Network Controllability Is Determined by the Density of Low In-Degree and Out-Degree Nodes

TL;DR

This paper reveals that the controllability of networks is primarily influenced by the density of low in-degree and out-degree nodes, and introduces an algorithm to enhance network controllability based on these insights.

Contribution

It demonstrates that the density of low-degree nodes determines the number of driver nodes and provides a new algorithm to improve network controllability.

Findings

Networks with minimum in-degree and out-degree greater than 2 are fully controllable with few driver nodes.

The density of nodes with in-degree and out-degree 0, 1, and 2 influences the number of driver nodes.

An algorithm is proposed to enhance network controllability based on degree distribution.

Abstract

The problem of controllability of the dynamical state of a network is central in network theory and has wide applications ranging from network medicine to financial markets. The driver nodes of the network are the nodes that can bring the network to the desired dynamical state if an external signal is applied to them. Using the framework of structural controllability, here we show that the density of nodes with in-degree and out-degree equal to $0$, $1$ and $2$ determines the number of driver nodes of random networks. Moreover we show that networks with minimum in-degree and out-degree greater than 2, are always fully controllable by an infinitesimal fraction of driver nodes, regardless on the other properties of the degree distribution. Finally, based on these results, we propose an algorithm to improve the controllability of networks.