Pólya urns on hypergraphs

TL;DR

This work analyzes Pólya urns on finite hypergraphs by recasting the urn dynamics as a stochastic approximation with a gradient-like vector field F driven by the hypergraph's incidence structure. A key Lyapunov function L(v) certifies gradient-like behavior and enables a dynamical systems view of the limiting behavior: if the incidence restriction I(H)|_Γ is injective, the urn proportions converge almost surely to a deterministic point v(H); if not, the limit set lies in an affine subspace parallel to K=ker(I(H)|_Γ), with convergence to a point of a non-unstable equilibria set Λ_{[m]} under suitable conditions. The paper also establishes that unstable equilibria cannot attract the process and provides a partial description of the non-injective case through a convex Lyapunov construction and invariant-manifold arguments, correcting previous gaps in related graph-theoretic work. The results unify and extend the understanding of Pólya urns from graphs to hypergraphs, linking limiting behavior to combinatorial structure and opening questions about boundary behavior and full convergence in general.

Abstract

We study Pólya urns on hypergraphs and prove that, when the incidence matrix of the hypergraph is injective, there exists a point $v=v(H)$ such that the random process converges to $v$ almost surely. We also provide a partial result when the incidence matrix is not injective.

Figures (2)

• Figure 1: (a) Four simulations on the cube with initial condition $(1,1,1,1,1,1,1,1)$. (b) Four simulations on the cube with initial condition $(10,1,1,1,1,1,1,1)$. (c) Four simulations on the cube with initial condition $(100,1,1,1,1,1,1,1)$. The almost vertical components in the graphs in the first iterations are due to large oscillations in the beginning of the process.
• Figure :

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: BBCL-15CL-14Lim-16
• Conjecture 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3: Benaim-96Benaim-99
• Proposition 2.4