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A. C. C. Coolen

Publications and source records attributed to A. C. C. Coolen.

At least 37 records · Page 2Linked to original sources
arXiv2007

Generating functional analysis of minority games with inner product strategy definitions

We use generating functional methods to solve the so-called inner product versions of the minority game (MG), with fake and/or real market histories, by generalizing the theory developed recently for look-up table MGs with real histories. The phase diagrams of the lookup table and inner product MG versions are generally found to be identical, with the exception of inner product MGs where histories are sampled linearly, which are found to be structurally critical. However, we encounter interesting differences both in the theory (where the role of the history frequency distribution in lookup table MGs is taken over by the eigenvalue spectrum of a history covariance matrix in inner product MGs) and in the static and dynamic phenomenology of the models. Our theoretical predictions are supported by numerical simulations.

physics.soc-ph↗
arXiv2006

Generating functional analysis of batch minority games with arbitrary strategy numbers

Both the phenomenology and the theory of minority games (MG) with more than two strategies per agent are different from those of the conventional and extensively studied case S=2. MGs with $S>2$ exhibit nontrivial statistics of the frequencies with which the agents select from their available decision making strategies, with far-reaching implications. In the few theoretical MG studies with $S>2$ published so far, these statistics could not be calculated analytically. This prevented solution even in ergodic stationary states; equations for order parameters could only be closed approximately, using simulation data. Here we carry out a generating functional analysis of fake history batch MGs with arbitary values of $S$, and give an analytical solution of the strategy frequency problem. This leads to closed equations for order parameters in the ergodic regime, exact expressions for strategy selection statistics, and phase diagrams. Our results find perfect confirmation in numerical simulations.

cond-mat.dis-nn↗
arXiv2005

Dynamical replica analysis of disordered Ising spin systems on finitely connected random graphs

We study the dynamics of macroscopic observables such as the magnetization and the energy per degree of freedom in Ising spin models on random graphs of finite connectivity, with random bonds and/or heterogeneous degree distributions. To do so we generalize existing implementations of dynamical replica theory and cavity field techniques to systems with strongly disordered and locally tree-like interactions. We illustrate our results via application to the dynamics of e.g. $\pm J$ spin-glasses on random graphs and of the overlap in finite connectivity Sourlas codes. All results are tested against Monte Carlo simulations.

cond-mat.dis-nn↗
arXiv2004

Generating functional analysis of Minority Games with real market histories

It is shown how the generating functional method of De Dominicis can be used to solve the dynamics of the original version of the minority game (MG), in which agents observe real as opposed to fake market histories. Here one again finds exact closed equations for correlation and response functions, but now these are defined in terms of two connected effective non-Markovian stochastic processes: a single effective agent equation similar to that of the `fake' history models, and a second effective equation for the overall market bid itself (the latter is absent in `fake' history models). The result is an exact theory, from which one can calculate from first principles both the persistent observables in the MG and the distribution of history frequencies.

cond-mat.dis-nn↗
arXiv2004

Diagonalization of replicated transfer matrices for disordered Ising spin systems

We present an alternative procedure for solving the eigenvalue problem of replicated transfer matrices describing disordered spin systems with (random) 1D nearest neighbor bonds and/or random fields, possibly in combination with (random) long range bonds. Our method is based on transforming the original eigenvalue problem for a $2^n\times 2^n$ matrix (where $n\to 0$) into an eigenvalue problem for integral operators. We first develop our formalism for the Ising chain with random bonds and fields, where we recover known results. We then apply our methods to models of spins which interact simultaneously via a one-dimensional ring and via more complex long-range connectivity structures, e.g. $1+\infty$ dimensional neural networks and `small world' magnets. Numerical simulations confirm our predictions satisfactorily.

cond-mat.dis-nn↗
arXiv2004

Asymmetrically extremely dilute neural networks with non-trivial dynamics

We study graded response attractor neural networks with asymmetrically extremely dilute interactions and Langevin dynamics. We solve our model in the thermodynamic limit using generating functional analysis, and find (in contrast to the binary neurons case) that even in statics one cannot eliminate the non-persistent order parameters. The macroscopic dynamics is driven by the (non-trivial) joint distribution of neurons and fields, rather than just the (Gaussian) field distribution. We calculate phase transition lines and present simulation results in support of our theory.

