[Paper Review] On quadratic orbital networks

Oliver Knill|arXiv (Cornell University)|Dec 2, 2013

Graph theory and applications7 references3 citations

TL;DR

This paper studies quadratic orbital networks over finite fields 𝔽ₚ, defined by dynamical systems generated by quadratic maps Tᵢ(x) = x² + aᵢ. It analyzes connectivity, Euler characteristic, clique structure, planarity, diameter, and inductive dimension, proving that for d=1 generators, the Euler characteristic is non-negative, while for d=2 and large p, it becomes negative. As p→∞, the inductive dimension of these networks converges to 1, and for d≥2, the networks are conjectured to be non-planar for sufficiently large p.

ABSTRACT

These are some informal remarks on quadratic orbital networks over finite fields. We discuss connectivity, Euler characteristic, number of cliques, planarity, diameter and inductive dimension. We find a non-trivial disconnected graph for d=3. We prove that for d=1 generators, the Euler characteristic is always non-negative and for d=2 and large enough p the Euler characteristic is negative. While for d=1, all networks are planar, we suspect that for d larger or equal to 2 and large enough prime p, all networks are non-planar. As a consequence on bounds for the number of complete sub graphs of a fixed dimension, the inductive dimension of all these networks goes 1 as p goes to infinity.

Motivation & Objective

• To investigate the structural and topological properties of finite dynamical networks generated by quadratic maps over finite fields.
• To understand the asymptotic behavior of key graph invariants—Euler characteristic, clique count, diameter, and inductive dimension—as the field size p→∞.
• To determine connectivity, planarity, and dimensionality of quadratic orbital networks for varying numbers of generators d and primes p.
• To explore the existence and rarity of exceptional disconnected graphs with d=3 generators, and to analyze their structural properties.

Proposed method

• Define orbital networks as finite simple graphs with vertex set 𝔽ₚ and edges between x and y if Tᵢ(x)=y or Tᵢ(y)=x for some generator Tᵢ.
• Use monoid dynamics to model time evolution via compositions of quadratic maps Tᵢ(x) = x² + aᵢ.
• Apply inductive dimension via unit sphere recursion: dim(G) = 1 + (1/|V|)∑ᵥ dim(S(v)), where S(v) is the link of v.
• Employ Diophantine equation bounds to show that the number of Kₘ₊₁ subgraphs is uniformly bounded for fixed d and m.
• Use Kuratowski’s theorem to analyze planarity by searching for K₃,₃ or K₅ minors.
• Perform computational experiments on small primes and smooth numbers to observe patterns in connectivity, Euler characteristic, and dimension.

Experimental results

Research questions

• RQ1 For d=1, is the Euler characteristic of quadratic orbital networks always non-negative over 𝔽ₚ?
• RQ2 For d=2 and large p, does the Euler characteristic become negative, and what is its asymptotic behavior?
• RQ3 Are quadratic orbital networks with d≥2 and large p always non-planar, and if so, what structural features enforce this?
• RQ4 Does the inductive dimension of these networks converge to 1 as p→∞, and for which d and p is it maximized?
• RQ5 Are there only finitely many disconnected quadratic orbital networks with d=3 generators, and what characterizes such exceptional cases?

Key findings

• For d=1, the Euler characteristic of quadratic orbital networks over 𝔽ₚ is always non-negative.
• For d=2 and sufficiently large p, the Euler characteristic becomes negative.
• The inductive dimension of quadratic orbital networks converges to 1 as p→∞, regardless of d.
• The number of Kₘ₊₁ subgraphs in any network from Xₚᵈ is uniformly bounded by a constant depending only on d and m.
• No K₆ subgraphs exist in any network with d=3, and K₅ subgraphs are absent for p≤47.
• The largest known planar quadratic orbital network with d=2 generators occurs at p=23; no larger planar examples are found for p>23.