Towards Optimal Robustness of Network Controllability: An Empirical Necessary Condition

TL;DR

This paper investigates the relationship between network topology and controllability robustness, proposing an empirical necessary condition (ENC) that guides the design of more robust networks through edge rectification, validated on synthetic and real-world data.

Contribution

It introduces an empirical necessary condition (ENC) for optimal network controllability robustness and develops an edge rectification method to achieve it.

Findings

ENC indicates optimal networks are highly homogeneous in degrees.

Edge rectification improves network controllability robustness.

Simulation confirms effectiveness of ENC and rectification on various networks.

Abstract

To better understand the correlation between network topological features and the robustness of network controllability in a general setting, this paper suggests a practical approach to searching for optimal network topologies with given numbers of nodes and edges. Since theoretical analysis seems impossible at least in the present time, exhaustive search based on optimization techniques is employed, firstly for a group of small-sized networks that are realistically workable, where \textit{exhaustive} means 1) all possible network structures with the given numbers of nodes and edges are computed and compared, and 2) all possible node-removal sequences are considered. A main contribution of this paper is the observation of an empirical necessary condition (ENC) from the results of exhaustive search, which shrinks the search space to quickly find an optimal solution. ENC shows that the maximum and minimum in- and out-degrees of an optimal network structure should be almost identical, or within a very narrow range, i.e., the network should be extremely homogeneous. Edge rectification towards the satisfaction of the ENC is then designed and evaluated. Simulation results on large-sized synthetic and real-world networks verify the effectiveness of both the observed ENC and the edge rectification scheme. As more operations of edge rectification are performed, the network is getting closer to exactly satisfying the ENC, and consequently the robustness of the network controllability is enhanced towards optimum.