arXiv subjects

Kádmo Laxa

Publications and source records attributed to Kádmo Laxa.

5 recordsLinked to original sources

arXiv2026

Metastability and phase transition in a social network model with multiple opinions

We consider a stochastic opinion dynamics model on a fully connected social network with $N$ actors interacting by expressing opinions from a set of $M$ opinions. At any time $t\geq 0$, each actor is associated to an $M$-tuple representing the social pressure exerted on this actor for each opinion. The evolution of the matrix containing the social pressure of all actors for all opinions is a Markov jump process. Each actor tends to express opinions according to their social pressure vector and this tendency is modulated by a polarization coefficient. When an actor expresses an opinion $o$, its social pressure for all opinions is reset to zero, while for other actors the social pressure for $o$ increases by 1 and the social pressure for other opinions decreases by $1/(M-1)$. In this setting, we prove fast consensus formation, existence of a unique invariant measure and metastability in a highly polarized network. Moreover, by considering a communication bias parameter, the system exhibits a phase transition described as follows. With a negative communication bias parameter, all actors except one stop expressing in a finite time almost surely. Otherwise, no actor stops expressing opinions.

math.PR↗

arXiv2026

The Pathwise Approach to Metastability and its Applications to Galves--Löcherbach Models

Metastability is the tendency of a system to dwell for a very long time near an apparently stable equilibrium before a rare fluctuation drives it, on a comparatively short time scale, towards another. Among the rigorous frameworks developed to capture this phenomenon, the pathwise approach proceeds by identifying the ``typical'' trajectories of the stochastic dynamics at hand and estimating their probabilities. In this article we review the pathwise approach and its application to the Galves--Löcherbach (GL) class of stochastic models of spiking neural networks. After recalling the conceptual and historical roots of the theory -- which goes from chemistry to rigorous probability theory, with fundamental ideas coming mainly from statistical physics -- and illustrating them on two classical examples, we give a general definition encompassing the known variants of the GL model and survey the metastability results already established for some of these variants. As far as we can, we do so in a self-contained fashion, and we sketch the proofs when possible, highlighting their common structure. We close with a discussion on open problems and point to possible further directions.

math.PR↗

arXiv2025

A new look at perfect simulation for chains with infinite memory

In this article we introduce two new perfect simulation algorithms for chains with infinite memory. Both algorithms belong to the coupling of past procedures. The novelty of our approach is that it allows to include unknown states to the possible past symbols such that we can also deal with sparsely distributed past dependencies. In our first algorithm, spontaneous occurrence of symbols is possible. This means that there is a positive probability that the chain chooses the next symbol independently of the past. Our second algorithm deals with the case in which spontaneous occurrence of symbols is not possible. Chains with infinite memory are discrete-time stochastic processes in which the distribution of the next symbol depends on all past symbols. These transition probabilities are described by a probability kernel. Our results give conditions on the way the dependency of the transition kernel on long past strings decays, guaranteeing that our algorithms stop after a finite number of steps almost surely. Strengthening these conditions, we show that the mean number of steps of our algorithms is finite. We discuss the consequence of having a coupling from the past algorithm with such properties and we present examples in which our results can be applied while others result in the literature cannot be applied.

math.PR↗

arXiv2024

Fast Consensus and Metastability in a Highly Polarized Social Network

A polarized social network is modeled as a system of interacting marked point processes with memory of variable length. Each point process indicates the successive times in which a social actor expresses a "favorable" or "contrary" opinion. After expressing an opinion, the social pressure on the actor is reset to 0, waiting for the group's reaction. The orientation and the rate at which an actor expresses an opinion is influenced by the social pressure exerted on it, modulated by a polarization coefficient. We prove that the network reaches an instantaneous but metastable consensus, when the polarization coefficient diverges.

math.PR↗

arXiv2024

Propagation of chaos and phase transition in a stochastic model for a social network

We consider a model for a social network with N interacting social actors. This model is a system of interacting marked point processes in which each point process indicates the successive times in which a social actor expresses a "favorable" (+1) or "contrary" (-1) opinion. The orientation and the rate at which an actor expresses an opinion is influenced by the social pressure exerted on this actor. The social pressure of an actor is reset to 0 when the actor expresses an opinion, and simultaneously the social pressures on all the other actors change by h/N in the direction of the opinion that was just expressed. We prove propagation of chaos of the system, as N diverges to infinity, to a limit nonlinear jumping stochastic differential equation. Moreover, we prove that under certain conditions the limit system exhibits a phase transition described as follows. If h is smaller or equal than a certain threshold, the limit system has only the null Dirac measure as an invariant probability measure, corresponding to a vanishing social pressure on all actors. However, if h is greater than the threshold, the system has two additional non-trivial invariant probability measures. One of these measures has support on the positive real numbers and the other is obtained by symmetrization with respect to 0, having thus support on the negative real numbers.

math.PR↗