prisoner s dilemma on a stochastic nongrowth network evolution model we investigate the evolution of cooperation on a nongrowth dynamic network model with a death birth dynamics based on tournament selection in the limit of large population size the equilibrium cooperator density is well described by our mean field approximations and inversely related to the average degree in the system small populations are also examined and found to deviate considerably from their expected mean field behavior an expanded replicator equation incorporating gaussian fluctuations in the strategy densities is then constructed with its output agreeing well with our simulation data for all sizes we also briefly comment on the role of strategy mutation in sustaining polymorphic populations in small systems

similarity based cooperation and spatial segregation we analyze a cooperative game where the cooperative act is not based on the previous behavior of the coplayer but on the similarity between the players this system has been studied in a mean field description recently phys rev e 68 046129 2003 here the spatial extension to a two dimensional lattice is studied where each player interacts with eight players in a moore neighborhood the system shows a strong segregation independent of parameters the introduction of a local conversion mechanism towards tolerance allows for four state cycles and the emergence of spiral waves in the spatial game in the case of asymmetric costs of cooperation a rich variety of complex behavior is observed depending on both cooperation costs finally we study the stabilization of a cooperative fixed point of a forecast rule in the symmetric game which corresponds to cooperation across segregation borders this fixed point becomes unstable for high cooperation costs but can be stabilized by a linear feedback mechanism

evolutionary dynamics of spatial games abstract a lattice model of coevolution of strategies for two person 2 2 matrix games is introduced the model allows evolution in an unbounded space of strategies in particular we explore the region of parameter space corresponding to the prisoner s dilemma and classify the types of dynamical and evolutionary behavior that appear for certain restricted strategy spaces we explore the complete space of lattice games

stochastic stability in spatial games we discuss similarities and differences between systems of interacting players maximizing their individual payoffs and particles minimizing their interaction energy long run behavior of stochastic dynamics of spatial games with multiple nash equilibria is analyzed in particular we construct an example of a spatial game with three strategies where stochastic stability of nash equilibria depends on the number of players and the kind of dynamics

functorial qft gauge anomalies and the dirac determinant bundle using properties of the determinant line bundle for a family of elliptic boundary value problems we explain how the fock space functor defines an axiomatic quantum field theory which formally models the fermionic path integral the sewing axiom of the theory arises as an algebraic pasting law for the determinant of the dirac operator we show how representations of the boundary gauge group fit into this description and that this leads to a fock functor description of certain gauge anomalies

a metric between probability distributions on finite sets of different cardinalities and applications to order reduction in this paper we define a metric distance between probability distributions on two distinct finite sets of possibly different cardinalities the metric is defined in terms of a joint distribution on the product of the two sets which has the two given distributions as its marginals and has minimum entropy computing the metric exactly turns out to be np hard therefore an efficient greedy algorithm is presented for finding an upper bound on the distance we then study the problem of optimal order reduction in the metric defined here it is shown that every optimal reduced order approximation must be an aggregation of the original distribution and that optimal reduced order approximation is equivalent to finding an aggregation with maximum entropy this problem also turns out to be np hard so again a greedy algorithm is constructed for finding a suboptimal reduced order approximation taken together all the results presented here permit the approximation of an independent and identically distributed i i d process over a set of large cardinality by another i i d process over a set of smaller cardinality in future work attempts will be made to extend this work to markov processes over finite sets

cooperation in spatial prisoner s dilemma with two types of players for increasing number of neighbors we study a spatial two strategy cooperation and defection prisoner s dilemma game with two types a and b of players located on the sites of a square lattice the evolution of strategy distribution is governed by iterated strategy adoption from a randomly selected neighbor with a probability depending on the payoff difference and also on the type of the neighbor the strategy adoption probability is reduced by a prefactor w 1 from the players of type b we consider the competition between two opposite effects when increasing the number of neighbors k 4 8 and 24 within a range of the portion of influential players type a the inhomogeneous activity in strategy transfer yields a relevant increase dependent on w in the density of cooperators the noise dependence of this phenomenon is also discussed by evaluating phase diagrams

