63 research outputs found
Attractor Modulation and Proliferation in 1+ Dimensional Neural Networks
We extend a recently introduced class of exactly solvable models for
recurrent neural networks with competition between 1D nearest neighbour and
infinite range information processing. We increase the potential for further
frustration and competition in these models, as well as their biological
relevance, by adding next-nearest neighbour couplings, and we allow for
modulation of the attractors so that we can interpolate continuously between
situations with different numbers of stored patterns. Our models are solved by
combining mean field and random field techniques. They exhibit increasingly
complex phase diagrams with novel phases, separated by multiple first- and
second order transitions (dynamical and thermodynamic ones), and, upon
modulating the attractor strengths, non-trivial scenarios of phase diagram
deformation. Our predictions are in excellent agreement with numerical
simulations.Comment: 16 pages, 15 postscript figures, Late
Slowly evolving random graphs II: Adaptive geometry in finite-connectivity Hopfield models
We present an analytically solvable random graph model in which the
connections between the nodes can evolve in time, adiabatically slowly compared
to the dynamics of the nodes. We apply the formalism to finite connectivity
attractor neural network (Hopfield) models and we show that due to the
minimisation of the frustration effects the retrieval region of the phase
diagram can be significantly enlarged. Moreover, the fraction of misaligned
spins is reduced by this effect, and is smaller than in the infinite
connectivity regime. The main cause of this difference is found to be the
non-zero fraction of sites with vanishing local field when the connectivity is
finite.Comment: 17 pages, 8 figure
Dynamic rewiring in small world networks
We investigate equilibrium properties of small world networks, in which both
connectivity and spin variables are dynamic, using replicated transfer matrices
within the replica symmetric approximation. Population dynamics techniques
allow us to examine order parameters of our system at total equilibrium,
probing both spin- and graph-statistics. Of these, interestingly, the degree
distribution is found to acquire a Poisson-like form (both within and outside
the ordered phase). Comparison with Glauber simulations confirms our results
satisfactorily.Comment: 21 pages, 5 figure
Analysis of common attacks in LDPCC-based public-key cryptosystems
We analyze the security and reliability of a recently proposed class of
public-key cryptosystems against attacks by unauthorized parties who have
acquired partial knowledge of one or more of the private key components and/or
of the plaintext. Phase diagrams are presented, showing critical partial
knowledge levels required for unauthorized decryptionComment: 14 pages, 6 figure
Thermodynamics of spin systems on small-world hypergraphs
We study the thermodynamic properties of spin systems on small-world
hypergraphs, obtained by superimposing sparse Poisson random graphs with p-spin
interactions onto a one-dimensional Ising chain with nearest-neighbor
interactions. We use replica-symmetric transfer-matrix techniques to derive a
set of fixed-point equations describing the relevant order parameters and free
energy, and solve them employing population dynamics. In the special case where
the number of connections per site is of the order of the system size we are
able to solve the model analytically. In the more general case where the number
of connections is finite we determine the static and dynamic
ferromagnetic-paramagnetic transitions using population dynamics. The results
are tested against Monte-Carlo simulations.Comment: 14 pages, 7 figures; Added 2 figures. Extended result
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 matrix (where ) 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. dimensional neural networks and `small world' magnets.
Numerical simulations confirm our predictions satisfactorily.Comment: 24 pages, LaTex, IOP macro
A Solvable Model of Secondary Structure Formation in Random Hetero-Polymers
We propose and solve a simple model describing secondary structure formation
in random hetero-polymers. It describes monomers with a combination of
one-dimensional short-range interactions (representing steric forces and
hydrogen bonds) and infinite range interactions (representing polarity forces).
We solve our model using a combination of mean field and random field
techniques, leading to phase diagrams exhibiting second-order transitions
between folded, partially folded and unfolded states, including regions where
folding depends on initial conditions. Our theoretical results, which are in
excellent agreement with numerical simulations, lead to an appealing physical
picture of the folding process: the polarity forces drive the transition to a
collapsed state, the steric forces introduce monomer specificity, and the
hydrogen bonds stabilise the conformation by damping the frustration-induced
multiplicity of states.Comment: 24 pages, 14 figure
Magnetization enumerator of real-valued symmetric channels in Gallager error-correcting codes
Using the magnetization enumerator method, we evaluate the practical and
theoretical limitations of symmetric channels with real outputs. Results are
presented for several regular Gallager code constructions.Comment: 5 pages, 1 figure, to appear as Brief Report in Physical Review
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. spin-glasses on random graphs and of the
overlap in finite connectivity Sourlas codes. All results are tested against
Monte Carlo simulations.Comment: 4 pages, 14 .eps file
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