Sufficient Control of Complex Networks

TL;DR

This paper introduces the concept of sufficient control in complex networks, transforming the problem into a minimum cost flow problem, and proposes algorithms for both small and large-scale instances, with validation on synthetic and real networks.

Contribution

It is the first to investigate sufficient controllability, providing polynomial algorithms and addressing NP-hardness with heuristic methods for large networks.

Findings

Polynomial-time algorithm for sufficient controllability

NP-hardness of minimum-cost sufficient control problem

Effective algorithms demonstrated on real and synthetic networks

Abstract

In this paper, we propose to study on sufficient control of complex networks which is to control a sufficiently large portion of the network, where only the quantity of controllable nodes matters. To the best of our knowledge, this is the first time that such a problem is investigated. We prove that the sufficient controllability problem can be converted into a minimum cost flow problem, for which an algorithm can be easily devised with polynomial complexity. Further, we study the problem of minimum-cost sufficient control, which is to drive a sufficiently large subset of the network nodes to any predefined state with the minimum cost using a given number of controllers. It is proved that the problem is NP-hard. We propose an ``extended $L_{\mathrm{0}}$-norm-constraint-based Projected Gradient Method" (eLPGM) algorithm which may achieve suboptimal solutions for the problems at small or medium sizes. To tackle the large-scale problems, we propose to convert the control problem into a graph algorithm problem, and devise an efficient low-complexity ``Evenly Divided Control Paths" (EDCP) algorithm to tackle the graph problem. Simulation results on both synthetic and real-life networks are provided, demonstrating the satisfactory performance of the proposed methods.