arXiv subjects

Edgardo Ugalde

Publications and source records attributed to Edgardo Ugalde.

13 recordsLinked to original sources

arXiv2026

Dominant vertices and attractors' landscape for Boolean networks

In previous works, we introduced the notion of dominant vertices in the context of dynamical systems on networks. This is a set of nodes in the underlying network whose evolution determines the whole network's dynamics after a transient time. In this paper, we focus on the case of Boolean networks. We define a reduced graph on the dominant vertices and an induced dynamics on this graph, which we prove is asymptotically equivalent to the original Boolean dynamics. Asymptotic conjugacy ensures that the systems, restricted to their respective attractors, are dynamically equivalent. For a significant class of networks, the induced dynamics is indeed a reduction of the original system. In these cases, the reduction, which is obtained from the structure of dominant vertices, supplies a more tractable system with the same structure of attractors as the original one. Furthermore, the structure of the induced system allows us to establish bounds on the number and period of the attractors, as well as on the reduction of the basin's sizes and transient lengths. We illustrate this reduction by considering a class of networks, which we call clover networks, whose dominant set is a singleton. To get insight into the structure of the basins of attraction of Boolean networks with a single dominant vertex, we complement this work with a numerical exploration of the behavior of a parametrized ensemble of systems of this kind.

math.DS↗

arXiv2025

Gaussian concentration bounds for probabilistic cellular automata

We study lattice spin systems and analyze the evolution of Gaussian concentration bounds (GCB) under the action of probabilistic cellular automata (PCA), which serve as discrete-time analogues of Markovian spin-flip dynamics. We establish the conservation of GCB and, in the high-noise regime, demonstrate that GCB holds for the unique stationary measure. Additionally, we prove the equivalence of GCB for the space-time measure and its spatial marginals in the case of contractive probabilistic cellular automata. Furthermore, we explore the relationship between (non)-uniqueness and GCB in the context of space-time Gibbs measures for PCA and illustrate these results with examples.

math.PR↗

arXiv2022

On the correspondence between Subshifts of Finite Type and Statistical Mechanics Models

R. Burton and J. Steif developed a strategy to construct examples of strongly irreducible subshifts of finite type admitting several measures of maximal entropy. This strategy exploits a correspondence between equilibrium statistical mechanics and symbolic dynamics, correspondence which was later formalized by O. Häggström. In this paper, we revisit and discuss this correspondence with the aim of presenting a simplified version of it, and to expose some applications of rigorous results concerning the Potts model and the six-vertex model to symbolic dynamics, illustrating in this way of the possibilities of this correspondence.

math.DS↗

arXiv2019

Dynamical regimes and finite time behavior in a trapped random walk: a direct iterative approach

We consider a basic one-dimensional model of diffusion which allows to obtain a diversity of diffusive regimes whose speed depends on the moments of the per-site trapping time. This model is closely related to the continuous time random walks widely studied in the literature. The model we consider lends itself to a detailed treatment, making it possible to study deviations from normality due to finite time effects.

math.PR↗

arXiv2017

Constant-length random substitutions and Gibbs measures

This work is devoted to the study of processes generated by random substitutions over a finite alphabet. We prove, under mild conditions on the substitution's rule, the existence of a unique process which remains invariant under the substitution, and exhibiting polynomial decay of correlations. For constant-length substitutions, we go further by proving that the invariant state is precisely a Gibbs measure which can be obtained as the projective limit of its natural Markovian approximations. We close the paper with a class of substitutions whose invariant state is the unique Gibbs measure for a hierarchical two-body interaction.

math-ph↗

arXiv2015

Projective distance and $g$-measures

We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to the $\bar{d}$-distance and we prove that in general they are not comparable. Our projective distance is inspired on Hilbert's projective metric, and in the framework of $g$-measures, it allows to assess the continuity of the entropy at $g$-measures satisfying uniqueness. It also allows to relate the speed of convergence and the regularity of sequences of locally finite $g$-functions, to the preservation at the limit, of certain ergodic properties for the associate $g$-measures.

math.DS↗

arXiv2013

Exact scaling in the expansion-modification system

This work is devoted to the study of the scaling, and the consequent power-law behavior, of the correlation function in a mutation-replication model known as the expansion-modification system. The latter is a biology inspired random substitution model for the genome evolution, which is defined on a binary alphabet and depends on a parameter interpreted as a \emph{mutation probability}. We prove that the time-evolution of this system is such that any initial measure converges towards a unique stationary one exhibiting decay of correlations not slower than a power-law. We then prove, for a significant range of mutation probabilities, that the decay of correlations indeed follows a power-law with scaling exponent smoothly depending on the mutation probability. Finally we put forward an argument which allows us to give a closed expression for the corresponding scaling exponent for all the values of the mutation probability. Such a scaling exponent turns out to be a piecewise smooth function of the parameter.

