[Paper Review] Coloring Fibonacci-Cayley tree: An application to neural networks

Jung‐Chao Ban, Chih-Hung Chang|arXiv (Cornell University)|Jul 5, 2017

Neural dynamics and brain function15 references3 citations

TL;DR

This paper investigates the entropy of neural networks on Fibonacci-Cayley trees, modeling neuronal dysfunction such as in Alzheimer’s disease. By reducing entropy computation to a nonlinear recursive system and applying a novel algorithm, it proves that only two entropy values—0 or ln g—are possible, with a precise formula for the critical boundary in parameter space.

ABSTRACT

This paper investigates the coloring problem on Fibonacci-Cayley tree, which is a Cayley graph whose vertex set is the Fibonacci sequence. More precisely, we elucidate the complexity of shifts of finite type defined on Fibonacci-Cayley tree via an invariant called entropy. It comes that computing the entropy of a Fibonacci tree-shift of finite type is equivalent to studying a nonlinear recursive system. After proposing an algorithm for the computation of entropy, we apply the result to neural networks defined on Fibonacci-Cayley tree, which reflect those neural systems with neuronal dysfunction. Aside from demonstrating a surprising phenomenon that there are only two possibilities of entropy for neural networks on Fibonacci-Cayley tree, we reveal the formula of the boundary in the parameter space.

Motivation & Objective

• To model neural networks with neuronal dysfunction using Fibonacci-Cayley tree topologies, reflecting pathological spread in diseases like Alzheimer’s.
• To measure the complexity of such neural systems via topological entropy, a key invariant for shift spaces on trees.
• To develop an algorithm for computing the entropy of tree-shifts of finite type on Fibonacci-Cayley trees.
• To determine the full parameter space where entropy transitions between zero and positive values, identifying critical boundaries.
• To extend results from one-dimensional and multidimensional shifts to tree-shifts, revealing a surprising dichotomy in entropy values.

Proposed method

• Models neural networks on Fibonacci-Cayley trees as tree-shifts of finite type (SFT), where forbidden local patterns represent dysfunctional neuron states.
• Reduces entropy computation to analyzing a nonlinear recursive system derived from adjacency matrices of the underlying graph structure.
• Applies the Perron-Frobenius theorem to compute the dominant eigenvalue of transition matrices, yielding entropy as the logarithm of the spectral radius.
• Introduces a region-based indexing system on the (a,z)-plane into 25 equivalent regions [p,q], enabling classification of entropy behavior.
• Uses simple recurrence representations and matrix reduction techniques (e.g., deleting rows/columns) to analyze entropy in subcases with one or two essential symbols.
• Derives a critical condition: (a,z) is critical iff a−1 = ||z|−m|−M, where m=min{|a₁|,|a₂|}, M=max{|a₁|,|a₂|}, linking parameter space to entropy phase transitions.

Experimental results

Research questions

• RQ1 What is the entropy of a tree-shift of finite type defined on a Fibonacci-Cayley tree, and how does it relate to the underlying recursive structure?
• RQ2 Can the entropy of neural networks on Fibonacci-Cayley trees be computed algorithmically, and what is the computational complexity?
• RQ3 Does the entropy of such neural networks exhibit a dichotomous behavior—only zero or positive values—under the separation property?
• RQ4 What is the exact geometric boundary in the (a,z)-parameter space that separates zero-entropy from positive-entropy regimes?
• RQ5 How does neuronal dysfunction, modeled as dead nodes in the tree, affect the information-carrying capacity of the network?

Key findings

• The entropy of a neural network on a Fibonacci-Cayley tree is either zero or ln g, revealing a surprising dichotomy in information capacity.
• Entropy is zero if min{p,q}=0 or max{p,q}=1 in the region index [p,q], indicating restricted pattern growth.
• Entropy is ln g when p,q>0 and max{p,q}≥2, corresponding to maximal pattern diversity under the given constraints.
• The critical boundary in the (a,z)-parameter space is given by a−1 = ||z|−m|−M, where m and M are the min and max of |a₁| and |a₂|.
• The entropy computation reduces to analyzing a nonlinear recursive system, with the dominant eigenvalue of a transition matrix determining the entropy.
• The result extends Lind’s one-dimensional SFT entropy result to tree-shifts, showing that tree-shift entropy is the logarithm of a Perron number.