the atiyah patodi singer index and domain wall fermion dirac operators we introduce a mathematician friendly formulation of the physicist friendly derivation of the atiyah patodi singer index in a previous paper motivated by the study of lattice gauge theory the physicist half of the authors derived a formula expressing the atiyah patodi singer index in terms of the eta invariant of domain wall fermion dirac operators when the base manifold is a flat 4 dimensional torus in this paper we generalise this formula to any even dimensional closed riemannian manifolds and prove it mathematically rigorously our proof uses a witten localisation argument combined with a devised embedding into a cylinder of one dimension higher our viewpoint sheds some new light on the interplay among the atiyah patodi singer boundary condition domain wall fermions and edge modes

the ζ determinant and the additivity of the η invariant on the smooth self adjoint grassmannian in this paper we discuss the existence of the determinant of a dirac operator on a compact manifold with boundary we show that the determinant is well defined in the case of the operator with a domain determined by a boundary condition from the smooth self adjoint grassmannian discussed in the papers 5 13 29 we prove a generalization of a pasting formula for the invariant see 34 the results of the paper are used in the recent proof of the projective equality of the determinant and quillen determinant on see 30 31

on the entropy of couplings abstract in this paper some general properties of shannon information measures are investigated over sets of probability distributions with restricted marginals certain optimization problems associated with these functionals are shown to be np hard and their special cases are found to be essentially information theoretic restatements of well known computational problems such as the subset sum and the 3 partition the notion of minimum entropy coupling is introduced and its relevance is demonstrated in information theoretic computational and statistical contexts finally a family of pseudometrics on the space of discrete probability distributions defined by these couplings is studied in particular their relation to the total variation distance and a new characterization of the conditional entropy is given

effects of strategy migration direction and noise in the evolutionary spatial prisoner s dilemma spatial games are crucial for understanding patterns of cooperation in nature and to some extent society they are known to be more sensitive to local symmetries than e g spin models this paper concerns the evolution of the prisoner s dilemma game on regular lattices with three different types of neighborhoods char22 the von neumann moore and kagom e types we investigate two kinds of dynamics for the players to update their strategies that can be unconditional cooperator or defector depending on the payoff difference an individual can adopt the strategy of a random neighbor a voter model like dynamics vmld or impose its strategy on a random neighbor i e invasion process like dynamics ipld in particular we focus on the effects of noise in combination with the strategy dynamics on the evolution of cooperation we find that vmld compared to ipld better supports the spreading and sustaining of cooperation we see that noise has nontrivial effects on the evolution of cooperation maximum cooperation density can be realized either at a medium noise level in the limit of zero noise or in both these regions the temptation to defect and the local interaction structure determine the outcome especially in the low noise limit the local interaction plays a crucial role in determining the fate of cooperators we elucidate these both by numerical simulations and mean field cluster approximation methods

aggregation of graph models and markov chains by deterministic annealing we consider the problem of simplifying large weighted directed graphs by aggregating nodes and edges this problem is recast as a clustering resource allocation problem and a solution method that incorporates features of the deterministic annealing da algorithm is proposed the novelty in our method is a quantitive measure of dissimilarity that allows us to compare directed graphs of possibly different sizes i e the original and the aggregated graphs the approach we propose is insensitive to initial conditions and less likely to converge to poor local minima than lloyd type algorithms we apply our graph aggregation clustering method to markov chains where low order markov chains that approximate high order chains are obtained through appropriate aggregation of state transition matrices we further develop a decentralized computational scheme to improve tractability of the algorithm

quadratic functions in geometry topology and m theory we describe an interpretation of the kervaire invariant of a riemannian manifold of dimension 4k 2 in terms of a holomorphic line bundle on the abelian variety h 2k 1 m otimes r z our results are inspired by work of witten on the fivebrane partition function in m theory hep th 9610234 hep th 9609122 our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics and is based on our theory of differential functions these differential functions generalize the differential characters of cheeger simons and the bulk of this paper is devoted to their study

chern simons theory with finite gauge group we construct in detail a 2 1 dimensional gauge field theory with finite gauge group in this case the path integral reduces to a finite sum so there are no analytic problems with the quantization the theory was originally introduced by dijkgraaf and witten without details the point of working it out carefully is to focus on the algebraic structure and particularly the construction of quantum hilbert spaces on closed surfaces by cutting and pasting this includes the verlinde formula the careful development may serve as a model for dealing with similar issues in more complicated cases

