# Diversity of structural controllability of complex networks with given degree sequence

## TL;DR

This paper explores how the degree sequence of directed networks constrains the number of driver nodes needed for control, introducing algorithms to find bounds and a new measure called structural control complexity.

## Contribution

It develops algorithms to determine bounds on driver nodes based on degree sequences and introduces the concept of structural control complexity.

## Key findings

- Algorithms achieve bounds for most networks.
- Degree distribution features influence control bounds.
- Structural control complexity varies across networks.

## Abstract

We investigate to what extent the degree sequence of a directed network constrains the number of driver nodes. We develop a pair of algorithms that take a directed degree sequence as input and aim to output a network with the maximum or minimum number of driver nodes. We find an upper bound for the maximum and a lower bound for the minimum, and show that the algorithms achieve these bounds for all real and model networks, with few exceptions characterized by tiny system size and heterogeneous degree distributions. Applying these algorithms to a broad range of real networks, we show the gap between the upper and lower bounds can vary dramatically across different degree sequences. Thus, we introduce the notion of \bl{structural control} complexity to capture how much more difficult it is to control a specific network beyond what is required given its degree sequence, suggesting additional structure is present. Using model networks, we numerically and analytically investigate how typical features of the degree distribution affect the range of required driver nodes. We find that the minimum is determined by the number of sources or sinks, while the maximum is strongly affected by the presence of hubs.