Control core of undirected complex networks

TL;DR

This paper introduces a leaf removal process to identify a control core in undirected networks, maintaining controllability while simplifying the network structure, and analyzes how the core's properties vary with network density.

Contribution

It presents a mathematically proven method to find a control core that preserves controllability and offers theoretical predictions for core properties based on network average degree.

Findings

Control core has the same controllability as the original network.

Node and link densities in the control core vary nonmonotonously with average degree.

Finding driver nodes in the control core is more efficient than in the original network.

Abstract

With the development of complex networks, many researchers have paid greater attention to studying the control of complex networks over the last decade. Although some theoretical breakthroughs allow us to identify all driver nodes, we still lack an efficient method to identify the driver nodes and understand the roles of individual nodes in contributing to the control of a large complex network. Here, we apply a leaf removal process (LRP) to find a substructure of an undirected network, which is considered as the control core of the original network. Based on a strict mathematical proof, the control core obtained by the LRP has the same controllability as the original network, and it contains at least one set of driver nodes. With this method, we systematically investigate the structural property of the control core with respect to different average degrees of the original networks ($\langle k \rangle$). We denote the node density ($n_\text{core}$) and link density ($l_\text{core}$) to characterize the control core when applying the LRP, and we study the impact of $\langle k \rangle$ on $n_\text{core}$ and $l_\text{core}$ in two artificial networks: undirected Erd\"os-R\'enyi (ER) random networks and undirected scale-free (SF) networks. We find that $n_\text{core}$ and $l_\text{core}$ both change nonmonotonously with increasing $\langle k \rangle$ in the two typical undirected networks. With the aid of core percolation theory, we can offer the theoretical predictions for both $n_\text{core}$ and $l_\text{core}$ as a function of $\langle k \rangle$. Then, we recognize that finding the driver nodes in the control core is much more efficient than in the original network by comparing $n_{\text{D}}$, the controllability of the original network, and $n_{\text{core}}$, regardless of how $\langle k \rangle$ increases.