evolutionary markovian strategies in 2 2 spatial games abstract evolutionary spatial 2 2 games between heterogeneous agents are analyzed using different variants of cellular automata ca agents play repeatedly against their nearest neighbors 2 2 games specified by a rescaled payoff matrix with two parameters each agent is governed by a binary markovian strategy bms specified by four conditional probabilities p r p s p t p p that take values 0 or 1 the initial configuration consists in a random assignment of strategists among the 2 4 16 possible bms the system then evolves within strategy space according to the simple standard rule each agent copies the strategy of the neighbor who got the highest payoff besides on the payoff matrix the dominant strategy and the degree of cooperation depend on i the type of the neighborhood von neumann or moore ii the way the cooperation state is actualized deterministically or stochastically and iii the amount of noise measured by a parameter e however a robust winner strategy is 1 0 1 1

stochastic stability in spatial three player games we discuss long run behavior of stochastic dynamics of many interacting agents in particular three player spatial games are studied the effect of the number of players and the noise level on the stochastic stability of nash equilibria is investigated

η invariants and determinant lines the invariant of an odd dimensional manifold with boundary is investigated the natural boundary condition for this problem requires a trivialization of the kernel of the dirac operator on the boundary the dependence of the invariant on this trivialization is best encoded by the statement that the exponential of the invariant lives in the determinant line of the boundary our main results are a variational formula and a gluing law for this invariant these results are applied to reprove the formula for the holonomy of the natural connection on the determinant line bundle of a family of dirac operators also known as the global anomaly formula the ideas developed here fit naturally with recent work in topological quantum field theory in which gluing which is a characteristic formal property of the path integral and the classical action is used to compute global invariants on closed manifolds from local invariants on manifolds with boundary

phase diagrams for an evolutionary prisoner s dilemma game on two dimensional lattices the effects of payoffs and noise on the maintenance of cooperative behavior are studied in an evolutionary prisoner s dilemma game with players located on the sites of different two dimensional lattices this system exhibits a phase transition from a mixed state of cooperators and defectors to a homogeneous one where only the defectors remain alive using monte carlo simulations and the generalized mean field approximations we have determined the phase boundaries critical points separating the two phases on the plane of the temperature noise and temptation to choose defection in the zero temperature limit the cooperation can be sustained only for those connectivity structures where three site clique percolation occurs

cooperation in the noisy case prisoner s dilemma game on two types of regular random graphs we have studied an evolutionary prisoner s dilemma game with players located on two types of random regular graphs with a degree of 4 the analysis is focused on the effects of payoffs and noise temperature on the maintenance of cooperation when varying the noise level and or the highest payoff the system exhibits a second order phase transition from a mixed state of cooperators and defectors to an absorbing state where only defectors remain alive for the random regular graph and bethe lattice the behavior of the system is similar to those found previously on the square lattice with nearest neighbor interactions although the measure of cooperation is enhanced by the absence of loops in the connectivity structure for low noise the optimal connectivity structure is built up from randomly connected triangles

the low energy tqft of the generalized double semion model the generalized double semion gds model introduced by freedman and hastings is a lattice system similar to the toric code with a gapped hamiltonian whose definition depends on a triangulation of the ambient manifold m but whose space of ground states does not depend on the triangulation but only on the underlying manifold in this paper we use topological quantum field theory tqft to investigate the low energy limit of the gds model we define and study a functorial tqft z mathrm gds in every dimension n such that for every closed n 1 manifold m z mathrm gds m is isomorphic to the space of ground states of the gds model on m the isomorphism can be chosen to intertwine the actions of the mapping class group of m that arise on both sides throughout this paper we compare our constructions and results with their known analogues for the toric code

