamics of evolutionary games.
Infinite population size: Assuming an infinitely large population makes the equation easier to understand and analyze, as the dynamics of populations become deterministic when an infinite number of individuals is assumed.
Continuous time: The dynamics in a replicator equation are defined by the rates of birth and death of individuals, resulting in differential equations
en.wikipedia.org /wiki/Replicator_equation   (420 words)

 iApplianceWeb-iApplianceReview   (Site not responding. Last check: 2007-10-26)
Rather than global functions, the agents were programmed with a model of their environment without sharing knowledge.
Using the rock-scissors-papers game interaction as a model, the researchers showed that coupled replicator equations explain the macroscopic behaviors of a network of learning agents.
The researchers said they plan to quantify the collective functions of large multiagent systems, developing statistical equations of motion to account for fluctuations in the finite models and histories among agents.
www.iapplianceweb.com /story/OEG20020528S0062.htm   (245 words)

 SSRN-On the Evolution of Strategy: Ito Vector Fields and Replicator Flows by Dominick Samperi
An interdisciplinary study is presented where replicator equations from evolutionary biology are applied to the problem of modeling the evolution of strategy in finance.
It is shown that replicator equations can be viewed as integral curves for vector fields on a space of measures, the so-called Ito vector fields.
Connections to opinion formation models, agent-based simulations, porous medium equations, and the Tsallis entropy are made.
papers.ssrn.com /sol3/papers.cfm?abstract_id=870076   (248 words)

 Conference Materials   (Site not responding. Last check: 2007-10-26)
The approach is centered around a remarkable result proved by Motzkin and Straus in the mid-1960s, and recently expanded in various ways, which allows us to formulate the maximum clique problem in terms of a standard quadratic program.
To solve the program we use``replicator'' equations, a class of simple continuous- and discrete-time dynamical systems developed in various branches of theoretical biology.
We show how, despite their inability to escape from local solutions, they nevertheless provide experimental results which are competitive with those obtained using more elaborate mean-field annealing heuristics.
cognet.mit.edu /library/conferences/paper?paper_id=2627   (109 words)

 Citebase - Heteroclinic Chaos, Chaotic Itinerancy and Neutral Attractors in Symmetrical Replicator Equations with ...
Heteroclinic Chaos, Chaotic Itinerancy and Neutral Attractors in Symmetrical Replicator Equations with Mutations
Authors: Hashimoto, K. Ikegami, T. A replicator equation with mutation processes is numerically studied.
Without any mutations, two characteristics of the replicator dynamics are known: an exponential divergence of the dominance period, and hierarchical orderings of the attractors.
citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:nlin/0005036   (153 words)

 Occidental College Math Department
Fractal image compression applies fractal geometry by using systems of equations known as affine transformations to scale, rotate, or translate a digital image in order to fit in just a small fraction of a computer’s hard drive.
She lived there for 26 years before coming to Southern California where she is currently a Junior in the Math department at Occidental College.
The replicator equations, which are also fundamental in describing population dynamics, govern the dynamics of this game and give rise to interesting behavior.
departments.oxy.edu /math/Activities/s05speakers.htm   (989 words)

 Title: "Functional properties of neural circuits for vision"
  The distributed connectionist implementation of this recognition process can be construed as an indirect implementation of Pelillo's (1999) approximate graph matching via replicator equations, by embedding his algorithm in a fixed high-dimensional vector space.
A., Anderson, C. H., and Van Essen, D. A neurobiological model of visual attention and invariant pattern recognition based on dynamic routing of information.
Pelillo, M. Replicator equations, maximal cliques, and graph isomorphism.
www.rni.org /Seminars/Gayler10_22_03.html   (610 words)

 Payoff-Monotonic Game Dynamics and the Maximum Clique Problem -- Pelillo and Torsello 18 (5): 1215 -- Neural Computation
replicator equations have recently proven to be remarkably effective
of which replicator dynamics are a special instance, and show
faster than and at least as accurate as replicator equations.
neco.mitpress.org /cgi/content/abstract/18/5/1215   (338 words)

 Coupled Replicator Equations for the Dynamics of Learning in Multiagent Systems   (Site not responding. Last check: 2007-10-26)
Starting with a group of reinforcement-learning agents we derive coupled replicator equations that describe the dynamics of collective learning in multiagent systems.
We show that, although agents model their environment in a self-interested way without sharing knowledge, a game dynamics emerges naturally through the environment.
These include quasiperiodicity, stable limit cycles, intermittency, and deterministic chaos---behaviors that are to be expected in the multiagent, heterogeneous setting described by the general replicator equations.
www.santafe.edu /sfi/publications/wpabstract/200204017   (137 words)

