Source-linked AI summary
Exact Controllability of Complex Networks
Zhengzhong Yuan, Chen Zhao, Zengru Di, Wen-Xu Wang, Ying-Cheng Lai
TL;DR
Controllability theory lacks a universal framework for complex networks with arbitrary structures and link weights, particularly undirected networks. This paper introduces an exact-controllability framework based on eigenvalue multiplicity, showing that minimum driver nodes are determined by maximum eigenvalue multiplicity and extending applicability across network types.
Problem
A universal framework for controlling complex networks with arbitrary structures and link weights, especially undirected networks, remains lacking.
Method
The framework uses the PBH rank condition and eigenvalue multiplicities to identify the minimum number and configuration of independent driver nodes.
Results
The minimum number of independent driver nodes equals the maximum geometric multiplicity, reducing to maximum algebraic multiplicity for undirected networks.
Takeaways & Limitations
Exact controllability provides a unified way to analyze directed, undirected, weighted, and unweighted networks with or without self-loops.
Takeaways & Limitations
When exact link weights are unknown, the method has numerical-error and computational-efficiency disadvantages relative to structural controllability for directed networks.
Abstract
from arXiv · showhide
Controlling complex networks is of paramount importance in science and engineering. Despite the recent development of structural-controllability theory, we continue to lack a framework to control undirected complex networks, especially given link weights. Here we introduce an exact-controllability paradigm based on the maximum multiplicity to identify the minimum set of driver nodes required to achieve full control of networks with arbitrary structures and link-weight distributions. The framework reproduces the structural controllability of directed networks characterized by structural matrices. We explore the controllability of a large number of real and model networks, finding that dense networks with identical weights are difficult to be controlled. An efficient and accurate tool is offered to assess the controllability of large sparse and dense networks. The exact-controllability framework enables a comprehensive understanding of the impact of network properties on controllability, a fundamental problem towards our ultimate control of complex systems.
Additional information
Additional information defines exact-controllability measures for real unweighted, directed, and weighted networks, and reports equivalence with structural-controllability values under random weights. The authors also declare no competing financial interests.
- Declaration: The authors declare no competing financial interests.
- Real unweighted and directed networks: Table II reports exact-controllability measures for real unweighted and directed networks, including node number N and directed-link number L.It includes nMMT D from Eq. (5) and nLSB D as the structural-controllability measure.
- Random link weights: For directed networks with random link weights, nMMT D and nLSB D are exactly the same.nMMT D is computed from maximum geometric multiplicity, Eq. (3).
- Real weighted networks: Table III reports exact-controllability measures for real weighted networks, using the same legends as Table II but without nLSB D.Data sources and references are provided in Supplementary Table S1 and Note 7.