self organization in the spatial battle of the sexes with probabilistic updating abstract the dynamics of a spatial formulation of the iterated battle of the sexes with probabilistic updating is assessed in this work the game is played in the cellular automata manner i e with local and synchronous interaction the effect of memory of past encounters is also taken into account it is concluded that the spatial structure enables the emergence of clusters of coincident choices leading to the mean payoff per encounter to values that are accessible only in the cooperative two person game scenario which constitutes a notable case of self organization with probabilistic updating of choices both kinds of players reach mean payoffs per encounter that are notably higher than those reached with a deterministic updating mechanism albeit the evolutionary dynamics does not stabilize and one of the two possible choices tends to prevail for both kinds of players memory of past iterations induces an inertial effect that moderates this tendency so that intermediate levels of the memory charge tend to favor fairly stable high egalitarian payoffs without impeding the necessary recovering from their initial plummeting

optimal kullback leibler aggregation via information bottleneck in this paper we present a method for reducing a regular discrete time markov chain dtmc to another dtmc with a given typically much smaller number of states the cost of reduction is defined as the kullback leibler divergence rate between a projection of the original process through a partition function and a dtmc on the correspondingly partitioned state space finding the reduced model with minimal cost is computationally expensive as it requires an exhaustive search among all state space partitions and an exact evaluation of the reduction cost for each candidate partition our approach deals with the latter problem by minimizing an upper bound on the reduction cost instead of minimizing the exact cost the proposed upper bound is easy to compute and it is tight if the original chain is lumpable with respect to the partition then we express the problem in the form of information bottleneck optimization and propose using the agglomerative information bottleneck algorithm for searching a suboptimal partition greedily rather than exhaustively the theory is illustrated with examples and one application scenario in the context of modeling bio molecular interactions

on the cobordism classification of symmetry protected topological phases in the framework of atiyah s axioms of topological quantum field theory with unitarity we give a direct proof of the fact that symmetry protected topological phases without hall effects are classified by cobordism invariants we first show that the partition functions of those theories are cobordism invariants after a tuning of the euler term conversely for a given cobordism invariant we construct a unitary topological field theory whose partition function is given by the cobordism invariant assuming that a certain bordism group is finitely generated two theories having the same cobordism invariant partition functions are isomorphic

the similarity metric a new class of distances appropriate for measuring similarity relations between sequences say one type of similarity per distance is studied we propose a new normalized information distance based on the noncomputable notion of kolmogorov complexity and show that it is in this class and it minorizes every computable distance in the class that is it is universal in that it discovers all computable similarities we demonstrate that it is a metric and call it the similarity metric this theory forms the foundation for a new practical tool to evidence generality and robustness we give two distinctive applications in widely divergent areas using standard compression programs like gzip and gencompress first we compare whole mitochondrial genomes and infer their evolutionary history this results in a first completely automatic computed whole mitochondrial phylogeny tree secondly we fully automatically compute the language tree of 52 different languages

coevolutionary dynamics in large but finite populations coevolving and competing species or game theoretic strategies exhibit rich and complex dynamics for which a general theoretical framework based on finite populations is still lacking recently an explicit mean field description in the form of a fokker planck equation was derived for frequency dependent selection with two strategies in finite populations based on microscopic processes a traulsen j c claussen and c hauert phys rev lett 95 238701 2005 here we generalize this approach in a twofold way first we extend the framework to an arbitrary number of strategies and second we allow for mutations in the evolutionary process the deterministic limit of infinite population size of the frequency dependent moran process yields the adjusted replicator mutator equation which describes the combined effect of selection and mutation for finite populations we provide an extension taking random drift into account in the limit of neutral selection i e whenever the process is determined by random drift and mutations the stationary strategy distribution is derived this distribution forms the background for the coevolutionary process in particular a critical mutation rate uc is obtained separating two scenarios above uc the population predominantly consists of a mixture of strategies whereas below uc the population tends to be in homogeneous states for one of the fundamental problems in evolutionary biology the evolution of cooperation under darwinian selection we demonstrate that the analytical framework provides excellent approximations to individual based simulations even for rather small population sizes this approach complements simulation results and provides a deeper systematic understanding of coevolutionary dynamics

statistical mechanics of spatial evolutionary games we discuss the long run behaviour of stochastic dynamics of many interacting players in spatial evolutionary games in particular we investigate the effect of the number of players and the noise level on the stochastic stability of nash equilibria we discuss similarities and differences between systems of interacting players maximizing their individual payoffs and particles minimizing their interaction energy we use concepts and techniques of statistical mechanics to study game theoretic models in order to obtain results in the case of the so called potential games we analyse the thermodynamic limit of the appropriate models of interacting particles

