arXiv subjects

Daniel Miranda Machado

Publications and source records attributed to Daniel Miranda Machado.

5 recordsLinked to original sources

arXiv2026

Extinction, Survival and Fluctuations for the Spatial Maki--Thompson Model on Infinite Graphs

We study the spatial Maki--Thompson rumor model on infinite, connected graphs of bounded degree. Spreaders transmit the rumor to ignorant neighbors but become stiflers upon contacting non-ignorant neighbors. We prove extinction on Cayley graphs of linear growth, for every $\lambda,\alpha>0$ and every initial configuration with finitely many non-ignorant vertices, and establish an explicit extinction criterion on arbitrary bounded-degree graphs for processes started from finitely many spreaders. On Cayley graphs of superlinear growth, we prove survival from a single spreader whenever the ratio of the stifling to the transmission rate lies below an explicit threshold depending only on the maximum degree. On Cayley graphs of polynomial growth of degree $D\ge2$, we further show that the range has positive lower density with positive probability. Under a stronger condition, macroscopic annuli contain a surface-order number of simultaneously active spreaders for a total duration bounded uniformly away from zero. When $\alpha>0$, every finite region eventually contains no spreaders, so global survival forces the rumor to move continually into new regions. Under either subcriticality or a sufficiently small stifling rate, we prove central limit theorems for the final stifler density and the total spreader occupation time, together with a functional central limit theorem for the empirical survival function. These results follow from a central limit theorem for stationary stabilizing functionals of i.i.d.fields on polynomial-growth Cayley graphs; the functionals may depend on the field outside the observation set.

math.PR↗

arXiv2026

Topology of Percolation Clusters: Central Limit Theorems beyond the Lattice

We prove central limit theorems (CLTs) for topological functionals of Bernoulli bond percolation on infinite graphs beyond the Euclidean lattice $\mathbb{Z}^{d}$. For quasi-transitive graphs of subexponential growth, we show that the number $K_{r}$ of open clusters intersecting the metric ball $B_{r}$ satisfies a CLT as $r\to\infty$. For amenable Cayley graphs, we prove a general CLT for stationary percolation functionals along Folner sequences under sequential stabilization and a finite-moment assumption, provided the group admits a left-orderable finite-index subgroup. This applies in particular to groups of polynomial growth. As an application, we obtain CLTs for Betti numbers of graph-generated random simplicial complexes, including clique and neighbor complexes. The proofs combine invariant edge orderings, martingale decompositions, and stabilization estimates for single-edge perturbations.

math.PR↗

arXiv2026

Sharkovskiis theorem under small random perturbations

We establish a Sharkovskii-type theorem for a class of discrete random dynamical systems via the random Conley index. Using the continuation property of the Conley index, we extend classical forcing results to random systems obtained from small random perturbations of one-dimensional maps. In contrast to earlier measure-theoretic results, which are typically subject to an inherent period-doubling ambiguity (realizing period $n$ or $2n$), our topological approach allows us to detect random periodic points and orbits with precise minimal periods. This yields realisation results for arbitrary finite tails of the Sharkovskii ordering. These results are illustrated by constructing random periodic orbits for perturbed versions of the tent map and the logistic map.

math.DS↗

arXiv2026

Symmetry Breaking, Hysteresis, and Convergence to the Mean Voter in two-party Spatial Competition

Classical spatial models of two-party competition typically predict convergence to the median voter, yet real-world party systems often exhibit persistent and asymmetric polarization. We develop a spatial model of two-party competition in which voters evaluate parties through general satisfaction functions, and a width parameter $q$ captures how tolerant they are of ideological distance. This parameter governs the balance between centripetal and centrifugal incentives and acts as the bifurcation parameter governing equilibrium configurations. Under mild regularity assumptions, we characterize Nash equilibria through center-distance coordinates, which separate the endogenous political center from polarization. When the voter density is symmetric, the reduced equilibrium condition exhibits a generic supercritical pitchfork bifurcation at a critical value $q_{c}$. Above $q_{c}$, the unique stable equilibrium features convergence to the center, recovering the classical median voter result, whereas below it two symmetric polarized equilibria arise. Asymmetry in the voter distribution unfolds the pitchfork, producing drift in the endogenous center and asymmetric polarized equilibria. The resulting equilibrium diagram has an S-shaped geometry that generates hysteresis, allowing polarization to persist even after tolerance returns to levels that would support convergence in a symmetric environment. In the high-tolerance regime, we show that the unique non-polarized equilibrium converges to the mean of the voter distribution, while the median is recovered only under symmetry. Hence, unlike the Hotelling--Downs model, where convergence to the median is universal, the median voter appears here as an asymptotic benchmark rather than a robust predictor.

physics.soc-ph↗

arXiv2025

The Maki-Thompson Model with Spontaneous Stifling on Symmetric Networks

We investigate rumor spreading in a generalized Maki-Thompson model with spontaneous stifling, evolving on quasi-transitive networks. Individuals are either ignorants, spreaders, or stiflers; spreaders stop by contact with other spreaders or stiflers or after an independent random waiting time sampled from a given distribution, modeling a spontaneous loss of interest. The topology of the underlying population network is incorporated by modeling it as a broad class of symmetric networks, whose vertices are partitioned into finitely many orbit types. This yields a unified framework for homogeneous and heterogeneous networks. For sequences of finite quasi-transitive graphs, and for infinite quasi-transitive graphs with subexponential growth, we establish a Functional Law of Large Numbers and a Functional Central Limit Theorem for the densities of each vertex type for the three states. The mean-field limit is described by a system of nonlinear integral equations, while fluctuations are asymptotically Gaussian and governed by a system of stochastic integral equations with explicit covariance. Our results show how the topology and the law of spontaneous stifling jointly shape the speed and variability of rumor outbreaks. As a special case, our model reduces to the classical Maki-Thompson model when spontaneous stifling is absent.

math.PR↗