cond-mat.dis-nn↗
arXiv2004

Analytic solution of attractor neural networks on scale-free graphs

We study the influence of network topology on retrieval properties of recurrent neural networks, using replica techniques for diluted systems. The theory is presented for a network with an arbitrary degree distribution $p(k)$ and applied to power law distributions $p(k) \sim k^{-γ}$, i.e. to neural networks on scale-free graphs. A bifurcation analysis identifies phase boundaries between the paramagnetic phase and either a retrieval phase or a spin glass phase. Using a population dynamics algorithm, the retrieval overlap and spin glass order parameters may be calculated throughout the phase diagram. It is shown that there is an enhancement of the retrieval properties compared with a Poissonian random graph. We compare our findings with simulations.

cond-mat.dis-nn↗
arXiv2004

Slowly evolving geometry in recurrent neural networks I: extreme dilution regime

We study extremely diluted spin models of neural networks in which the connectivity evolves in time, although adiabatically slowly compared to the neurons, according to stochastic equations which on average aim to reduce frustration. The (fast) neurons and (slow) connectivity variables equilibrate separately, but at different temperatures. Our model is exactly solvable in equilibrium. We obtain phase diagrams upon making the condensed ansatz (i.e. recall of one pattern). These show that, as the connectivity temperature is lowered, the volume of the retrieval phase diverges and the fraction of mis-aligned spins is reduced. Still one always retains a region in the retrieval phase where recall states other than the one corresponding to the `condensed' pattern are locally stable, so the associative memory character of our model is preserved.

cond-mat.dis-nn↗
arXiv2004

Parallel dynamics of disordered Ising spin systems on finitely connected random graphs

We study the dynamics of bond-disordered Ising spin systems on random graphs with finite connectivity, using generating functional analysis. Rather than disorder-averaged correlation and response functions (as for fully connected systems), the dynamic order parameter is here a measure which represents the disorder averaged single-spin path probabilities, given external perturbation field paths. In the limit of completely asymmetric graphs our macroscopic laws close already in terms of the single-spin path probabilities at zero external field. For the general case of arbitrary graph symmetry we calculate the first few time steps of the dynamics exactly, and we work out (numerical and analytical) procedures for constructing approximate stationary solutions of our equations. Simulation results support our theoretical predictions.

cond-mat.dis-nn↗
arXiv2004

Replicated Transfer Matrix Analysis of Ising Spin Models on `Small World' Lattices

We calculate equilibrium solutions for Ising spin models on `small world' lattices, which are constructed by super-imposing random and sparse Poissonian graphs with finite average connectivity c onto a one-dimensional ring. The nearest neighbour bonds along the ring are ferromagnetic, whereas those corresponding to the Poisonnian graph are allowed to be random. Our models thus generally contain quenched connectivity and bond disorder. Within the replica formalism, calculating the disorder-averaged free energy requires the diagonalization of replicated transfer matrices. In addition to developing the general replica symmetric theory, we derive phase diagrams and calculate effective field distributions for two specific cases: that of uniform sparse long-range bonds (i.e. `small world' magnets), and that of (+J/-J) random sparse long-range bonds (i.e. `small world' spin-glasses).

cond-mat.dis-nn↗
arXiv2003

Finite Connectivity Attractor Neural Networks

We study a family of diluted attractor neural networks with a finite average number of (symmetric) connections per neuron. As in finite connectivity spin glasses, their equilibrium properties are described by order parameter functions, for which we derive an integral equation in replica symmetric (RS) approximation. A bifurcation analysis of this equation reveals the locations of the paramagnetic to recall and paramagnetic to spin-glass transition lines in the phase diagram. The line separating the retrieval phase from the spin-glass phase is calculated at zero temperature. All phase transitions are found to be continuous.