math.PR↗

arXiv2009

On the preservation of Gibbsianness under symbol amalgamation

Starting from the full--shift on a finite alphabet $A$, mingling some symbols of $A$, we obtain a new full shift on a smaller alphabet $B$. This amalgamation defines a factor map from $(A^{\mathbb N},T_A)$ to $(B^{\mathbb N},T_B)$, where $T_A$ and $T_B$ are the respective shift maps. According to the thermodynamic formalism, to each regular function (`potential') $ψ:A^{\mathbb N}\to{\mathbb R}$, we can associate a unique Gibbs measure $μ_ψ$. In this article, we prove that, for a large class of potentials, the pushforward measure $μ_ψ\circπ^{-1}$ is still Gibbsian for a potential $ϕ:B^{\mathbb N}\to{\mathbb R}$ having a `bit less' regularity than $ψ$. In the special case where $ψ$ is a `2--symbol' potential, the Gibbs measure $μ_ψ$ is nothing but a Markov measure and the amalgamation $π$ defines a hidden Markov chain. In this particular case, our theorem can be recast by saying that a hidden Markov chain is a Gibbs measure (for a Hölder potential).

math.DS↗

arXiv2009

Open Regulatory Networks and Modularity

We study the dynamical properties of small regulatory networks treated as non autonomous dynamical systems called modules when working inside larger networks or, equivalently when subject to external signal inputs. Particular emphasis is put on the interplay between the internal properties of the open systems and the different possible inputs on them to deduce new functionalities of the modules. We use discrete-time, piecewise-affine and piecewise-contracting models with interactions of a regulatory nature to perform our study.

q-bio.MN↗

arXiv2008

Regulatory Dynamics on Random Networks: Asymptotic Periodicity and Modularity

We study the dynamics of discrete-time regulatory networks on random digraphs. For this we define ensembles of deterministic orbits of random regulatory networks, and introduce some statistical indicators related to the long-term dynamics of the system. We prove that, in a random regulatory network, initial conditions converge almost surely to a periodic attractor. We study the subnetworks, which we call modules, where the periodic asymptotic oscillations are concentrated. We proof that those modules are dynamically equivalent to independent regulatory networks.

math.DS↗

arXiv2007

Dominant Vertices in Regulatory Networks Dynamics

Discrete-time regulatory networks are dynamical systems on directed graphs, with a structure inspired on natural systems of interacting units. There is a natural notion of determination amongst vertices, which we use to classify the nodes of the network, and to determine what we call "sets of dominant vertices". In this paper we prove that in the asymptotic regime, the projection of the dynamics on a dominant set allows us to determine the state of the whole system at all times. We provide an algorithm to find sets of dominant vertices, and we test its accuracy on three families of theoretical examples. Then, by using the same algorithm, we study the relation between the structure of the underlying network and the corresponding dominant set of vertices. We also present a result concerning the inheritability of the dominance between strongly connected networks.

nlin.AO↗

arXiv2006

A Spatially Extended Model for Residential Segregation

In this paper we analyze urban spatial segregation phenomenon in terms of the income distribution over a population, and inflationary parameter weighting the evolution of housing prices. For this, we develop a discrete, spatially extended model based in a multi--agent approach. In our model, the mobility of socioeconomic agents is driven only by the housing prices. Agents exchange location in order to fit their status to the cost of their housing. On the other hand, the price of a particular house changes depends on the status of its tenant, and on the neighborhood mean lodging cost, weighted by a control parameter. The agent's dynamics converges to a spatially organized configuration, whose regularity we measured by using an entropy--like indicator. With this simple model we found a nontrivial dependence of segregation on both, the initial inequality of the socioeconomic agents and the inflationary parameter. In this way we supply an explanatory model for the segregation--inequality thesis putted forward by Douglas Massey.

nlin.AO↗

arXiv2005

Dynamical complexity of discrete time regulatory networks

Genetic regulatory networks are usually modeled by systems of coupled differential equations and by finite state models, better known as logical networks, are also used. In this paper we consider a class of models of regulatory networks which present both discrete and continuous aspects. Our models consist of a network of units, whose states are quantified by a continuous real variable. The state of each unit in the network evolves according to a contractive transformation chosen from a finite collection of possible transformations, according to a rule which depends on the state of the neighboring units. As a first approximation to the complete description of the dynamics of this networks we focus on a global characteristic, the dynamical complexity, related to the proliferation of distinguishable temporal behaviors. In this work we give explicit conditions under which explicit relations between the topological structure of the regulatory network, and the growth rate of the dynamical complexity can be established. We illustrate our results by means of some biologically motivated examples.

math.DS↗