Controllability of giant connected components in a directed network.

Area of Science:

• Network science
• Complex systems analysis
• Control theory

Background:

• Controlling large-scale biological, technological, and social systems is challenging due to their immense size and complexity.
• Giant connected components (GCCs) encapsulate essential system information, making them key targets for control.
• Methods for controlling GCCs in complex networks remain an open research area.

Purpose of the Study:

• To develop an analytical tool for assessing the controllability of giant connected components (GCCs) in complex networks.
• To investigate how network structure, specifically Erdős-Rényi (ER) and scale-free (SF) networks, influences the minimum driver node density required for GCC control.
• To compare the controllability of GCCs in ER and SF networks under varying conditions.

Main Methods:

• Derivation of mathematical expressions for degree distributions of four types of GCCs.
• Development of an analytical tool to study the controllability of GCCs.
• Analysis of minimum driver node density in ER and SF networks with a fraction 'p' of remaining nodes.
• Comparative analysis of GCC controllability between ER and SF network models.

Main Results:

• The minimum driver node density for controlling GCCs in both ER and SF networks exhibits a peak at a critical fraction of remaining nodes (p=p_m).
• For ER networks, the peak driver node density is independent of average degree and determined by p_m * .
• For SF networks, higher degree distribution exponents correlate with lower minimum driver node densities for GCC control.
• At low fractions of remaining nodes (p), ER networks demonstrate lower controllability for their GCCs compared to SF networks.

Conclusions:

• The controllability of complex networks is significantly influenced by the properties of their giant connected components.
• The developed analytical tool provides insights into optimizing control strategies for large-scale systems.
• Scale-free networks generally offer better controllability for their giant components than Erdős-Rényi networks, particularly under sparse conditions.