the analysis of elliptic families i metrics and connections on determinant bundles in this paper we construct the quillen metric on the determinant bundle associated with a family of elliptic first order differential operators we also introduce a unitary connection on and calculate its curvature our results will be applied to the case of dirac operators in a forthcoming paper

least informative bayesian prior distributions for finite samples based on information theory a procedure is presented based on shannon information theory for producing least informative prior distributions for bayesian estimation and identification this approach relies on constructing an optimal mixture distribution and applies in small sample sizes unlike certain approaches based on asymptotic theory the procedure is illustrated in a small scale numerical study and is contrasted with an approach based on maximum entropy

optimal kullback leibler aggregation via spectral theory of markov chains this paper is concerned with model reduction for complex markov chain models the kullback leibler divergence rate is employed as a metric to measure the difference between the markov model and its approximation for a certain relaxation of the bi partition model reduction problem the solution is shown to be characterized by an associated eigenvalue problem the form of the eigenvalue problem is closely related to the markov spectral theory for model reduction this result is the basis of a heuristic proposed for the m ary partition problem resulting in a practical recursive algorithm the results are illustrated with examples

coevolutionary games on networks we study agents on a network playing an iterated prisoner s dilemma against their neighbors the resulting spatially extended coevolutionary game exhibits stationary states which are nash equilibria after perturbation of these equilibria avalanches of mutations reestablish a stationary state scale free avalanche distributions are observed that are in accordance with calculations from the nash equilibria and a confined branching process the transition from subcritical to critical avalanche dynamics can be traced to a change in the degeneracy of the cooperative macrostate and is observed for many variants of this game

higher dimensional algebra and topological quantum field theory the study of topological quantum field theories increasingly relies upon concepts from higher dimensional algebra such as n categories and n vector spaces we review progress towards a definition of n category suited for this purpose and outline a program in which n dimensional topological quantum field theories tqfts are to be described as n category representations first we describe a suspension operation on n categories and hypothesize that the k fold suspension of a weak n category stabilizes for k n 2 we give evidence for this hypothesis and describe its relation to stable homotopy theory we then propose a description of n dimensional unitary extended tqfts as weak n functors from the free stable weak n category with duals on one object to the n category of n hilbert spaces we conclude by describing n categorical generalizations of deformation quantization and the quantum double construction

setting the quantum integrand of m theory in anomaly free quantum field theories the integrand in the bosonic functional integral the exponential of the effective action after integrating out fermions is often defined only up to a phase without an additional choice we term this choice setting the quantum integrand in the low energy approximation to m theory the e8 model for the c field allows us to set the quantum integrand using geometric index theory we derive mathematical results of independent interest about pfaffians of dirac operators in 8k 3 dimensions both on closed manifolds and manifolds with boundary these theorems are used to set the quantum integrand of m theory for closed manifolds and for compact manifolds with either temporal global or spatial local boundary conditions in particular we show that m theory makes sense on arbitrary 11 manifolds with spatial boundary generalizing the construction of heterotic m theory on cylinders

time evolution of the rule 150 cellular automaton activity from a fibonacci iteration the rule 150 cellular automaton is a remarkable discrete dynamical system as it shows 1 f spectra if started from a single seed j nagler and j c claussen phys rev e 71 067103 2005 despite its simplicity a feasible solution for its time behavior is not obvious its self similarity does not follow a one step iteration like other elementary cellular automata here it is shown how its time behavior can be solved as a two step vectorial or string iteration which can be viewed as a generalization of fibonacci iteration generating the time series from a sequence of vectors of increasing length this allows us to compute the total activity time series more efficiently than by simulating the whole spatiotemporal process or even by using the closed expression the results are further extended to the generalization of rule 150 to the two dimensional case and to bethe lattices and the relation to corresponding integer sequences is discussed

string structures and trivialisations of a pfaffian line bundle the present paper is a contribution to categorial index theory its main result is the calculation of the pfaffian line bundle of a certain family of real dirac operators as an object in the category of line bundles furthermore it is shown how string structures give rise to trivialisations of that pfaffian