28 research outputs found
The Number of Different Binary Functions Generated by NK-Kauffman Networks and the Emergence of Genetic Robustness
We determine the average number , of \textit{NK}-Kauffman
networks that give rise to the same binary function. We show that, for , there exists a connectivity critical value such that () for and
for . We find that is not a
constant, but scales very slowly with , as . The problem of genetic robustness emerges as a statistical property
of the ensemble of \textit{NK}-Kauffman networks and impose tight constraints
in the average number of epistatic interactions that the genotype-phenotype map
can have.Comment: 4 figures 18 page
Multi-fidelity regression using a non-parametric relationship
International audienceWhen the precision of the output of a heavy computer code can be tuned, it is possible to incorporate responses with different levels of fidelity to enhance the prediction of output of the most accurate simulation. This is usually done by adding several imprecise responses instead of a few precise ones. The main example for this type of computer experiments are the numerical solutions of differential equations. This problem has been studied by many authors, most notably by LeGratiet (2012) and Kennedy and O'Hagan (2001). In the present work, we propose a new approach that is different from the existing ones and based on a non-parametric relationship between two consecutive levels of fidelity
Phase Transition in NK-Kauffman Networks and its Correction for Boolean Irreducibility
In a series of articles published in 1986 Derrida, and his colleagues studied
two mean field treatments (the quenched and the annealed) for
\textit{NK}-Kauffman Networks. Their main results lead to a phase transition
curve () for the
critical average connectivity in terms of the bias of
extracting a "" for the output of the automata. Values of bigger than
correspond to the so-called chaotic phase; while , to an
ordered phase. In~[F. Zertuche, {\it On the robustness of NK-Kauffman networks
against changes in their connections and Boolean functions}. J.~Math.~Phys.
{\bf 50} (2009) 043513], a new classification for the Boolean functions, called
{\it Boolean irreducibility} permitted the study of new phenomena of
\textit{NK}-Kauffman Networks. In the present work we study, once again the
mean field treatment for \textit{NK}-Kauffman Networks, correcting it for {\it
Boolean irreducibility}. A shifted phase transition curve is found. In
particular, for the predicted value by Derrida {\it
et al.} changes to We support our results with
numerical simulations.Comment: 23 pages, 7 Figures on request. Published in Physica D: Nonlinear
Phenomena: Vol.275 (2014) 35-4
On the Robustness of NK-Kauffman Networks Against Changes in their Connections and Boolean Functions
NK-Kauffman networks {\cal L}^N_K are a subset of the Boolean functions on N
Boolean variables to themselves, \Lambda_N = {\xi: \IZ_2^N \to \IZ_2^N}. To
each NK-Kauffman network it is possible to assign a unique Boolean function on
N variables through the function \Psi: {\cal L}^N_K \to \Lambda_N. The
probability {\cal P}_K that \Psi (f) = \Psi (f'), when f' is obtained through f
by a change of one of its K-Boolean functions (b_K: \IZ_2^K \to \IZ_2), and/or
connections; is calculated. The leading term of the asymptotic expansion of
{\cal P}_K, for N \gg 1, turns out to depend on: the probability to extract the
tautology and contradiction Boolean functions, and in the average value of the
distribution of probability of the Boolean functions; the other terms decay as
{\cal O} (1 / N). In order to accomplish this, a classification of the Boolean
functions in terms of what I have called their irreducible degree of
connectivity is established. The mathematical findings are discussed in the
biological context where, \Psi is used to model the genotype-phenotype map.Comment: 17 pages, 1 figure, Accepted in Journal of Mathematical Physic
Discrete Dynamical Systems Embedded in Cantor Sets
While the notion of chaos is well established for dynamical systems on
manifolds, it is not so for dynamical systems over discrete spaces with
variables, as binary neural networks and cellular automata. The main difficulty
is the choice of a suitable topology to study the limit . By
embedding the discrete phase space into a Cantor set we provided a natural
setting to define topological entropy and Lyapunov exponents through the
concept of error-profile. We made explicit calculations both numerical and
analytic for well known discrete dynamical models.Comment: 36 pages, 13 figures: minor text amendments in places, time running
top to bottom in figures, to appear in J. Math. Phy
The Number of Different Binary Functions Generated by NK-Kauffman Networks and the Emergence of Genetic Robustness
We determine the average number , of \textit{NK}-Kauffman
networks that give rise to the same binary function. We show that, for , there exists a connectivity critical value such that () for and
for . We find that is not a
constant, but scales very slowly with , as . The problem of genetic robustness emerges as a statistical property
of the ensemble of \textit{NK}-Kauffman networks and impose tight constraints
in the average number of epistatic interactions that the genotype-phenotype map
can have.Comment: 4 figures 18 page
The Asymptotic Number of Attractors in the Random Map Model
The random map model is a deterministic dynamical system in a finite phase
space with n points. The map that establishes the dynamics of the system is
constructed by randomly choosing, for every point, another one as being its
image. We derive here explicit formulas for the statistical distribution of the
number of attractors in the system. As in related results, the number of
operations involved by our formulas increases exponentially with n; therefore,
they are not directly applicable to study the behavior of systems where n is
large. However, our formulas lend themselves to derive useful asymptotic
expressions, as we show.Comment: 16 pages, 1 figure. Minor changes. To be published in Journal of
Physics A: Mathematical and Genera
Homotopy Invariants and Time Evolution in (2+1)-Dimensional Gravity
We establish the relation between the ISO(2,1) homotopy invariants and the
polygon representation of (2+1)-dimensional gravity. The polygon closure
conditions, together with the SO(2,1) cycle conditions, are equivalent to the
ISO(2,1) cycle conditions for the representa- tions of the fundamental group in
ISO(2,1). Also, the symplectic structure on the space of invariants is closely
related to that of the polygon representation. We choose one of the polygon
variables as internal time and compute the Hamiltonian, then perform the
Hamilton-Jacobi transformation explicitly. We make contact with other authors'
results for g = 1 and g = 2 (N = 0).Comment: 34 pages, Mexico preprint ICN-UNAM-93-1