 Untitled Document
I will show how the correspondences are constructed, write the polynomial equations they satisfy and the combinatorial identities that follow.
Abstract: The replicator equation is a system of ordinary differential equations on an n-simplex that arise in the study of evolutionary game theory.
I will show how the Poincare'-Hopf theorem can be used to prove that in case the game matrix has a special form, the replicator system has a unique local attractor.
www.math.uga.edu /seminars_conferences/Oct_09_06.htm   (570 words)

 Geometry Seminar @ University of Georgia
Abstract: After a brief introduction to a nonlinear system of ordinary differential equations known as the replicator equations of evolutionary game theory, I will describe the solutions for a large class of such systems.
Abstract: This is an informal talk on a theorem of Scherrer from 1940 that says that if all the closed curves on a surface have total twist zero, the surface must be a plane or a sphere.
The proof will also be sketched and involves a geometric interpretation of the Hirota Bilinear Difference Equation (HBDE) as a statement about linear maps between Grassmann cones.
www.math.uga.edu /~parsley/geom-sem.html   (1181 words)

 EUI - ECO - Ianni, Antonella Abstract
Specifically, the paper shows that: a) up to an error term, the stochastic process is driven by a system of discrete time difference equations of the replicator type.
b) if the trajectories of the system of replicator equations converge sufficiently fast, then the probability that all realizations of the learning process over a possibly infinite spell of time lie within a given small distance of the solution path of the replicator dynamics becomes, from some time on, arbitrarily close to one.
Fast convergence, in the form of exponential convergence, is shown to hold for any strict Nash equilibrium of the underlying game.
www.iue.it /ECO/ResearchActivities/ResearchWorkshops/PastResearchWorkshops/Papers05-06/AbstractIanni.shtml   (213 words)

 Replicator Equations, Maximal Cliques, and Graph Isomorphism - Pelillo (ResearchIndex)   (Site not responding. Last check: 2007-10-26)
0.6: Annealed Replication: A New Heuristic for the Maximum..
0.4: A Replicator Equations-Based Evolutionary Algorithm for the..
@misc{ pelillo-replicator, author = "M. Pelillo", title = "Replicator equations, maximal cliques, and graph isomorphism", text = "M. Pelillo, Replicator equations, maximal cliques, and graph isomorphism, Neural Computation, to appear.", url = "citeseer.ist.psu.edu/article/pelillo98replicator.html" }
citeseer.ist.psu.edu /130456.html   (1036 words)

 Matching Free Trees with Replicator Equations (ResearchIndex)   (Site not responding. Last check: 2007-10-26)
Abstract: Motivated by our recent work on rooted tree matching, in this paper we provide a solution to the problem of matching two free (i.e., unrooted) trees by constructing an association graph whose maximal cliques are in one-to-one correspondence with maximal common subtrees.
0.6: A Replicator Equations-Based Evolutionary Algorithm for the..
7 Approximating the maximum weight clique using replicator dyn..
citeseer.ist.psu.edu /540988.html   (285 words)

 Citebase - Coupled Replicator Equations for the Dynamics of Learning in Multiagent Systems   (Site not responding. Last check: 2007-10-26)
Coupled Replicator Equations for the Dynamics of Learning in Multiagent Systems
Authors: Sato, Yuzuru; Crutchfield, James P. Starting with a group of reinforcement-learning agents we derive coupled replicator equations that describe the dynamics of collective learning in multiagent systems.
These include quasiperiodicity, stable limit cycles, intermittency, and deterministic chaos--behaviors that should be expected in heterogeneous multiagent systems described by the general replicator equations we derive.
citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:nlin/0204057   (186 words)

 Dynamics of Autocatalytic Replicator Networks Based on Higher Order Ligation Reactions
A class of autocatalytic reaction networks based on template dependent ligation and higher order catalysis is analyzed.
The over-all dynamics can be understood by means of replicator equations with highly non-linear interaction functions.
The dynamics depends crucially on the total concentration c0 of replicating material.
ideas.repec.org /p/wop/safiwp/99-09-065.html   (242 words)

 IngentaConnect On a Notion of Linear Replicator Equations
IngentaConnect On a Notion of Linear Replicator Equations
Source: Journal of Dynamics and Differential Equations, Volume 17, Number 2, April 2005, pp.
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www.ingentaconnect.com /content/klu/jody/2005/00000017/00000002/00004574   (166 words)

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