cond-mat.dis-nn↗
arXiv2003

Dynamics of a spherical minority game

We present an exact dynamical solution of a spherical version of the batch minority game (MG) with random external information. The control parameters in this model are the ratio of the number of possible values for the public information over the number of agents, and the radius of the spherical constraint on the microscopic degrees of freedom. We find a phase diagram with three phases: two without anomalous response (an oscillating versus a frozen state), and a further frozen phase with divergent integrated response. In contrast to standard MG versions, we can also calculate the volatility exactly. Our study reveals similarities between the spherical and the conventional MG, but also intriguing differences. Numerical simulations confirm our analytical results.

cond-mat.dis-nn↗
arXiv2002

Coupled dynamics of sequence selection and compactification in mean-field hetero-polymers

We study a simple solvable model describing the genesis of monomer sequences for hetero-polymers (such as proteins), as the result of the equilibration of a slow stochastic genetic selection process which is assumed to be driven by the competing demands of functionality and reproducibility of the polymer's folded structure. Since reproducibility is defined in terms of properties of the folding process, one is led to the analysis of the coupled dynamics of (fast) polymer folding and (slow) genetic sequence selection. For the present mean-field model this analysis can be carried out using the finite-dimensional replica method, leading to exact results for (first- and second-order) transitions and to rich phase diagrams.

cond-mat.dis-nn↗
arXiv2002

Non-equilibrium statistical mechanics of Minority Games

In this paper I give a brief introduction to a family of simple but non-trivial models designed to increase our understanding of collective processes in markets, the so-called Minority Games, and their non-equilibrium statistical mathematical analysis. Since the most commonly studied members of this family define disordered stochastic processes without detailed balance, the canonical technique for finding exact solutions is found to be generating functional analysis a la De Dominicis, as originally developed in the spin-glass community.

cond-mat.dis-nn↗
arXiv2001

Solvable Lattice Gas Models of Random Heteropolymers at Finite Density: II. Dynamics and Transitions to Compact States

In this paper we analyse both the dynamics and the high density physics of the infinite dimensional lattice gas model for random heteropolymers recently introduced in \cite{jort}. Restricting ourselves to site-disordered heteropolymers, we derive exact closed deterministic evolution equations for a suitable set of dynamic order parameters (in the thermodynamic limit), and use these to study the dynamics of the system for different choices of the monomer polarity parameters. We also study the equilibrium properties of the system in the high density limit, which leads to a phase diagram exhibiting transitions between swollen states, compact states, and regions with partial compactification. Our results find excellent verification in numerical simulations, and have a natural and appealing interpretation in terms of real heteropolymers.

cond-mat.dis-nn↗
arXiv2001

Dynamics of the Batch Minority Game with Inhomogeneous Decision Noise

We study the dynamics of a version of the batch minority game, with random external information and with different types of inhomogeneous decision noise (additive and multiplicative), using generating functional techniques à la De Dominicis. The control parameters in this model are the ratio $α=p/N$ of the number $p$ of possible values for the external information over the number $N$ of trading agents, and the statistical properties of the agents' decision noise parameters. The presence of decision noise is found to have the general effect of damping macroscopic oscillations, which explains why in certain parameter regions it can effectively reduce the market volatility, as observed in earlier studies. In the limit $N\to\infty$ we (i) solve the first few time steps of the dynamics (for any $α$), (ii) calculate the location $α_c$ of the phase transition (signaling the onset of anomalous response), and (iii) solve the statics for $α>α_c$. We find that $α_c$ is not sensitive to additive decision noise, but we arrive at non-trivial phase diagrams in the case of multiplicative noise. Our theoretical results find excellent confirmation in numerical simulations.

cond-mat.dis-nn↗
arXiv2001

Dynamics of on-line Hebbian learning with structurally unrealizable restricted training sets

We present an exact solution for the dynamics of on-line Hebbian learning in neural networks, with restricted and unrealizable training sets. In contrast to other studies on learning with restricted training sets, unrealizability is here caused by structural mismatch, rather than data noise: the teacher machine is a perceptron with a reversed wedge-type transfer function, while the student machine is a perceptron with a sigmoidal transfer function. We calculate the glassy dynamics of the macroscopic performance measures, training error and generalization error, and the (non-Gaussian) student field distribution. Our results, which find excellent confirmation in numerical simulations, provide a new benchmark test for general formalisms with which to study unrealizable learning processes with restricted training sets.

cond-mat.dis-nn↗