Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2606.20688

On the Structure and Stability of Boundary Mixed Steady States in Evolutionary Games on Networks

Chiara Mocenni·Jean Carlo Moraes·Matheus C. Santos
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study steady states of evolutionary games on networks in which some players adopt pure strategies while others play mixed strategies. We refer to these configurations as boundary mixed steady states. Such states arise naturally in structured populations and have no counterpart in the classical well-mixed setting. We introduce a relaxed equilibrium notion, called boundary Nash equilibrium, in which the Nash condition is imposed only on non-pure players. In two-strategy systems, this notion characterizes boundary mixed steady states, while this correspondence breaks down in higher dimensions. The stability of these states is governed by the interaction structure among mixed players. When mixed players do not interact, the system exhibits continua of equilibria. In contrast, any nontrivial interaction generically produces instability. In particular, boundary mixed steady states that are not fully degenerate are never asymptotically stable. Degeneracies are further linked to the rank properties of the underlying interaction. These results reveal a structural instability mechanism specific to networked replicator dynamics, highlighting a qualitative gap with respect to the classical well-mixed case and showing how network topology influences the local behavior of equilibria.

Explore related subjects

math.DSmath.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Chiara Mocenni, Jean Carlo Moraes, Matheus C. Santos. 2026-06-14. On the Structure and Stability of Boundary Mixed Steady States in Evolutionary Games on Networks. https://arxiv.org/abs/2606.20688

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