Complexity of neural networks on Fibonacci-Cayley tree

• Main Authors: Jung-Chao Ban, Chih-Hung Chang
• Format: Article
• Language: English
• Published: Yildiz Technical University 2019-05-01
• Series: Journal of Algebra Combinatorics Discrete Structures and Applications
• Online Access: http://jacodesmath.com/index.php/jacodesmath/article/view/259

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. We demonstrate that computing the entropy of a Fibonacci tree-shift of finite type is equivalent to studying a nonlinear recursive system and reveal an algorithm for the computation. What is more, the entropy of a Fibonacci tree-shift of finite type is the logarithm of the spectral radius of its corresponding matrix. 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 address the formula of the boundary in the parameter space.