The role of directed cycles in a directed neural network

Qinrui Dai; Jin Zhou; Zhengmin Kong

doi:10.1016/j.neunet.2024.106329

The role of directed cycles in a directed neural network

Neural Netw. 2024 Apr 19:176:106329. doi: 10.1016/j.neunet.2024.106329. Online ahead of print.

Authors

Qinrui Dai¹, Jin Zhou², Zhengmin Kong³

Affiliations

¹ School of Mathematics and Statistics, Wuhan University, Hubei 430072, China.
² School of Mathematics and Statistics, Wuhan University, Hubei 430072, China; Hubei Key Laboratory of Computational Science, Wuhan University, Hubei 430072, China. Electronic address: jzhou@whu.edu.cn.
³ School of Electrical Engineering and Automation, Wuhan University, Hubei 430072, China.

PMID: 38653127
DOI: 10.1016/j.neunet.2024.106329

Abstract

This paper investigates the dynamics of a directed acyclic neural network by edge adding control. We find that the local stability and Hopf bifurcation of the controlled network only depend on the size and intersection of directed cycles, instead of the number and position of the added edges. More specifically, if there is no cycle in the controlled network, the local dynamics of the network will remain unchanged and Hopf bifurcation will not occur even if the number of added edges is sufficient. However, if there exist cycles, then the network may undergo Hopf bifurcation. Our results show that the cycle structure is a necessary condition for the generation of Hopf bifurcation, and the bifurcation threshold is determined by the number, size, and intersection of cycles. Numerical experiments are provided to support the validity of the theory.

Keywords: Cycle; Directed acyclic network; Hopf bifurcation; Neural network; Stability.